The 10th Annual Meeting on Inverse Problems

The 10th Annual Meeting

on Inverse Problems

Tianyuan Mathematical Center in Northeast China Jilin University·Changchun 2018.5.28-2018.5.31

Contents

Welcome Letter ............................................................................................................ 1

About the Conference .................................................................................................. 2

Conference Committees ............................................................................................. 5

Schedule at a Glance .................................................................................................. 6

Title of Minisymposia .................................................................................................. 7

Conference Schedule .................................................................................................... 8

Minisymposia Agenda ................................................................................................. 9

List of talks of Minisymposia .................................................................................. 11

Abstract of Plenary Talks ........................................................................................ 18

Abstract of talks of Minisymposia ............................................................................ 23

M1-1 ..................................................................................................................... 23

M1-2 ..................................................................................................................... 26

M2-1 ..................................................................................................................... 28

M2-2 ..................................................................................................................... 31

M3-1 ..................................................................................................................... 33

M3-2 ..................................................................................................................... 36

M3-3 ..................................................................................................................... 38

M3-4 ..................................................................................................................... 40

M4-1 ..................................................................................................................... 43

M4-2 ..................................................................................................................... 46

M5-1 ..................................................................................................................... 49

M5-2 ..................................................................................................................... 51

M6-1 ..................................................................................................................... 53

M6-2 ..................................................................................................................... 56

Introduction to Tianyuan Mathematical Center in Northeast China ................. 58

1

Welcome letter

※Welcome to the 10th Annual Meeting on Inverse Problems

Inverse problems arise from the demand to interpret indirect measurements. Such

situations are common in many applications including medical imaging,

nondestructive testing, underground prospecting, astronomical imaging, remote

sensing, image processing, and data mining. Scientific research of inverse problems is

multidisciplinary and involves many fields for example mathematics, physics,

engineering and signal processing.

The main topics of the 2018 annual inverse problem meeting include the inverse

problem of electromagnetic scattering, elastic wave inverse problem, geophysical

detection inverse problem, inverse problem in imaging, etc.

※Welcome to Changchun

Changchun (simplified Chinese: 长春; traditional Chinese: 長春; pinyin: Chángchūn)

is the capital and largest city of Jilin Province, and is also the core city of Northeast

Asia. The name of the city means "long spring" in Chinese.

Known locally as China's "City of Automobiles", Changchun is an important

industrial base with a particular focus on the automotive sector. Because of its key

role in the domestic automobile industry, Changchun was sometimes referred to as the

"Detroit of China." Apart from this industrial aspect, Changchun is also one of four

"National Garden Cities" awarded by the Ministry of Construction of P.R. China in

2001 due to its high urban greening rate. Besides, there are many famous scenic spots

in Changchun such as Changchun film century city, Oriental Hollywood, Changchun

sculpture park, Changchun cultural plaza, Jingyuetan forest park, and Puppet palace.

2

About the Conference

※Conference Venue

1. The plenary talks will take place at Dongrong Conference Center in Jilin University

on May 28, 29, 30, 31.

2. The Minisymposia will be held in the lecture hall of the Math Building May 28, 29,

30.

3

June Hotel

The East Entrance

of Jilin University

Dongrong

Builnding Math Building

4

※Registration

Registration Opens: 2:00pm to 10:00pm on May 27, at June Hotel

8:00am to 12:00pm on May 28，at Dongrong Conference Center.

13:30pm to 17:00pm on May 28, the lecture hall of math building

For other time registration, please contact us.

※Dining

The conference Banquet will be held at 6pm of May 28, 2018. Transportation is

available at the Math Building.

We will also provide each conference participant 5 tickets for lunch and dinner buffet

at June Hotel.

※Contact us

Questions: Please consult volunteers or contact us directly.

Xiuping Gao 13596117968

Danhong Wang 13404775711

Xingwu Sun

13504468895

5

Conference Committees

※Scientific Committee

Gang Bao

Jin Cheng

Bo Han

Jijun Liu

Fuming Ma

Jianwei Ma

Yanfei Wang

Ting Wei

Haijun Wu

Zhaofa Zeng

Bo Zhang

Weiying Zheng

Jun Zou

※Organizing Committee

Jun Lai

Ming Li

Shuai Lu

Xiang Xu

Deyue Zhang

Ran Zhang

Zhejiang University

Fudan University

Harbin Institute of Technology

Southeast University

Jilin University


Chinese Academy of Sciences

Lanzhou University

Nanjing University

Jilin University



Chinese University of Hong Kong

Zhejiang University

Taiyuan University of Technology

Fudan University

Zhejiang University

Jilin University

Jilin University

6

Registration Opens: 2:00pm to 10:00pm on Sunday May 27, at June Hotel

8:00am to 12:00pm on Monday May 28, Dongrong Conference Center.

13:30pm to 17:00pm on May 28, the lecture hall of the Math building

Questions: Please consult volunteers or contact us directly.

May 28

Monday

May 29

Tuesday

May 30

Wednesday

May 31

Thursday

Location

9:00 - 9:30

Opening Ceremony

9:00 - 9:55

Plenary talk:

Yanfei Wang

9:00 - 9:55

Plenary talk:

Changchun Yin

9:00 - 9:55

Plenary talk:

Masahiro

Yamamoto

Dongrong

Conference

Center

9:50 -10:45

Plenary talk:

Peijun Li

9:55 - 10:50

Plenary talk:

Jinghuai Gao

9:55 - 10:05

Laudations for Prize

Winners 9:55 - 10:10

Coffee Break 10:05 -10:55

Award Lecture 1

10:45 - 11:05

Coffee Break

10:50 - 11:10

Coffee Break

10:55 - 11:15

Coffee Break

11:05 - 12:00

Plenary talk:

Junshan Lin

11:10 - 12:00

Plenary talk:

Jianwei Ma

11:15 - 12:00

Award Lecture 2

10:10 - 11:05

Plenary talk:

Faouzi Triki

12:00 - 13:30

Buffet at June Hotel

12:00 - 13:30

Buffet at June

Hotel

12:00 - 13:30


13:30 - 17:50

Minisymposia

M4-1,M2-1,M3-1

M4-2,M2-2,M3-2

13:30 - 17:50

Minisymposia

M3-3,M1-1

M3-4,M6-1

13:30 - 17:50

Minisymposia

M5-1,M1-2

M5-2,M6-2

The Math

Building

17:50-

Banquet

Yunuo Restaurant

17:50-

Buffet at June

hotel

17:50-


Schedule at a Glance

7

Title of Minisymposia

M1 Forward and inverse scattering problems and their applications

Jun Lai, Zhejiang University and Wangtao Lu, Zhejiang University

M1-1:M1 Session1; M1-2:M1 Session2

M2 Recent Advances and Applications in Regularization

Xiang Xu, Zhejiang University and Haibing Wang, Southeast University


M3 Recent advances in inverse scattering theory

Guanghui Hu, Beijing Computational Science Research Center

Deyue Zhang, Jilin University and Xiaodong Liu, Chinese Academy of Sciences



M4 Computational inverse problems and their applications in atmospheric and

oceanic sciences

Shuai Lu, Fudan University, and Xiliang Lv Wuhan University

Yi Heng, Sun Yat-Sen University


M5 Inverse Problems in Imaging Science

Bin Dong, Peking University

Xiaoqun Zhang, Shanghai Jiao Tong University


M6 Contributed talks


8

Conference Schedule

Day 1 – May 28, Monday, 9:00-12:00 , chaired by Gang Bao

Location Dongrong Conference Center, Jilin University

9:00 - 9:10 Opening Ceremony- Youhong Sun

9:10 - 9:20 Opening Ceremony- Yanzhao Cao

9:20 - 9:30 Opening Ceremony- Gang Bao

9:30 - 9:50 Photos

9:50 - 10:45 Peijun Li, Purdue University

Direct and Inverse Scattering for Elastic Wave Propagation

10:45 - 11:05 Coffee Break

11:05 - 12:00

Junshan Lin, Auburn University

A Mathematical Perspective on Extraordinary Optical Transmission Through

Small Holes: Perfect Conductors and Plasmonic Metals

Day 2 – May 29, Tuesday, 9:00-12:00, chaired by Jun Zou


9:00 - 9:55 Yanfei Wang, Institute of Geology and Geophysics, CAS

Joint matrix optimization method for seismic data recovery

9:55 - 10:50

Jinghuai Gao, Xi'an Jiaotong University

A data driven representation method for nonstationary convolution seismic

trace model with its applications for enhancing the resolution of seismic data

and Q estimation


11:10 - 12:00 Jianwei Ma, Harbin Institute of Technology

Shape optimization-based image data assimilation with Wasserstein distance

Day 3 – May 30, Wednesday, 9:00-12:00, chaired by Fuming Ma


9:00 - 9:55 Changchun Yin, Jilin University

3D Time-Domain Airborne EM Inversion with Finite-Volume Method

9:55 - 10:05 Laudations for Prize Winners

10:05 -10:55 Award Lecture 1


11:15 - 12:00 Award Lecture 2

Day 4 – May 31, Thursday，9:00-11:05, chaired by Haijun Wu


9:00 - 9:55 Masahiro Yamamoto, The University of Tokyo

Inverse source problems for a parabolic and a hyperbolic equations


10:10 - 11:05 Faouzi Triki, Grenoble Alpes University

On the inverse conductivity problem with a single internal measurement

Minisymposia Agenda

9

Minisymposia Agenda

Day 1 –

May 28

Monday

Location The meeting room, the third floor of the

Math building, Jilin University Room1, the lecture hall of the Math

Building, Jilin University

Room2, the lecture hall of the Math

Building, Jilin University

Organizer Shuai Lu, Xiliang Lv and Yi Heng Xiang Xu and Haibing Wang Guanghui Hu, Deyue Zhang

and Xiaodong Liu

Minisymposia

Computational inverse problems and

their applications in atmospheric and

oceanic sciences

Recent Advances and Applications in

Regularization

Recent advances in inverse scattering

theory

Session 1

13:30 - 15:30

M4-1

Yi Heng, Liang Yan,

Chuan Gao, Min Zhong

M2-1

Yuanxiang Zhang, Yu Jiang

Wen Zhang, Zhiyuan Li

M3-1

JunXiong Jia, Keji Liu

Yue Zhao, Hongwei Zhou

Session 2

15:50 - 17:50

M4-2

Jinglai Li, Xuefeng Zhang

Huadong Du, Pingping Niu

M2-2

Rongfang Gong, Wei Wang

Liang Yan, Hai Zhang

M3-2

Fenglong Qu, Xiaoli Liu

Xiaoxu Xu, Guanghui Hu

Day 2 –

May 29

Tuesday

Location Room2, the lecture hall of the Math Building, Jilin University Room1, the lecture hall of the Math Building, Jilin University

Organizer Guanghui Hu, Deyue Zhang and Xiaodong Liu Jun Lai and Wangtao Lu

Minisymposia Recent advances in inverse scattering theory Forward and inverse scattering problems and their applications

Session 1

13:30 - 15:30

M3-3

Jiguang Sun, Xiaodong Liu, Yukun Guo, Heping Dong

M1-1

Yayan Lu, Wenbin Li, Yuliang Wang, Xinming Zhang

Session 2

15:50 - 17:50

M3-4

Jingzhi Li, Xia Ji, Jiaqing Yang, Haiwen Zhang

M6-1 Contributed Talks

Xiaomei Yang, Fangfang Dou, Shanshan Tong, Hongpeng Sun

Day 3 –

May 30 Wednesday

Location Room2, the lecture hall of the Math Building, Jilin University Room1, the lecture hall of the Math Building, Jilin University

Organizer Bin Dong and Xiaoqun Zhang Jun Lai and Wangtao Lu

Minisymposia Inverse Problems in Imaging Science Forward and inverse scattering problems and their applications

Session 1

13:30 - 15:30

M5-1

Zuoqiang Shi, Ke Wei, Chenglong Bao, Liyan Ma

M1-2

Leung Shingyu, Lijun Yuan, Zhen Hu, Ke Yin

Session 2

15:50 - 17:50

M5-2 Bin Dong, Chunlin Wu, Jianfeng Cai, Jae Kyu Choi

M6-2 Huaian Diao, Xiaoliang Song, Wenxiu Gong

List of talks of Minisymposia

10


M4-1 Computational inverse problems and their applications in atmospheric and

oceanic sciences-Session 1

Time: May 28, Monday，13:30 - 15:30

Venue: The meeting room, the third floor of the Math building, Jilin University

Organizers: Shuai Lu, Fudan University

Xiliang Lv, Wuhan University


Speaker Time Title

Yi Heng 13:30 –14:00

Volcanic eruption case study - Nabro: Supercomputer

implementation for identifying source terms of the

atmospheric pollutants

Liang Yan 14:00 – 14:30

Multi-fidelity method using polynomial chaos:

analysis and applications

Chuan Gao 14:30 – 15:00

Application of an adjoint method to an intermediate

coupled model and its improvement for real-time

ENSO prediction

Min Zhong 15:00 – 15:30

A Multiscale SVR Method On Spheres with Data

Compression

M2-1 Recent Advances and Applications in Regularization-Session 1

Time: May 28, Monday，13:30 - 15:30

Venue: Room1, the lecture hall of the Math Building, Jilin University

Organizers: Xiang Xu, Zhejiang University

Haibing Wang, Southeast University

Speaker Time Title

Yuanxiang

Zhang 13:30 –14:00

Bayesian Approach to A Nonlinear Inverse Problem

for Time-Space Fractional Diffusion Equation

Yu Jiang 14:00 – 14:30

A Hybrid Inversion Scheme for Diffuse Optical

Tomography

Wen Zhang 14:30 – 15:00

A coupled model of partial differential equations

for Uranium ores heap leaching and its parameters

identification

Zhiyuan Li 15:00 – 15:30

Unique continuation principle for the

time-fractional diffusion equation


11

M3-1 Recent advances in inverse scattering theory-Session 1

Time: May 28, Monday，13:30 - 15:30


Organizers: Guanghui Hu, Beijing Computational Science Research Center

Deyue Zhang, Jilin University

Xiaodong Liu, Institute of Applied Mathematics, Academy of

Mathematics and Systems Science, Chinese Academy of Sciences

Speaker Time Title

Junxiong Jia 13:30 –14:00

Complex Gaussian mixture based model error

learning for inverse medium scattering problems

with multi-frequencies

Keji Liu 14:00 – 14:30

The application of multilevel sampling method in

the inverse scattering problems

Yue Zhao 14:30 – 15:00 Inverse Source Problem in Electrodynamics

Hongwei

Zhou 15:00 – 15:30

Imaging perfectly conducting cylinders with

experimental data

M4-2 Computational inverse problems and their applications in atmospheric and

oceanic sciences-Session 1 -Session 2

Time: May 28, Monday，15:50 - 17:50

Venue: The meeting room, the third floor of the Math Building, Jilin University

Organizers: Shuai Lu, Fudan University

Xiliang Lv, Wuhan University


Speaker Time Title

Jinglai Li 15:50 - 16:20

A hybrid marginal sequential Monte Carlo method

for data assimilation

Xuefeng

Zhang 16:20 - 16:50

Correction of Biased Climate Simulated by Biased

Physics through Parameter Estimation in an

Intermediate Coupled Model

Huadong Du 16:50 - 17:20

Channel selection method for high spectral

resolution infrared data based on relative entropy

Pingping

Niu 17:20 - 17:50

On periodic parameter identification in stochastic

differential equations


12

M2-2 Recent Advances and Applications in Regularization-Session 2

Time: May 28, Monday，15:50 - 17:50


Organizers: Xiang Xu, Zhejiang University

Haibing Wang, Southeast University

Speaker Time Title

Rongfang

Gong 15:50 - 16:20

A dynamical regularization algorithm for solving

inverse source problems of elliptic partial differential

equations

Wei Wang 16:20 - 16:50

A regularizing multilevel approach for nonlinear

inverse problems.

Liang Yan 16:50 - 17:20

Adaptive multi-fidelity polynomial chaos approach

to Bayesian inference in inverse problems

Hai Zhang 17:20 - 17:50 Shape reconstruction by using plasmonic resonance

Note：Zhang Hai belongs to M1-1 Forward and inverse scattering problems and their

applications


Time: May 28, Monday，15:50 - 17:50






Speaker Time Title

Fenglong

Qu 15:50 - 16:20

The inverse scattering problems for an

inhomogeneous cavity

Xiaoli Liu 16:20 - 16:50

Near-field imaging of an unbounded elastic rough

surface with a direct imaging method

Xiaoxu Xu 16:50 - 17:20

Uniqueness in Inverse Scattering Problems with

Phaseless Far-field Data at A Fixed Frequency

Guanghui

Hu 17:20 - 17:50 Inverse source problems in elasticity


13


Time: May 29, Tuesday，13:30 - 15:30






Speaker Time Title

Jiguang

Sun 13:30 –14:00 Extended Sampling Method in Inverse Scattering

Xiaodong

Liu 14:00 – 14:30

Target reconstruction with a reference point scatterer

using phaseless far field patterns

Yukun

Guo 14:30 – 15:00

Reconstruction of acoustic sources from

multi-frequency phaseless data

Heping

Dong 15:00 – 15:30

A reference ball based iterative algorithm for

phaseless inverse obstacle scattering problem

M1-1 Forward and inverse scattering problems and their applications- Session 1



Organizers: Jun Lai, Zhejiang University

Wangtao Lu, Zhejiang University

Speaker Time Title

Yayan Lu 13:30 –14:00 Exceptional Points and Novel Wave Phenomena

Wenbin Li 14:00 – 14:30

A Newton-type linewise Lax-Friedrichs sweeping

method for generalized eikonal equation

Yuliang

Wang 14:30 – 15:00

A joint reconstruction scheme for inverse scattering

problems with limited-aperture data

Xinming

Zhang 15:00 – 15:30

Optimization of drug controlled release from

multi-laminated devices based on the modified

Tikhonov regularization method

Hai

Zhang 15:50 - 17:50 Shape reconstruction by using plasmonic resonance

Note: Zhang Hai's report is on May 28, Monday，15:50 - 17:50



14








Speaker Time Title

Jingzhi Li 15:50 - 16:20

Shape derivatives-new perspective and applications

in scattering

Xia Ji 16:20 - 16:50

Direct sampling methods for inverse elastic

scattering problems

Jiaqing

Yang 16:50 - 17:20

Recovering an elastic obstacle containing

embedded objects by the acoustic far-field

measurements

Haiwen

Zhang 17:20 - 17:50

Inverse scattering problem from phaseless far-field

data

M6-1 Contributed talks-Session 1




Speaker Time Title

Xiaomei

Yang 15:50 - 16:20

q-Gauss prior and spectral likelihood

approximation in Bayesian inversion

Fangfang

Dou 16:20 - 16:50

Logarithmic stability in a coefficient inverse

problem for coupled Schrödinger equations by

arbitrary internal observation

Shanshan

Tong 16:50 - 17:20

Edge-guided $TV^p$ regularization for diffuse

optical tomography based on radiative transfer

equation

Hongpeng

Sun 17:20 - 17:50

Preconditioned Alternating Direction Method of

Multipliers with Relaxation in Hilbert Spaces


15

M5-1 Inverse Problems in Imaging Science -Session 1

Time: May 30, Wednesday，13:30 - 15:30


Organizers:Bin Dong, Peking University


Speaker Time Title

Zuoqiang

Shi 13:30 –14:00

Low Dimensional Manifold Model for Image

Processing

Ke Wei 14:00 – 14:30

Spectral Compressed Sensing via Projected

Gradient Descent

Chenglong

Bao 14:30 – 15:00

A mathematical investigation of phase space

tomography

Liyan Ma 15:00 – 15:30 Sparsity driven image recovery

M1-2 Forward and inverse scattering problems and their applications- Session 2




Wangtao Lu, Zhejiang University

Speaker Time Title

Shingyu

Leung 13:30 –14:00

Adjoint State Methods for Inverse Problems in

Seismology

Lijun Yuan 14:00 – 14:30

Nonlinear Diffraction Problems Based on Bound

States in the Continuum

Zhen Hu 14:30 – 15:00

Sensitivity Analysis for Photonic Crystal

Devices

Ke Yin 15:00 – 15:30

A new smoothing technique for non-smooth

optimization


16

M5-2 Inverse Problems in Imaging Science -Session 2



Organizers:Bin Dong, Peking University


Speaker Time Title

Bin Dong 15:50 - 16:20 PDE-Net: Learning PDEs from Data

Chunlin Wu 16:20 - 16:50

A General Truncated Regularization Framework

for Contrast-Preserving Variational Signal and

Image Restoration: Motivation and

Implementation

Jianfeng Cai 16:50 - 17:20

Accelerated Alternating Projections for Robust

Principal Component Analysis

Jae Kyu

Choi 17:20 - 17:50

HIRE: Harmonic Incompatibility REmoval Model

for Whole Brain Susceptibility Imaging

M6-2 Contributed talks-Session 2




Speaker Time Title

Huaian Diao 15:50 - 16:20

Inverse elastic surface scattering with far-field

data

Xiaoliang

Song 16:20 - 16:50

An efficient duality-based approach for

PDE-constrained sparse optimization

Wenxiu Gong 16:50 - 17:20

Non-recombining Trinomial Tree Pricing Model

and Calibration for the Volatility Smile

Day 1 – May 28, Tuesday, 9:50 - 12:00 Abstract of Plenary talks

17

Abstract of Plenary Talks

Direct and Inverse Scattering for Elastic Wave Propagation

Peijun Li

Purdue University

The scattering problems for elastic waves have received great attention for their

significant applications in diverse scientific areas such as geophysics and seismology.

In this talk, I shall present our recent studies on analysis and computation for some

direct and inverse elastic scattering problems, which include the acoustic-elastic

interaction problems, the inverse obstacle and source scattering problems, and the

time-domain scattering problems.

A Mathematical Perspective on Extraordinary Optical Transmission Through

Small Holes: Perfect Conductors and Plasmonic Metals

Junshan Lin

Auburn University

The remarkable phenomenon of extraordinary optical transmission (EOT) through

metallic nanoholes has important applications in bio-sensing and near-field imaging,

etc. Its experimental and theoretical investigation has also triggered extensive

research in modern plasmonics. The mechanisms contributing to the EOT

phenomenon can be complicated due to the multiscale nature of the underlying

structure. In this talk, I will present rigorous mathematical analysis and numerical

approaches for the studies of such phenomenon in two-dimensional structures. Both

the perfect electric conductors and plasmonic metals will be discussed.


18

Joint matrix optimization method for seismic data recovery

Yanfei Wang

Institute of geology and Geophysics,Chinese Academy of Sciences

Reconstruction of the seismic wavefield from sub-sampled data is important in

seismic image processing. To make the best of the observed seismic signals, we

propose a joint matrix minimization model to recover the seismic wavefield.

Employing matrix instead of vector as weight variable can express all the

sub-sampled traces simultaneously. This scheme utilizes the collective representation

rather than an individual one to recover a given set of sub-samples. To finish this job,

an L2,p (0 < p ≤ 1) regularized joint matrix minimization is proposed and a unified

algorithm is developed. Convergence analysis is given. Experimental results are

presented to show the efficient performance of the joint technique.


19

A data driven representation method for nonstationary convolution seismic

trace model with its applications for enhancing the resolution of seismic data

and Q estimation

Jinghuai Gao1,2, Lingling Wang1,3, Bing Zhang1 and Zongben Xu3

1: National Engineering Laboratory for Offshore Oil Exploration, Xi’an Jiaotong University;

2: Centre for Mathematics and Geodetection Technology, Xi’an Jiaotong University;

3: School of Mathematics and Statistics, Xi’an Jiaotong University;

The convolution model is the theoretical basis for seismic data processing and

enhancing vertical resolution of seismic trace and inversing the reservoir features.

This model is most useful when applied to a band limited vertical section of the waves

with a limited emergent angle. In particular, if the 3-D prestack seismic data is imaged

and corrected properly, or imaged with true amplitude, each seismic trace in the post

stack seismic data could be considered approximately to meet the convolution model.

In the convolution model, it is generally assumed that the seismic wavelet is not

varying with seismic wave traveling-time. This hypothesis however, fails under some

conditions. As a matter of fact, the seismic trace meets the varying-wavelet model, i.e.

non-stationary convolution model. Therefor we propose a data driven representation

method for nonstationary convolution seismic trace. The proposed method contains

two steps. Step1: splitting the nonstationary seismic trace into several segments, each

of them is stationary. To this end, we propose two approaches, one is deterministic

method and another is statistics one. Step 2: Constructing molecular frame transform

based on the segments which has an exact inverse transform.

In order to estimate the Q factor from the seismic trace or enhance the resolution of

seismic trace, the Fourier amplitude spectrum of “equivalent wavelet” for each

molecular is required. Therefor we construct a constructive operator to obtain the

amplitude spectrum.

As an application 1, we propose a method to enhance the resolution of seismic trace.

The effectiveness of the proposed method is by synthetic and field data. We compare

our method with spectrum –whitening method and the results suggest that the

proposed method can produce true reflector coefficients, which is very important for

lithology reservoir identification.

As an application 2, based on the constructive – operator –method, we propose a

estimation method of Q from reflection seismic data. The proposed method is used to

synthesize data and real data, and its validity is verified.


20

Shape optimization-based image data assimilation with Wasserstein distance

Jianwei Ma


In this paper, we introduce the optimal transport theory to level-set-based image data

assimilation method.Level-set method could describe the motion of the geometric

shape of a given system only using an initial contour.To obtain reliable prediction,

active contour containing in image observation data should be assimilated in order to

modify such model. However, position errors from the observation and background

will have great influence on the result when there is no prior information about the

weight parameters between them. In this regard, we use the Wasserstein distance in

optimal transport theory rather than the traditional Euclidean distance to measure the

misfit. The non-local metric combined with the level-set method is taken as a kind of

shape optimization which can deal with such position and shape error to some extent.

The decent performance of numerical tests demonstrate the efficiency of our proposed

method.

Day 3 – May 30, Wednesday, 9:00 - 12:00 Abstract of Plenary talks

21

3D Time-Domain Airborne EM Inversion with Finite-Volume Method

Changchun Yin

Jilin University

We investigate an algorithm for 3D time-domain AEM inversion with the

finite-volume and direct Gauss-Newton methods. We separate a spatially varying

secondary field from the 1D background in time-domain, and constrain the calculation

to be within the small volume of influence of airborne EM secondary source, resulting

in more compact discretization. To demonstrate the validity and merits of 3D

inversion, we first compare the results with 1D inversion on synthetic data for a

horizontal conductor and a dipping plate, which shows that both methods can well

recover the horizontal conductor, while only 3D inversion can offer good recovery for

the dipping plate. We apply our 3D algorithm to invert GEOTEM data obtained over

the Lisheen deposit in Ireland to map the sulphides at depth and obtain similar results

to 1D inversion but with better data fitting, further showing the effectiveness of our

3D inversion algorithm.

Day 4 – May 31, Thursday, 9:00 - 11:05 Abstract of Plenary talks

22

Inverse source problems for a parabolic and a hyperbolic equations

Masahiro Yamamoto, The University of Tokyo

We discuss inverse source problems of determining spatial or temporal functions of

source terms for initial-boundary value problems for a diffusion and a wave equations,

assuming that the source terms are given by products of spatial and temporal

functions. We consider several formulations for the inverse source problems and show

the uniqueness and the stability.

On the inverse conductivity problem with a single internal measurement.

Faouzi Triki, Grenoble Alpes University

In the talk I will present recent results on recovering the conductivity map from a

single internal measurement. This inverse problem is originated from multi-wave

imaging. The objective is to stabilize and improve the resolution in imaging biological

tissues. I will first show a stability estimate of Hölder type without any assumptions

on the conductivity map. Then, I will give a convergence result for the reconstruction

of the conductivity coefficient using discontinuous Galerkin method (DG). Finally, I

will present some numerical results on synthetic data to validate the theoretical

approach.

Tuesday, May 29, 2018, 13：30-15：30, Room 1, the lecture hall of the Math Building

M1-1 Forward and inverse scattering problems and their applications

23

Abstract of talks of Minisymposia

Exceptional Points and Novel Wave Phenomena

Yayan Lu

[email protected]，City University of Hong Kong

For non-Hermitian operators that depend on parameters, it is possible to have multiple

eigenvalues with only a single linearly independent eigenfunction. Such a degenerate

state is called an Exceptional Point (EP). In recent years, EPs have attracted much

attention in optics, since they give rise to many intriguing wave phenomena, some of

which are already observed in experiments, and some important applications of EPs

have already been realized. Most EPs in optics are related to PT-symmetric structures

where the real and imaginary parts of the dielectric function are even and odd with

respect to a spatial variable, respectively, but EPs are also observed in open dielectric

structures, where the dielectric function is real and positive. In this talk, we present

recent results about EPs in a simple dielectric periodic structure.

A Newton-type linewise Lax-Friedrichs sweeping method

for generalized eikonal equation

Wenbin Li

[email protected]，Harbin Institute of Technology – Shenzhen

We propose a Newton-type Lax-Friedrichs sweeping method to solve the generalized

eikonal equation arising from wave propagation in a moving fluid. The Lax-Friedrichs

numerical Hamiltonian is adopted in the discretization; different from traditional

methods, we develop a novel line-wise sweeping strategy, where the solutions on a

whole line of grid points are simultaneously updated by Newton method. The global

solution is achieved by alternatingly sweeping the domain line by line. We develop

first order algorithm as well as high order algorithm with the weighted essentially

non-oscillatory (WENO) approximations. Extensive 2-D and 3-D numerical examples

illustrate the effciency and accuracy of the new algorithm. The Newton-type sweeping

method converges faster than the traditional Lax-Friedrichs sweeping method.

mailto:[email protected]




24

Moreover, the new method manipulates solutions in a vectorized manner, and can be

effciently implemented by array programming.

A joint reconstruction scheme for inverse scattering problems

with limited-aperture data

Yuliang Wang

[email protected]，Hong Kong Baptist University

The talk is concerned with the inverse problem of reconstructing the shape of an

unknown/inaccessible scatterer from the corresponding acoustic probing. We are

particularly interested in the case with limited-aperture observation data, which arises

in a variety of important applications. Though it brings essentially no theoretical

difference, the lack of measurement information can cause severe deterioration for the

shape reconstruction in various imaging schemes. There have been some research

proposals in the literature to deal with this challenging issue that are mainly based on

data recovery. In this paper, from a different perspective, we propose a completely

novel scheme that concatenates the data recovery and the shape reconstruction. The

two processes are closely related, restricting each other and promoting each other. A

crucial ingredient for the concatenation is the localizing property of the direct imaging

method used for the shape reconstruction. The proposed joint scheme can also

incorporate any a prior knowledge of the underlying scatterer in a natural manner. We

provide theoretical explanations to the proposed joint scheme, and moreover we

conduct extensive numerical experiments to demonstrate the promising features of the

scheme in significantly enhancing both the data recovery and the shape

reconstruction.




25

Optimization of drug controlled release from multi-laminated devices based on

the modified Tikhonov regularization method

Xinming Zhang

[email protected]，Harbin Institute of Technolgoy (Shenzhen)

From the viewpoint of inverse problem, the optimization of drug release based on the

multi-laminated drug controlled release devices has been regarded as the solution

problem of the diffusion equation initial value inverse problem. In view of the

ill-posedness of the corresponding inverse problem, a modified Tikhonov

regularization method is proposed by constructing a new regularizing filter function

based on the singular value theory of compact operator. The convergence and the

optimal asymptotic order of the regularized solution are obtained. Then the classical

Tikhonov regularization method and the modified Tikhonov regularization method

are applied to the optimization problem of the initial drug concentration distribution.

For three various desired release profiles (constant release, linear decrease release and

linear increase followed by a constant release profiles), better results can be obtained

by using the modified Tikhonov regularization method. The numerical results

demonstrate that the modified Tikhonov regularization method not only has the

optimal asymptotic order, but also is suitable for the optimization and design of

multi-laminated drug controlled release devices.

Hong Kong University of Science and Technology

Hai Zhang,

[email protected]，Hong Kong University of Science and Technology

We will explore the applications of plasmonic resonance in bio-sensing in this talk.

We show that one can use the shift of plasmonic resonance frequency to reconstruct

the shapes of sub-wavelength targets. Both cases of intermediate regime and strong

interaction regime are considered.


Wednesday, May 30, 2018, 13：30-15：30, Room 1, the lecture hall of the Math Building


26

Adjoint State Methods for Inverse Problems in Seismology

Shingyu Leung

[email protected]，The Hong Kong University of Science and Technology

We discuss various applications of the adjoint state method for obtaining the

numerical solutions to various inverse problems originated from traveltime

tomography. We first formulate these inverse problems in variational formulations. To

minimize the energy in the variational formulation, we derive the gradient of the

nonlinear functional which can be efficiently computed using the adjoint state method.

We will also show various numerical examples to demonstrate the feasibility and the

robustness of these new formulations.

Nonlinear Diffraction Problems Based on Bound States in the Continuum

Lijun Yuan

[email protected]，Chongqing Technology and Business University

Bound states in the continuum (BICs) are localized or trapped modes with frequencies

in the frequency interval where out-going radiation modes exist. The BICs are related

to the non-uniqueness of corresponding boundary value problems. Nonlinear

diffraction problems based on BICs are ill-conditioned when incident waves are weak.

In this talk, a perturbation analysis is developed for the solution of a nonlinear

problem, and an efficient iterative method is introduced based on the perturbation

analysis. Our results show that the second harmonic wave fields can be very strong

comparing with the incident waves.

Sensitivity Analysis for Photonic Crystal Devices

Hu Zhen

[email protected] Hohai University

Photonic crystals (PhCs) are new kinds of artificial periodic materials with a period

on the wavelength scale. Due to their unusual ability to control and manipulate light,





27

PhCs have been widely used to design various photonic crystal devices. Sensitivity

analysis provides valuable information about fabrication tolerance for PhC devices. It

is also very useful in the optimal design process. In this paper, we develop an efficient

method for sensitivity analysis of PhC devices in idealized 2D PhCs with circular

cylinders. Our method is based on the Dirichlet-to-Neumann (DtN) map method,

which is a particular efficient numerical method for modeling 2D PhC devices. By

using the DtN maps of unit cells, the response function (such as the transmission

coefficient, the complex frequency of a resonant mode, etc.) and its partial derivatives

with respect to design parameters (such as the radii, refractive indices, positions of the

cylinders, etc.) can be efficiently calculated. Our method takes advantage of the

identical unit cells and the analytic solutions for circular cylindrical structures, and

can solve the problems in very small truncated domains. In particular, our method is

capable of rapidly computing the partial derivatives with respect to the parameters of

many different unit cells.

A new smoothing technique for non-smooth optimization

Ke Yin

[email protected]， Huazhong University of Science and Technology

Optimization problems involving non-smooth terms is prevalent in machine learning

and statistics, such as ridge regression, compressed sensing, multi-class multi-label

learning, to name a few. Usually technique for treating non-smooth terms includes

methods based on operator splitting, Frenchel duality and Moreau envelope. In this

project, we will study a new type of smoothing technique called compensated convex

transform, originally proposed by Kewei Zhang. This new type of approximation

technique provides analytical formula for many submodular functions including the

famous max function (possibly composed with a set of convex functions), squared

distance function to a finite set and upper transform for some non-smooth convex

functions in mathematical programming. The benefit of this is that it is a C^1,1

function and a tighter approximation than Moreau envelop, and it preserves convexity.

Therefore it allows first order optimization techniques that are unavailable for

non-smoothness functions, and the approximation error estimate can be conveniently

obtained. It should be competitive with other smoothing techniques。


Monday, May 28, 2018, 13：30-15：30, Room1, the lecture hall of the Math Building

M2-1 Recent Advances and Applications in Regularization

28

Bayesian Approach to A Nonlinear Inverse Problem for Time-Space Fractional

Diffusion Equation

Yuanxiang Zhang

[email protected]，Lanzhou University

The inverse problems for fractional partial differential equations has become a

promising research area because of its wide applications in many scientific and

engineering fields. Particularly, the correct orders of fractional derivative are hard to

know as they are usually determined by experimental data and contain non-negligible

uncertainty, therefore, the research on inverse problems involving the orders are in

necessary. Furthermore, the problems involving the inversion of orders are essentially

nonlinear, classical methods usually fail or quite expensive to provide satisfying

approximations, a natural way to solve such inverse problems is through Bayesian

approach. In this talk, we will consider a problem of simultaneously recovering the

source function and the orders of both time and space fractional derivative for

time-space fractional diffusion equation. The inverse problem will be formulated in

the Bayesian framework, the well-definedness and well-posedness of the

corresponding Bayesian inverse problem will be involved. In addition, the numerical

implementation for one dimensional case will be carried out via the iterative

regularizing Ensemble Kalman method (IREnKM), and the relevant numerical results

will be presented as well.




29

A Hybrid Inversion Scheme for Diffuse Optical Tomography

Yu Jiang

[email protected]，Shanghai University of Finance and Economics

Diffuse optical tomography is formulated as inverse coefficient problems for the

diffusion equation. Iterative inversion schemes such as the Levenberg-Marquardt

algorithm are known to fail when initial guesses are not close to the true value of the

coefficient to be reconstructed. In this talk, we investigate how this weakness of

iterative schemes is overcome by the use of Monte Carlo. We present a toy model of

diffuse optical tomography for which the Levenberg-Marquardt algorithm fails to

work but the Metropolis-Hastings Markov chain Monte Carlo works. We show that

our proposed hybrid scheme solves the inverse problem efficiently by preparing a

good initial guess by Monte Carlo and then computing the reconstructed value with

the Levenberg-Marquardt algorithm starting from the found initial guess.

A coupled model of partial differential equations for Uranium ores heap leaching

and its parameters identification

Wen Zhang

[email protected]，East China University of Technology

In this talk, we consider a mathematical modelling problem in engineering of

Uranium ores heap leaching. Firstly, we deduce a mathematical model of Uranium

ores heap leaching by combining solute transportation equations with microbial

chemical reactions. Secondly, an inverse problem, which is solved by the optimal

perturbation method together with the Tikhonov regularization, is considered for

identifying the parameters of the proposed mathematical model. Finally, numerical

simulations are given for the forward problem and the inverse problem to show the

pattern of Uranium ores microbial heap leaching and verify the effectiveness of

parameters identification, respectively.





30

Unique continuation principle for the time-fractional diffusion equation

Zhiyuan Li

[email protected]，Shandong University of Technology

In this talk, the diffusion equation with Caputo derivative is discussed. The Caputo

derivative is inherently nonlocal in time with history dependence, which makes the

crucial differences between fractional models and classical models, for example,

long-time asymptotic behavior. However, a maximum principle in the usual setting

still holds. Is there any other property retained from the parabolic equations? What

about the unique continuation (UC)? There is no affirmative answer to this problem

except for some special cases. Sakamoto-Yamamoto (2011) asserted that the

vanishment of a solution to a homogeneous problem in an open subset implies its

vanishment in the whole domain provided the solution vanishes on the whole

boundary. Lin-Nakamura (2016) obtained a UC by using a Carleman estimate

providing the homogeneous initial value. Both of these results are called as the weak

UC because the homogeneous condition is imposed on the boundary value or on the

initial value, which is absent in the parabolic prototype. In this talk, by using Theta

function method and Laplace transform argument, we will give a classical type unique

continuation.




31

A dynamical regularization algorithm for solving inverse source problems of

elliptic partial differential equations

Rongfang Gong

[email protected]，Nanjing University of Aeronautics and Astronautics

This study considers the inverse source problem for elliptic partial differential

equations with both Dirichlet and Neumann boundary data. The unknown source term

is to be determined by additional boundary conditions. Unlike the existing methods

found in the literature, which usually employ the first-order in time gradient-like

system (such as the steepest descent methods) for numerically solving the regularized

optimization problem with a fixed regularization parameter, we propose a novel

method with a second-order in time dissipative gradient-like system and a dynamical

selected regularization parameter. A damped symplectic scheme is proposed for the

numerical solution. Theoretical analysis is given for both the continuous model and

the numerical algorithm. Several numerical examples are provided to show the

robustness of the proposed algorithm.

A regularizing multilevel approach for nonlinear inverse problems

Wei Wang

[email protected]，Jiaxing University

In this paper, we propose a multilevel method for solving nonlinear ill-posed

problems $F(x) = y$ in Banach spaces. By minimizing the discretized version of the

regularized functionals for different levels of discretization, we define a sequence of

regularized approximations to the exact solution, which is shown to be stable and

globally convergent for arbitrary initial guess. The penalty terms $\Theta$ in

regularized functionals are allowed to be non-smooth in order to include

$L^p-L^1$ or $L^p-$TV (total variation) cases, which are important in reconstructing

special features of solutions such as sparsity and discontinuities. Two parameter

identification examples are presented to validate the theoretical analysis and to verify

the effectiveness of the method. This is a joint work with Min Zhong (Southeast

University)





32

Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in

inverse problems

Liang Yan

[email protected]，Southeast University

The polynomial chaos (PC) expansion is commonly used to construct a surrogate

model in the Bayesian inference to speed up Markov chain Monte Carlo (MCMC)

calculations. However, the use of a PC surrogate introduces a modeling error, that

may severely distort the estimate of the posterior distribution. This error can be

corrected by increasing the order of the PC to construct a very accurate surrogate, but

the cost may increase too fast. In this talk, we seek to address this challenge by

introducing an adaptive procedure to construct a multi-fidelity PC surrogate and

explore the posterior simultaneously. The new algorithm combines the high-fidelity

and low-fidelity model evaluations, where the low-fidelity evaluations arise from a

prior-based PC surrogate that approximates the forward model as the high-fidelity

model. The key idea is to speed up the MCMC by combing, instead of replacing, the

high-fidelity model with the low-fidelity PC model. We demonstrate the practical

performance of the proposed strategies through two nonlinear inverse problems. Both

of these examples show that the proposed adaptive multi-fidelity PC can improve

significantly the accuracy of the prior-based PC method without a significantly

increase in the computational time. Further, the numerical results also indicate that the

new algorithm can provide greater efficiency by several orders of magnitude

compared to a standard MCMC using high-fidelity model.

Hong Kong University of Science and Technology

Hai Zhang,


We will explore the applications of plasmonic resonance in bio-sensing in this talk.

We show that one can use the shift of plasmonic resonance frequency to reconstruct

the shapes of sub-wavelength targets. Both cases of intermediate regime and strong

interaction regime are considered.



Monday, May 28, 2018, 13：30-15：30, Room2, the lecture hall of the Math building

M3-1 Recent advances in inverse scattering theory

33

Complex Gaussian mixture based model error learning for inverse medium

scattering problems with multi-frequencies

Junxiong Jia

[email protected]，Xi'an Jiaotong University

This talk is concerned with the modeling errors appeared in the numerical methods of

inverse medium scattering problems (IMSP).Optimization based iterative methods are

wildly employed to solve IMSP, which are computationally intensive due to a series

of Helmholtz equations need to be solved numerically. Hence, rough approximations

of Helmholtz equations can significantly speed up the iterative procedure. However,

rough approximations will lead to instability and inaccurate estimations. Using the

Bayesian inverse methods, we incorporate the modelling errors brought by the rough

approximations. Modelling errors are assumed to be some complex Gaussian

mixture(CGM) random variables, and in addition, well-posedness of IMSP in the

statistical sense has been established by extending the general theory to involve CGM

noise. Then, we generalize the real valued expectation-maximization (EM) algorithm

used in the machine learning community to our complex valued case to learn

parameters in the CGM distribution. Based on these preparations, we generalize the

recursive linearization method (RLM) to a new iterative method named as Gaussian

mixture recursive linearization method (GMRLM) which takes modelling errors into

account. Finally, we provide two numerical examples to illustrate the effectiveness of

the proposed method.



34

The application of multilevel sampling method in the inverse scattering problems

Keji Liu

[email protected]，Shanghai University of Finance and Economics

The multilevel sampling method can be viewed as a direct sampling method, since it

only involves matrix–vector operations and does not need to solve any large-scale

ill-posed linear systems or any optimization process. Moreover, it is easy to

implement, highly tolerant to noise and computationally very cheap. Furthermore, it

can easily separate multiple disjoint medium components, usually with just a few

iterations to provide a satisfactory initial position of each object. And the technique

also works for the multiscale scatterers. With an effective initial location of each

obstacle, any existing efficient but computationally more demanding methods can be

applied for the further refinement of the estimated shape of each medium component

as well as for the recovery of the contrast profiles of different media.

Inverse Source Problem in Electrodynamics

Yue Zhao

[email protected]，Beijing Computational Science Research Center

This talk concerns inverse source problems for the time-dependent Maxwell equations.

The electric current density is assumed to be a separable function, which is the

product of a spatial function and a temporal function. We prove uniqueness and

stability in determining the spatial or temporal function from the electric field,which

is measured on a sphere or at a point over a finite time interval





35

Imaging perfectly conducting cylinders with experimental data

Hongwei Zhou

[email protected]， Northeast Forestry University

This talk concerns an inverse time-dependent electromagnetic scattering problem of

imaging perfectly conducting cylinders buried in an homogeneous isotropic medium.

The cross section of the cylinder is supposed to be detected by transient

electromagnetic pulses in the case of TE polarization.

We apply the Kirch-hoff migration scheme to locate the position of small objects

from both synthetic and experiment data.The multiple-input-multiple-out scheme are

used for imaging extended scatterers from the data generated by the software

gprMax.Numerics show that the Kirchhoff migration method is not only efficient but

also robust with respect to polluted data at high noise levels.Experimental results

show good quantitative agreement to numerical simulations.




36

The inverse scattering problems for an inhomogeneous cavity

Fenglong Qu

[email protected]，Yantai University

In this talk I will introduce some uniqueness results for the inverse scattering

problems based on a novel method. I will show that the interior part, the refractive

index described by a piecewise constant function as well as the transmission

coefficient of an inhomogeneous cavity can be uniquely determined. The proof is

based on the arguments of constructions of some well-posed interior transmission

problems and some L_P (1<P<2) estimates for the direct scattering problem.

Near-field imaging of an unbounded elastic rough surface

with a direct imaging method

Xiaoli Liu

[email protected]，Academy of Mathematics and Systems Science, Chinese

Academy of Sciences

This talk is concerned with the inverse scattering problem of time-harmonic elastic

waves by an unbounded rigid rough surface. A direct imaging method is developed to

reconstruct the unbounded rough surface from the elastic scattered near-field Cauchy

data generated by point sources. A Helmholtz-Kirchhoff-type identity is derived and

then used to provide a theoretical analysis of the direct imaging algorithm. Numerical

experiments are presented to show that the direct imaging algorithm is fast, accurate

and robust with respect to noise in the data.This is a joint work with Prof. Bo Zhang

and Dr. Haiwen Zhang.





37

Uniqueness in Inverse Scattering Problems with Phaseless Far-field Data

at A Fixed Frequency

Xiaoxu Xu

[email protected]，Academy of Mathematics and Systems Science,


This talk is concerned with uniqueness in inverse acoustic scattering with phaseless

far-field data at a fixed frequency. The main difficulty of this problem is the so-called

translation invariance property of the modulus of the far-field pattern generated by

one plane wave as the incident field. Based on B. Zhang and H. Zhang's

previous works on the numerical algorithms for phaseless inverse scattering problems,

the translation invariance property of the phaseless far-field pattern can be broken by

using infinitely many sets of superpositions of two plane waves as the incident fields

at a fixed frequency. In this talk, we will show that the location, shape and boundary

condition of the obstacle and the refractive index of an inhomogeneous medium can

be uniquely determined by the phaseless far-field patterns generated by infinitely

many sets of superpositions of two plane waves with different directions at a fixed

frequency under the condition that the obstacle is a priori known to be a sound-soft or

non-absorbing impedance obstacle and the index of refraction of the inhomogeneous

medium is real-valued and greater or less than 1 on its compact support. To the best of

our knowledge, this is the first uniqueness result in inverse scattering with phaseless

far-field data. The sketch of the proofs will be given. Our proofs are based essentially

on the limit of the normalized eigenvalues of the far-field operators which is also

established in this paper by using a factorization of the far-field operators. This talk is

based on a joint work with B. Zhang and H. Zhang.

Inverse source problems in elasticity

Guanghui, Hu

[email protected]，Beijing Computational Science Research Center (CSRC)

In this talk we consider uniqueness and stability to inverse source problems in linear

elasticity. Special attention will be paid to inverse problems with general source

terms and in inhomogeneous media.





38

Extended Sampling Method in Inverse Scattering

Jiguang Sun

[email protected]，MTU and UESTC

A new sampling method for inverse scattering problems is proposed to process far

field data of one incident wave.As the linear sampling method, the method sets up

ill-posed integral equations and uses the (approximate) solutions to reconstruct the

target. In contrast, the kernels of the associated integral operators are the far field

patterns of sound soft balls.The measured data is moved to right hand sides of the

equations, which gives the method the ability to process limit aperture data.

Furthermore, a multilevel technique is employed to improve the reconstruction.

Numerical examples show that the method can effectively determine the location and

approximate the support with little a priori information of the unknown target.

Target reconstruction with a reference point scatterer using phaseless far field

patterns

Xiaodong Liu

[email protected]，Institute of Applied Mathematics, Academy of Mathematics and

Systems Science, Chinese Academy of Sciences

An important property of the phaseless far field patterns with incident plane waves is

the translation invariance. Thus it is impossible to reconstruct the location of the

underlying scatterers. By adding a reference point scatterer into the scattering system,

we show that the nature of the scatterer can be uniquely determined by the phaseless

far field pattern.In particular, under a priori assumptions on the scatterer, uniqueness

for one incident wave can also be established. The reference point technique not only

overcomes the translation invariance, but also brings a practical phase retrieval

algorithm. We design a novel direct sampling method using the phaseless data

directly and propose a combination method with the novel phase retrieval algorithm

and the classical direct sampling methods. We provide rigorous theoretical

justifications for the proposed methods. Numerical examples in two dimensions are

also presented to demonstrate their feasibility and effectiveness.





39

Reconstruction of acoustic sources from multi-frequency phaseless data

Yukun Guo

[email protected]，Harbin Institute of Technology

This talk is concerned with the inverse source problem of reconstructing an unknown

acoustic excitation from phaseless measurements of the radiated fields away at

multiple frequencies. It is well known that the non-uniqueness issue is a major

challenge associated with such an inverse problem. We develop a novel strategy to

overcome this challenging problem by recovering the radiated fields via adding some

reference point sources as extra artificial sources to the inverse source system. This

novel reference source technique requires only a few extra data, and brings in a

simple phase retrieval formula. The stability of this phase retrieval approach is

rigorously analyzed. After the reacquisition of the phase information, the

multi-frequency inverse source problem with recovered phase information is solved

by the Fourier method, which is non-iterative, fast and easy to implement. Several

numerical examples will be presented to demonstrate the feasibility and effectiveness

of the proposed method.

A reference ball based iterative algorithm

for phaseless inverse obstacle scattering problem

Heping Dong

[email protected]，Jilin University

In this talk, we consider the inverse problem of determining the location and the shape

of a sound-soft obstacle from the phaseless far-field data for a single incident plane

wave. By adding a reference ball artificially to the inverse scattering system, we

propose an iterative scheme which involves a system of nonlinear and ill-posed

integral equations to reconstruct both the location and the shape of the obstacle. The

reference ball causes few extra computational costs, but breaks the translation

invariance and brings information about the location of the obstacle. Several

validating numerical examples are provided to illustrate the effectiveness and

robustness of the proposed inversion algorithm.





40

Shape derivatives — new perspective and applications in scattering

Jingzhi Li

[email protected]，Southern University of Science and Technology

This talk presents the “derivative” of solutions of second-order boundary value

problems with respect to the shape of the domain. A rigorous approach relies on

encoding shape variation by means of deformation vector fields, which will supply

the directions for taking shape derivatives. These derivatives and methods to compute

them numerically are key tools for studying shape sensitivity, performing gradient

based shape optimization, and small-variation shape uncertainty quantification. A

unifying view of second-order elliptic boundary value problems recasts them in the

language of differential forms (exterior calculus). Fittingly, the shape deformation

through vector fields matches the concept of Lie derivative in exterior calculus. This

paves the way for a unified treatment of shape differentiation in the framework of

exterior calculus. Applications in scattering problems reveals the extraordinary power

of the machinery.

Direct sampling methods for inverse elastic scattering problems

Xia Ji

[email protected]，Institute of Computational Mathematics， CAS

We consider the inverse elastic scattering of incident plane compressional and shear

waves from the knowledge of the far field patterns. Specifically,three direct sampling

methods for location and shape reconstruction are proposed using the different

component of the far field patterns. Only inner products are involved in the

computation, thus the novel sampling methods are very simple and fast to be

implemented. With the help of the factorization of the far field operator, we give a

lower bound of the proposed indicator functionals for sampling points inside the

scatterers. While for the sampling points outside the scatterers, we show that the

indicator functionals decay like the Bessel functions as the sampling point goes





41

away from the boundary of the scatterers. We also show that the proposed indicator

functionals continuously dependent on the far field patterns, which further implies

that the novel sampling methods are extremely stable with respect to data error. For

the case when the observation directions are restricted into the limited aperture, we

firstly introduce some data retrieval techniques to obtain those data that can not be

measured directly and then use the proposed direct sampling methods for location and

shape reconstructions. Finally, some numerical simulations in two dimensions are

conducted with noisy data, and the results further verify the effectiveness and

robustness of the proposed sampling methods, even for multiple multiscale cases and

limited-aperture problems.

Recovering an elastic obstacle containing embedded objects

by the acoustic far-field measurements

Jiaqing Yang

[email protected]，Xi'an Jiaotong University

This talk is concerned with the inverse scattering of time-harmonic acoustic waves by

a three-dimensional bounded elastic obstacle which may contain embedded

impenetrable obstacles inside. We propose a novel and simple technique to show that

the elastic obstacle can be uniquely recovered by the acoustic far-field pattern at a

fixed frequency, disregarding its contents. The method is based on constructing a

well-posed modified interior transmission problem on a small domain and makes use

of an a priori estimate for both the acoustic and elastic wave fields in the usual

H^1-norm. In the case when there is no obstacle embedded inside the elastic body,

our method gives a much simpler proof for the uniqueness result obtained previously

in the literature. This is a joint work with Prof. Bo Zhang and Dr. Fenglong Qu.




42

Inverse scattering problem from phaseless far-field data

Haiwen Zhang

[email protected]， Academy of Mathematics and Systems Science, Chinese

Academy of Sciences

It is well known that the modulus of the far-field pattern (or phaseless far-field pattern)

is invariant under translations of the scattering obstacle if only one plane wave is used

as the incident field, so the shape but not the location of the obstacle can be recovered

from the phaseless far-field data. In this talk, it is proved that the translation

invariance property of the phaseless far-field pattern can be broken if superpositions

of two plane waves are used as the incident fields.Based on this, a direct imaging

method is then developed to recover both the location and the shape of the obstacle

simultaneously from multi-frequency phaseless far-field data. Numerical examples

are also carried out to illustrate the validity of the approach and the effectiveness of

the inversion algorithm.


Monday, May 28, 2018, 13：30-15：30, the third floor of the Math Building

M4-1 Computational inverse problems and their applications in atmospheric and oceanic sciences

43

Volcanic eruption case study - Nabro: Supercomputer implementation for

identifying source terms of the atmospheric pollutants

Yi Heng

[email protected]，Sun Yat-sen University

The global air pollution problem is becoming more and more prominent. As one of

the main sources of natural pollution, the air pollutants due to strong volcanic

eruptions will have a big impact on the atmosphere. The study of volcanic eruption

events is of great significance for ensuring the aviation safety, revealing the causes of

global climate change and understanding the mechanisms of complex atmospheric

motion. Currently, it is still a key problem concerning the lack of computational

modeling techniques to meet the prediction requirements for the high-resolution

identification of the source terms of atmospheric pollutants. We focus on the study of

the inverse source problems of atmospheric pollution transport by considering the

Nabro volcanic eruption as a benchmark problem. By using available remote sensing,

lidar and other observation data, we are developing a high-resolution and stable

inversion method, and establishing a supercomputing platform that can predict the

transport of pollutants in events such as volcanic eruption, large-scale industrial

accidents. Recently, we further develop our Lagrangian particle dispersion model that

can handle large amounts of data. The goodness-of-fit of the simulations is studied

and the inversion algorithm that is based on the sequential importance sampling is

being developed. Besides, we also conduct the scalability of parallel codes and the

optimization of the environment of hardware and software on state-of-the-art

supercomputing systems. In this way, it aims at providing a scientific basis for

practical applications, and a general, widely applicable, quantitative research method

for commonly occurring inverse source problems of atmospheric pollution transport.




44

Multi-fidelity method using polynomial chaos: analysis and applications

Liang Yan

[email protected]，Southeast University

In many situations across computational science, such as optimization, inference and

uncertainty quantification, it often requires a large number of model evaluations. This

leads to long runtimes if the model is expensive to evaluate. One strategy is to replace

the computationally expensive high-fidelity model with a computationally cheap

surrogate model; however, simply replacing the high-fidelity model with a

low-fidelity model can result in significant speedups but leads to a lower

approximation quality result. It is often possible to construct a multi-fidelity model

having accuracy comparable with the high-fidelity model and computational cost

comparable with the low-fidelity model. In this talk, we will introduce a multi-fidelity

surrogate modeling approach based on polynomial chaos (PC), which combines

evaluations of both the high fidelity and the PC surrogate model. This method

generally relies on a relatively large number of low-fidelity samples along with a

smaller number of high-fidelity samples to build an additive correction of the

low-fidelity model. The method is demonstrated on some artificial test problems in

the context of UQ.

Application of an adjoint method to an intermediate coupled model and its

improvement for real-time ENSO prediction

Chuan Gao

[email protected]，Institute of Oceanology, CAS

The El Niño and Southern Oscillation (ENSO) has great influence on climate and

weather worldwide. It is of great practical significance to accurately and effectively

make real-time ENSO prediction using coupled models. An intermediate coupled

model (ICM) is used at the Institute of Oceanology, Chinese Academy of Sciences

(IOCAS), named the IOCAS ICM, to predict the sea surface temperature (SST)

evolution in the tropical Pacific. Recently, the IOCAS ICM has been routinely used to

predict the real-time ENSO event, which is collected by the International Research

Institute for Climate and Society (IRI) for further analyses and application. However,





45

there exist large uncertainties and model biases in real-time ENSO predictions. One of

the reasons is the uncertainties in model initial condition and model parameters. To

improve the real-time ENSO prediction, the conditional nonlinear optimal

perturbation approach supported by the adjoint method is used to IOCAS ICM to

explore ENSO predictability. The four dimentional variational (4D-Var) data

assimilation method supported by the adjoint method is used to IOCAS ICM to

optimize initial condition and model parameters for ENSO prediction. It is expected

to provide modeling tool and theoretical guidance for the improvement of real-time

ENSO prediction using the ICM, and the innovative modeling platform can have wide

applications for future studies on ENSO analyses and predictions. It is also providing

valuable methodologies and guidance for other modeling studies.

A Multiscale SVR Method On Spheres with Data Compression

Min Zhong

[email protected]，School of Mathematics, Southeast University

We propose and analyze a multiscale support vector regression (SVR) algorithm for

noisy scattered data on the unit sphere. To this end, the algorithm uses

Wendland's radial basis functions with different scales and the Vapnik

$\epsilon$-intensive loss function to compute a regularized approximation at each

step. A data compression method was applied to discard small coefficients

dynamically. We discuss the convergence of the algorithm and prove additional errors

can be controlled so that the discarding strategy does not lead to significant errors.




46

A hybrid marginal sequential Monte Carlo method for data assimilation

Jinglai Li

[email protected]，Shanghai Jiao Tong University

We present a marginal sequential Monte Calro method for data assimilation. The

method aims to compute the marginal posterior distribution at each given time,

instead of the joint posterior. As a result, we can perform importance sampling in

the low dimensional marginal space, which is consdideably easier than doing it in the

joint space. Moreover, we use the ensemble Kalman fiter to construct the importance

distribution and we use the fast multipole method to compute the importance weight.

Finally we use numerical examples to demonstrate the performance of the proposed

method.

Correction of Biased Climate Simulated by Biased Physics through Parameter

Estimation in an Intermediate Coupled Model

Xuefeng Zhang

[email protected]，Tianjin University

Imperfect physical parameterization schemes in a coupled climate model are an

important source of model biases that adversely impact climate prediction. Using an

intermediate coupled ocean-atmosphere model, we studied parameter optimization

when the assimilation model contains biased physics within a biased assimilation

experiment framework. While the stochastic physics, implemented by initially

perturbing the physical parameters, can significantly enhance the ensemble spread and

improve the representation of the model ensemble, the parameter estimation is able to

mitigate the model biases induced by the biased physics. Further, better results for

climate estimation and prediction can be obtained when only the most-influential

physical parameters are optimized and allowed to vary geographically. In addition,

With a coupled ocean-atmosphere-land model of intermediate complexity, the impact

of imperfect parameter estimation on model simulation with biased physics has been

studied. While the traditional LSPF (least-squares parameter fitting) method is able to

improve the performance of coupled model simulations, the optimized parameter

values from the CMPE (coupled model parameter estimation), which uses the coupled





47

model dynamics to project observational information onto the parameters, further

reduce the bias of the simulated climate arising from biased physics. These results

suggest that the physical parameter estimation via the CMPE scheme is an effective

approach to restrain the model climate drift during decadal climate predictions using

coupled general circulation models.

Channel selection method for high spectral resolution infrared data based on

relative entropy

Huadong Du

[email protected]，National University of Defense Technology

It is necessary to select finite channels from the observation data of thousands of

channels to improve the computational efficiency during the retrieval of atmospheric

profiles. The relative entropy contains the information of the analysis field and the

background field, the covariance matrix of the analysis and background, which can be

quantified by degrees of freedom and the information content. It unifies these two

quantities through a weighted way. In this paper, a channel selection algorithm based

on relative entropy is proposed. For the temperature retrieval using high spectral

resolution data from atmospheric infrared sounder (AIRS), a corresponding channel

selection scheme is designed and the results under clear air conditions in two extreme

regions, warm and wet in the tropical area, cold and dry in the sub-Arctic area, are

given. And the differences between these two cases are analyzed. Then the

temperature inversion was carried out and the distribution of mean bias and the

standard deviation with height are obtained. These results show that the channel

selection algorithm based on relative entropy is applicable to combine the degrees of

freedom and the information content in retrievals using different channels according

to different observations and has further application value.




48

On periodic parameter identification in stochastic differential equations

Pingping Niu

[email protected]，Fudan University

Periodic parameters are common and important in stochastic differential equations

(SDEs) arising in many contemporary scientific and engineering fields involving

dynamical processes. These parameters contain the damping coefficient, the volatility

or diffusion coefficient and possibly an external force. Identification of these periodic

parameters allows us a better understanding of the dynamical processes and their

hidden intermittent instability. Conventional approaches usually focus on one of these

parameters and assume that the rests are known. By introducing the decorrelation time

and calculating the standard Gaussian statistics (mean, variance) explicitly for the

scalar Langevin equations with periodic parameters, we propose a parameter

identification approach to simultaneously recovering all these parameters by

observing a single trajectory of SDEs. Such an approach is summarized in forms of

regularization schemes with noisy operators and noisy right-hand sides and is further

extended to parameter identification of SDEs which are indirectly observed by

another random process. Numerical examples show that our approach performs well

in stable and weakly unstable regimes but may fail in strongly unstable regime which

is induced by the strong intermittent instability itself.



M5-1 Inverse Problems in Imaging Science

49

Low Dimensional Manifold Model for Image Processing

Zuoqiang Shi

[email protected]，Tsinghua University

In this talk, I will introduce a novel low dimensional manifold model for image

processing problem.This model is based on the observation that for many natural

images, the patch manifold usually has low dimension structure. Then, we use the

dimension of the patch manifold as a regularization to recover the original image.

Using some formula in differential geometry, this problem is reduced to solve

Laplace-Beltrami equation on a manifold. The Laplace-Beltrami equation is solved by

the point integral method. Numerical tests show that this method gives very good

results in image inpainting, denoising and super-resolution problem. This is joint

work with Stanley Osher and Wei Zhu.

Spectral Compressed Sensing via Projected Gradient Descent

Ke Wei

[email protected]，Fudan University

Let $x$ be a spectrally sparse signal consisting of $r$ complex sinusoids with or

without damping. We consider the spectral compressed sensing problem, which is

about reconstructing $x$ from its partial revealed entries. By utilizing the low rank

structure of the Hankel matrix corresponding to $x$, we develop a computationally

efficient algorithm for this problem. The algorithm starts from an initial guess

computed via one-step hard thresholding followed by projection, and then proceeds

by applying projected gradient descent iterations to a non-convex functional. Based

on the sampling with replacement model, we prove that $O(r^2\log(n))$ observed

entries are sufficient for our algorithm to achieve the successful recovery of a

spectrally sparse signal. Moreover, extensive empirical performance comparisons

show that our algorithm is competitive with other state-of-the-art spectral

compressed sensing algorithms in terms of phase transitions and overall

computational time. Joint with Jian-Feng Cai (HKUST) and Tianming Wang (U. of

Iowa).





50

A mathematical investigation of phase space tomography

Chenglong Bao

[email protected]，Tsinghua University

Phase space tomography is an important tool in the study of light propagation and

dynamics. In this talk, we firstly show that traditional measurement methods result in

the coherence loss due to ignoring the pixel contents. Besides, we propose a robust

model with trace regularization term to overcome the noise effects. Both simulated

and experimental results will be reported.

Sparsity driven image recovery

Liyan Ma

[email protected]，Institute of Microelectronics of Chinese Academy of Sciences

In this talk, we present some recent progress on sparsity driven approaches for image

recovery. First, the properties of the structured sparse model selection over a family of

learned orthogonal bases and the weighted nuclear norm minimization (WNNM) are

analyzed. We prove that the minimization problem of WNNM has a unique global

optimal solution in the closed form for the weights being in arbitrary order. Then, we

propose models for deblurring under Gaussian or impulse noise. Numerical results are

presented to demonstrate the good performance of our approach.





51

PDE-Net: Learning PDEs from Data

Bin Dong

[email protected]，Peking University

Deep learning continues to dominate machine learning. It is now widely used in many

research areas in science and engineering, and has major industrial impacts.

In this talk, I will start with a brief review of deep learning in image restoration and

image analysis. I will present my personal understanding of deep learning from the

perspective of applied mathematics, which inspired some of our recent work including

the one that I will present in this talk. In this work, we designed a transparent deep

feed-forward convolutional network, called the PDE-Net, to accurately predict

dynamics of complex systems and to uncover the underlying hidden PDE models

simultaneously. The design is inspired by our previous theoretical studies on bridging

wavelet frame transforms and differential operators in variational and PDE

frameworks. Promising numerical results will be presented in the end of the talk.

A General Truncated Regularization Framework for Contrast-Preserving

Variational Signal and Image Restoration: Motivation and Implementation

Chunlin Wu

[email protected]，Nankai University

Variational methods have become an important kind of methods in signal and image

restoration - a typical inverse problem. One important minimization model consists of

the squared L2 data fidelity (corresponding to Gaussian noise) and a regularization

term constructed by a potential function composed of first order difference operators.

It is well known that total variation (TV) regularization, although achieved great

successes, suffers from a contrast reduction effect. Using a typical signal, we show

that, actually all convex regularizers and most nonconvex regularizers have this effect.

With this motivation, we present a general truncated regularization framework. The

potential function is a truncation of existing nonsmooth potential functions and thus

flat from some positive t. Some analysis in 1D theoretically demonstrate the good

contrast-preserving ability of the framework. We also give optimization algorithms

with convergence verification in 2D, where global minimizers of each subproblem

(either convex or nonconvenx) are calculated. Experiments numerically show the

advantages of the framework.





52

Accelerated Alternating Projections for Robust Principal Component Analysis

Jianfeng Cai


We study robust principal component analysis for the fully observed setting, which is

about separating a low rank matrix L and a sparse matrix S from their sum D=L+S.

This talk presents a new non-convex algorithm for RPCA, dubbed accelerated

alternating projections. Exact recovery guarantee has been established which shows

linear convergence of the proposed algorithm. Empirical performance evaluations

establish the advantage of our algorithm over other state-of-the-art algorithms for

robust PCA.

HIRE: Harmonic Incompatibility REmoval Model

for Whole Brain Susceptibility Imaging

Jae Kyu Choi

[email protected]，Shanghai Jiao Tong University

It is well known that the inverse problem of quantitative susceptibility mapping (QSM)

is ill-posed as the integral kernel has the zeros in the frequency domain. While

numerous single system regularization based models have been proposed to overcome

this ill-posedness, they show drawbacks as the field data may contain an

incompatibility other than the additive noise. In this talk, we propose a novel

regularization based susceptibility reconstruction model from a given measured local

field data. Following the QSM reconstruction procedure, we characterize that the

measured data contains the harmonic incompatibility associated with the boundary

condition imposed on the Poisson's equation. This harmonic incompatibility is

embedded in our reconstruction model by means of sparse regularization under the

Laplacian to formulate a two system based QSM reconstruction model. To solve the

proposed reconstruction model, an alternating minimization algorithm is proposed

with the guaranteed convergence. Finally, the numerical experiments show that our

proposed model achieves better performance over the existing approaches.




M6-1 Contributed talks

53

q-Gauss prior and spectral likelihood approximation in Bayesian inversion

Xiaomei Yang

[email protected]，Southwest Jiaotong University

The central issues of Bayesian inversion algorithm lie in the construction of prior and

acceleration. In real applications, they are usually the bottleneck of this method that

limits its scope for engineers. In this talk, q-analogy of Gauss distribution, q-Gauss

distribution, is taken as the prior of inverse problems. And an acceleration algorithm

based on spectral likelihood approximation is discussed. We mainly focus on the

convergence of the posterior distribution in the sense of Kullback-Leibler divergence

when approximated likelihood function and truncated prior distribution are

used.Moreover, the convergence measured in TV metric and Hellinger metric are

obtained.

Logarithmic stability in a coefficient inverse problem for coupled Schrödinger

equations by arbitrary internal observation

Fangfang Dou

[email protected]，University of Electronic Science and Technology of China

In this talk, we present an inverse problem of determining the potential of coupled

Schrödinger equation in a bounded domain from the data of the solution in a

subdomain over a time interval. Assuming that in a neighborhood of a suitable part of

the boundary of the spatial domain, the potential is known and without any

assumption the observation domain, we prove a logarithmic stability estimate for this

inverse problem.





54

Edge-guided $TV^p$ regularization for diffuse optical tomography based on

radiative transfer equation

Shanshan Tong

[email protected]，Harbin Institute of Technology

In this work, we address the recovery of scattering and absorption coefficients in

steady radiative transfer equation (RTE) with the application in diffuse optical

tomography (DOT). The edge-guided $TV^p$ regularization scheme is proposed for

such a situation, which consists of a data fidelity term and an $\ell^p$-norm ($0<p<1$)

of the gradients of underlying optical coefficients, where the edge-guided step works

in the form of a weighted matrix. A normalizing technique is incorporated into our

algorithm in order to reduce the ‘cross-talk’ between scattering and absorption

coefficients. We focus our attention on the DOT application of imaging the region

containing low scattering or non-scattering layer, such as human head of a neonate.

The simulations are proceeded under two cases, one is to recover the targets with

prior information of the layer, the other is to identify both layer and targets without

prior information of the layer. The recovered images and quantitative results show

that the proposed method hold promise for recovering the targets in the region

containing low scattering or non-scattering layer.

This is a joint work with Bo Han.




55

Preconditioned Alternating Direction Method of Multipliers with Relaxation

in Hilbert Spaces

Hongpeng Sun

[email protected]， Renmin University of China

Alternating direction method of multipliers (ADMM) is a powerful first order

methods for various applications in signal processing and imaging. However, there is

no clear result on the weak convergence of ADMM with relaxation studied by

Eckstein and Bertsakas in infinite dimensional Hilbert spaces. In this paper, by

employing a kind of "partial" gap analysis, we prove the weak convergence of general

preconditioned and relaxed ADMM in infinite dimensional Hilbert spaces, with

preconditioning for solving all the involved implicit equations under mild conditions.

We also give the corresponding ergodic convergence rates respecting to the "partial"

gap function. Furthermore, the connections between certain preconditioned and

relaxed ADMM and the corresponding Douglas-Rachford splitting methods are also

discussed, following the idea of Gabay. Numerical tests also show the efficiency of

the proposed overrelaxation variants of preconditioned ADMM.




56

Inverse elastic surface scattering with far-field data

Huaian Diao

[email protected]，Northeast Normal University

A rigorous mathematical model and an efficient computational method are proposed

to solving the inverse elastic surface scattering problem which arises from the

near-field imaging of periodic structures. We demonstrate how an enhanced resolution

can be achieved by using more easily measurable far-field data. The surface is

assumed to be a small and smooth perturbation of an elastically rigid plane. By

placing a rectangular slab of a homogeneous and isotropic elastic medium with larger

mass density above the surface, more propagating wave mode scan be utilized from

the far-field data which contributes to the reconstruction resolution. Requiring only a

single illumination, the method begins with the far-to-near (FtN) field data conversion

and utilizes the transformed field expansion to derive an analytic solution for the

direct problem, which leads toan explicit inversion formula for the inverse problem.

Moreover, a nonlinear correction scheme is developed to improve the accuracy of the

reconstruction. Results show that the proposed method is capable of stably

reconstructing surfaces with resolution controlled by the slab's density. This is

the joint work with Peijun Li and Xiaokai Yuan.

An efficient duality-based approach for PDE-constrained sparse optimization

Xiaoliang Song

[email protected]，Dalian University of Technology

In this paper, elliptic optimal control problems involving the $L^1$-control cost

($L^1$-EOCP) is considered. To numerically discretize $L^1$-EOCP, the standard

piecewise linear finite element is employed. However, different from the finite

dimensional $l^1$-regularization optimization, the resulting discrete $L^1$-norm

does not have a decoupled form. A common approach to overcome this difficulty is

employing a nodal quadrature formula to approximately discretize the $L^1$-norm. It

is clear that this technique will incur an additional error. To avoid the additional error,

solving $L^1$-EOCP via its dual, which can be reformulated as a multi-block

unconstrained convex composite minimization problem, is considered. Motivated by




57

the success of the accelerated block coordinate descent (ABCD) method for solving

large scale convex minimization problems in finite dimensional space, we consider

extending this method to $L^1$-EOCP. Hence, an efficient inexact ABCD method is

introduced for solving $L^1$-EOCP. The design of this method combines an inexact

2-block majorized ABCD and the recent advances in the inexact symmetric

Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic

programming whose objective contains a nonsmooth term involving only the first

block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical

examples. Numerical results not only confirm the finite element error estimates, but

also show that our proposed algorithm is more efficient than (a) the ihADMM

(inexact heterogeneous alternating direction method of multipliers), (b) the APG

(accelerated proximal gradient) method.

Non-recombining Trinomial Tree Pricing Model and Calibration

for the Volatility Smile

Wenxiu Gong

[email protected]，Renmin University of China

In this paper, we consider the non-recombining trinomial tree pricing model under the

volatility that is a function of time, establish the option pricing model and give the

convergence rates of the non-recombining trinomial tree method. In addition, we

research the calibration problem of volatility, and adopt an exterior penalty method to

transform this problem into a nonlinear unconstrained optimization problem. For the

optimization problem, we use the quasi-Newton algorithm. Finally, we test our model

by numerical examples and options data on the SP 500 index. The results show the

effectiveness of the non-recombining trinomial tree pricing model.


58

Introduction to Tianyuan Mathematical Center in Northeast

China

国家天元数学东北中心是国家自然科学基金委数学天元基金在 2017 年首批设立

的三个平台类项目之一。

数学天元基金是为了推动我国数学尽早实现数学强国目标而在 1990 年设立的专

项基金，以实现老一辈数学家提出的“中国数学要在二十一世纪率先赶上世界先

进水平”的目标。

数学天元基金是源于财政拨款，由国家自然科学基金委员会管理的数学专项基金，

该基金是凝聚数学家集体智慧，探索符合数学特点和发展规律的资助方式，推动

建设数学强国而设立的专项科学基金。数学天元基金项目支持科学技术人员结合

数学学科特点和需求，开展科学研究，培养青年人才，促进学术交流，优化研究

环境，传播数学文化，从而提升中国数学创新能力。

经过近 30 年的发展，在国家自然科学基金委、历届学术领导小组和全国数学工

作者的共同努力下，数学天元基金在学科发展规划、学科方向调整、学科队伍建

设、青年人才培养、研究环境的改善、优秀数学家的培养等方面发挥了重要作用，

为推动我国数学学科迅速发展做出了重要贡献。

国家自然科学基金委数学天元基金领导小组为更好的促进区域数学学科平衡发

展，于 2017 年设立天元数学中心项目，该项目以构建交流平台促进合作和研究

为主旨，针对若干数学及其交叉领域或专题，通过多种形式的学术交流研讨活动，

凝聚相关研究队伍，聚焦科学问题，深化国内外多种领域专家间合作，培养青年

学术骨干，引导年轻人进入学科前沿，促进数学与其他学科、数学各分支间的交

叉融合，提升我国相关领域或专题的整体研究水平，形成优势研究方向，推动数

学学科发展。

首批天元中心包括西北、西南、东北三个中心，其中东北中心主要围绕计算数学、

大分析和统计学展开活动。国家天元数学东北中心以 17 位国内外有重要影响的

专家学者组成的学术委员会为核心，由吉林大学协同东北师范大学、大连理工大

学、哈尔滨工业大学等 23 所东北共建院校数学学科负责人组成的执行委员会共

建。天元数学东北中心既鼓励自由探索，也主动面向国家重大战略需求，通过多

种形式的学术交流研讨，创造良好的学术交流环境，加强国内外多领域科学家之

间的紧密合作，促进数学与其它学科、数学各分支间的交叉融合，培植新兴学科

增长点，打造在国际上有重要影响的学科方向，力争在相关研究领域取得重大突

破，切实提升我国数学研究的整体地位。同时，培养一批复合型、高素质的数学

59

后备人才和善于解决实际问题的交叉型人才，力争对中国数学，特别是对东北地

区数学整体水平的提升起到重要推动作用。

Documents

The 10th Annual Meeting on Inverse Problems