Low-rank Properties, Tree Structure, and Recursive Algorithms with Applications

Jingfang Huang

Department of Mathematics

UNC at Chapel Hill

Fundamentals of Fast Multipole (type) Method

Fundamentals: • Low rank? Low dimensional? Simple rules? …

• Process the “low rank/ low dimensional” or

“simple” data on the tree structure? • Recursive Implementations and parallelization

Example: (real parallel code) RECFMM

Joint work with Bo Zhang,

Xiaobai Sun, and Nikos Pitsianis

Open

source

code

available.

Cilk Plus

runtime

used.

Parallelizing Recursive Algorithms: Cilk Plus Runtime Scheduler

Existing results on parallelizing hierarchy tree structure based algorithms. E.g., Cilk plus runtime scheduler. (Joint work with B. Zhang, X. Sun, N. Pitsianis.) https://www.cilkplus.org/

Computational Software section in CiCP

1. Computational Software: Simple FMM Libraries for Electrostatics, Slow

Viscous Flow, and Frequency-Domain Wave Propagation, by Zydrunas Gimbutas and Leslie Greengard

2. Computational Software: PVFMM: A Parallel Kernel Independent FMM for

Particle and Volume Potentials, by

Dhairya Malhotra and George Biros Associate Editors: Prof. Benzhuo Lu (Chinese Academy of Sciences) and J. Huang.

L2 Low rank: Mathematical Analysis vs Numerical Approach

Matrix Analysis: PCA or SVD

A simple Matlab code

• clear all; clc; • m=400;n=400;x=rand(n); y=rand(m)+2; • for i=1:m, • for j=1:n, • A(i,j)=1/(y(i)-x(j)); • end • end • [U,S,V]=svd(A);

• format long e; • pca=diag(S); • pca(1:15)

And the results (numerical rank=9)

• 2.113086441965111e+02 • 5.114102631925020e+00 • 9.392747101949028e-02 • 1.790209461350161e-03 • 3.235709204090181e-05 • 6.168512015716861e-07 • 1.151573821662682e-08 • 1.982373081735641e-10 • 3.570020729013249e-12 • 1.596504915597923e-13 • 1.377179100162433e-13 • 1.206114412870406e-13

Taylor expansion:

Mathematical analysis: Low rank?

Note that

P=15 for single precision and p=30 for double precision.

Multipole coefficients: compressed data

Low Rank and Low Dimensional Representations

Separation of Variables: Low rank and low dimensional data

Question: Generalization?

Low Rank? Tree? Recursive Algorithm

Apply the fundamentals to model and algorithm design.

Today’s topic 1:

Recursive Tree Algorithms for Orthogonal Matrix

Generation and Matrix-Vector Multiplications in Rigid Body Dynamics Simulations

Background: Rigid body dynamics

Consider a molecular system modeled by rigid bodies, each rigid body consists of n “beads” Given the force on and location of each bead The resultant force (size 3) and torque (size 3)of the rigid body are given by

Matrix Language

Orthogonal Form

Problem Statement

Background: Brownian Dynamics with Hydrodynamic Interactions

The “bead model” 1. Molecular system is modeled by m rigid bodies, each with

given external forces and torques , j=1,…,m. 2. Each rigid body contains nj beads.

3. The bead-to-bead hydrodynamics interactions satisfy

where D is the Rotne-Prager-Yamakawa tensor (translation invariant Green’s function, particle method for the Stokes equation) Reference: JA McCammon and G Huber (UCSD)

Bead model in Brownian dynamics

For the given force field on each rigid body, the unknown rigid body velocity vectors satisfy: Or equivalently, Note: Need to solve this at each time marching step…

Numerical Difficulty

Numerical difficulty: calculation of Reminder: Question (1) (Alternative approaches: Schur complement)

Reformulation

Next Generation Brownian Dynamics Solver

We already have 1. Fast multipole method for evaluating 2. Fast direct solver based preconditioner for each rigid

body (Codes from Jianlin Xia, et. al. )

Building blocks needed: stable and efficient algorithms for evaluating A nice project for undergrad students (Fuhui Fang, UNC)

So the problems:

Divide-and-conquer: Recursive Algorithms on the Tree Structure

Consider the divide-and-conquer strategy Observations: and

Divide-and-conquer: Recursive Algorithms on the Tree Structure

Algorithm Complexity and Storage

How to go from “Tree code” to FMM?

An upward pass: Algorithm for computing Qv Info-set = multipole coefficients

Multiple->Multipole: Parent P and Children X and Y, then,

Translation Operator: Inertia Matrix

Parent’s inertia matrix and Children’s inertia matrices then,

Residue Vectors

We want to compute without explicit forming

How to compute residue vector? Then,

Computing R from inertia matrix

R is readily available from the inertia matrix

Recursive Algorithm

Downward Pass: Algorithm for QT v

Local Expansions?

Recursive Algorithm

A downward pass

Pseudo-code

Numerical Results

Storage

Implicit Q v CPU time

Details: backward stable algorithm?

How to make sure || QT Q v – v || is small? In progress…

Summary of Topic 1:

Use the fundamentals

Low rank (comes from rigid body assumption) Tree structure Recursive implementation.

Efficient algorithms can be developed.

Topic 2:

Half-space impedance Green’s function for the Helmholtz equation

Layered media Green’s function

“On the efficient representation of the half-space impedance Green's function for the Helmholtz equation”, by Michael O'Neil, Leslie Greengard, Andras Pataki

How to evaluate the convolution?

• Integral equation formulation

where is the Green’s function for the half-space with homogeneous impedance boundary conditions, and the resulting Fredholm 2nd kind integral equation become

Method of images

Using the free space Green’s function

Modified fast multipole method

Multipole-to-multipole: same as free space FMM, as if the images do NOT exist.

Multipole-to-local: modified translation which include all the image contributions.

Local-to-local: No change!

Error analysis: Low rank? Note that source images are always well-separated from the target box. M2L Translation operator: analytical formula is possible. Either precomputed, or by designing special functions.

Generalization to multiple layers?

Can we develop the corresponding approximation theory for general multi-layer Green’s function in the form of Sommerfeld integral representation? Answer: Yes! Approximation using special basis, current research.

Summary

(Maybe it is a good idea to) consider recursively (hierarchically) on the tree structure when designing models and algorithms Mathematical analysis may help us to find the special matrix structures.

Thanks!