seminar: data assimilation · ma1304 introduction to numerical linear algebra ma2304 numerical...

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


Supervision Team

• Prof. Dr. Elisabeth Ullmann

Email: [email protected]

• M. Sc. Jonas Latz


• M. Sc. Fabian Wagner



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


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


https://www.ub.tum.de/en/eaccess

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)


http://www-m2.ma.tum.de/bin/view/Allgemeines/DATA

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)


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.


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


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


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


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,. . .



ICON scheme ( c© Deutscher Wetterdienst)

Evolution map Φ: ICON (Icosahedral Nonhydrostatic) Model (systemof discretised partial differential equations)



Wind speed signal ( c© Deutscher Wetterdienst)

Data Y t : Wind speed & temperature at several positions on the globe,satellite images, precipitation radar, . . .


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



Currently used data assimilation method by DWD:

Ensemble method with 20 particles

More information in German and English:

https://www.dwd.de


https://www.dwd.de

(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]


(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]


(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]


(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]


(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]


(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]


(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]


(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]


(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]


Supervision

Supervisor TopicUllmann B1Ullmann B2Latz B3Wagner B4Wagner B5Ullmann B6Wagner B7Latz M1Latz M2


Tentative schedule

Date TopicB1, B2B3, B4B5, B6B7, M1M2


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.


seminar: data assimilation · ma1304 introduction to numerical linear algebra ma2304 numerical...

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