Seminar: Data Assimilation
Jonas Latz, Elisabeth Ullmann
Chair of Numerical Mathematics (M2)
Technical University of Munich
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 1 / 28
Prerequisites
Bachelor:
MA1304 Introduction to Numerical Linear Algebra
MA2304 Numerical Methods for ODEs
MA1401 Introduction to Probability Theory
Language: English
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Supervision Team
• Prof. Dr. Elisabeth Ullmann
Email: [email protected]
• M. Sc. Jonas Latz
Email: [email protected]
• M. Sc. Fabian Wagner
Email: [email protected]
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Seminar setup
• Each participant prepares a 60 min presentation (projector orblackboard, we recommend projector) followed by 30 mindiscussion and feedback
• One consultation meeting with your supervisor at least 2 weeksbefore the presentation is required (more meetings possible uponrequest; recommended for Master’s students)
• Attendance of every session and active participation in thediscussion is expected
• Before the presentation: each participant submits executablecomputer code (in a suitable language, e.g. MATLAB) and ahandout (2–4 pages) summarising the basic ideas andexperiments performed
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 4 / 28
Seminar setup• For the most part, this seminar is based on [LSZ15]
• This book and all further literature is available online through TUMeAccess
https://www.ub.tum.de/en/eaccess
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 5 / 28
More information
• Schedule, Material, etc:
http://www-m2.ma.tum.de/bin/view/Allgemeines/DATA
• Tips for preparing and delivering your presentation
• Simple slides for LaTeX
• Equipment for presentation (blackboard, projector, laptop)
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 6 / 28
Motivation
How can we fit data into a dynamical system?
• State estimation (prediction)
• Bayesian statistics
• Smoothing and Filtering
• Efficient algorithms
Combination of Statistics and Dynamical Systems (ODEs)
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 7 / 28
MotivationDynamical systems
• In the lecture Numerical Methods for ODEs we considered adiscrete-in-time dynamical system on X := Rn,
vt = Φ(vt−1), t ∈ N
for some evolution map Φ : X → X and some initial value v0 ∈ X .
• In this seminar, we consider such a dynamical system underuncertainties.
• Uncertainties are modelled using randomness.
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 8 / 28
MotivationAdding uncertainties
(A) Uncertain initial value and deterministic dynamics
vt = Φ(vt−1), t ∈ Nv0 ∼ N(m0,C0)
• corresponds to a discretised ODE with uncertain initial value
• Example: periodic motion of a pendulum with uncertain initialposition
• states (vt )t∈N are now uncertain as well
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 9 / 28
MotivationAdding uncertainties
(B) Uncertain initial value and stochastic dynamics
vt = Φ(vt−1) + ξt , ξt ∼ N(0,Ct ), t ∈ Nv0 ∼ N(m0,C0)
• corresponds to a discretised stochastic differential equation (SDE)with uncertain initial value
• Example: motion of a pendulum with uncertain initial position anduncertain time-dependent friction
• states (vt )t∈N are now uncertain as well
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 10 / 28
MotivationAdding observations
• Assumption: true underlying trajectory (v truet )t∈N0
• Observation: we observe the true trajectory in terms of a noisysignal (Yt )t∈N:
Yt := H(v truet ) + ηt , ηt ∼ N(0, Γ), t ∈ N
• Data assimilation: Identify the true trajectory (v truet )t∈N0 based on
(Yt )t∈N
I Forecast future states with current data
I Correct past states with current data
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 11 / 28
Motivation: Weather forecasting
Triangulation of the globe, actu-ally much finer and more irregu-lar ( c© Deutscher Wetterdienst)
• True trajectory (v truet (4))t∈N: weather averaged over 4 ∈ globe
• Weather: temperature, pressure, clouds, water vapour,. . .
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 12 / 28
Motivation: Weather forecasting
ICON scheme ( c© Deutscher Wetterdienst)
Evolution map Φ: ICON (Icosahedral Nonhydrostatic) Model (systemof discretised partial differential equations)
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 13 / 28
Motivation: Weather forecasting
Wind speed signal ( c© Deutscher Wetterdienst)
Data Y t : Wind speed & temperature at several positions on the globe,satellite images, precipitation radar, . . .
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 14 / 28
Motivation: Weather forecastingChallenges:
• X is very high-dimensional
I hundreds of millions of spatial grid points
I high memory requirement
• Φ requires a supercomputer
I cannot be solved on a regular fine grid (∼ 2km grid size)
I one solve takes 8 minutes
• Yt is high-dimensional, sparse
I high memory requirement
I not always informative
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Motivation: Weather forecasting
Currently used data assimilation method by DWD:
Ensemble method with 20 particles
More information in German and English:
https://www.dwd.de
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 16 / 28
(B1) Dynamical systems
Content:
• Background on probability, Bayes’ formula• Dynamical systems (stochastic, deterministic)• Guiding examples: linear and nonlinear dynamics• Lorenz-63 system
Programming:
• ODE solvers for the Lorenz-63 system
Literature: 1.1, 1.2, 2.2 in [LSZ15]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 17 / 28
(B2) The Smoothing Problem and the KalmanSmoother
Content:
• Data Assimilation – Setup• Smoothing problem (stochastic, deterministic dynamics)• Linear Gaussian problems• Kalman Smoother
Programming:
• Kalman Smoother for linear Gaussian smoothing problem
Literature: 2.1, 2.3, 2.8, 3.1 in [LSZ15]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 18 / 28
(B3) Nonlinear Smoothing with MCMC
Content:
• Markov Chain Monte Carlo (MCMC) methodology• Metropolis–Hastings MCMC• Random Walk Metropolis• Optional: Independence Sampler, pCN Sampler
Programming:
• MCMC for nonlinear smoothing problem
Literature: 3.2, 3.4 in [LSZ15]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 19 / 28
(B4) The Filtering Problem and the Kalman Filter
Content:
• Filtering problem• Relation of filtering and smoothing• Kalman Filter for linear Gaussian problems• Large-time behavior of the Kalman Filter
Programming:
• Kalman filter for linear Gaussian filtering problem
Literature: 2.4, 2.5, 4.1, 4.4.1, 4.5 in [LSZ15]
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(B5) Approximate Kalman Filters
Content:
• Approximate Gaussian Filters (Extended Kalman Filter)• Ensemble Kalman Filter (EnKF)• Ensemble Square-Root Kalman Filter• Convergence of the EnKF in the large ensemble limit
Programming:
• EnKF for linear Gaussian filtering problem
Literature: 4.2, 4.5 in [LSZ15] and [MCB11]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 21 / 28
(B6) A Fresh Look at the Kalman Filter
Content:
• State estimation• Two-step Kalman filter (based on Newton’s method)• Extended Kalman filter (based on Newton’s method)• Variations: Smoothing, fading memory
Programming:
• Exercises 1–5 in [HRW12]
Literature: [HRW12]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 22 / 28
(B7) Nonlinear Filtering with Particle Filters
Content:
• Basic idea of particle filters• Sequential Importance Resampling (SIR)• Bootstrap Filter• Improved proposals
Programming:
• SIR for nonlinear filtering problem
Literature: 4.3 in [LSZ15]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 23 / 28
(M1) Analysis of the EnKF for Inverse Problems
Content:
• EnKF for inverse problems• Continuous time limit• Asymptotic behavior in the linear setting• Variants of the EnKF
Programming:
• Source identification with an elliptic PDE and the EnKF
Literature: [SS17]
Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 24 / 28
(M2) Particle Filters for Option Pricing
Content:
• Hidden Markov models• Sequential Monte Carlo methods• Particle Filtering• Application to Option Pricing
Programming:
• Example 4 in [DJ11] with different particle filters (SIS, SMC, EnKF)
Literature: [DJ11]
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Supervision
Supervisor TopicUllmann B1Ullmann B2Latz B3Wagner B4Wagner B5Ullmann B6Wagner B7Latz M1Latz M2
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Tentative schedule
Date TopicB1, B2B3, B4B5, B6B7, M1M2
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References
[DJ11] A. Doucet, A. Johansen: A tutorial on particle filtering and smoothing: fifteenyears later. The Oxford handbook of nonlinear filtering, pp. 656–704, Oxford Univ.Press, Oxford, 2011.
[HRW12] J. Humpherys, P. Redd, J. West: A Fresh Look at the Kalman Filter. SIAMReview, 54, pp. 801–823, 2012
[LSZ15] K. Law, A. M. Stuart, K. Zygalakis: Data Assimilation. A MathematicalIntroduction. Springer-Verlag, 2015.
[MCB11] J. Mandel, L. Cobb, J. Beezley: On the convergence of the EnsembleKalman filter. Applications of Mathematics, 6, pp. 533–541, 2011.
[SS17] C. Schillings, A.M. Stuart: Analysis of the Ensemble Kalman Filter for InverseProblems. SIAM J. Numer. Anal., 55, pp. 1264–1290, 2017.
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