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 Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 2 / 28
Supervision Team Prof. Dr. Elisabeth Ullmann Email: elisabeth.ullmann@ma.tum.de M. Sc. Jonas Latz Email: jonas.latz@tum.de M. Sc. Fabian Wagner Email: fabian.wagner@ma.tum.de Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 3 / 28
Seminar setup Each participant prepares a 60 min presentation (projector or blackboard, we recommend projector) followed by 30 min discussion and feedback One consultation meeting with your supervisor at least 2 weeks before the presentation is required (more meetings possible upon request; recommended for Master s students) Attendance of every session and active participation in the discussion is expected Before the presentation: each participant submits executable computer code (in a suitable language, e.g. MATLAB) and a handout (2 4 pages) summarising the basic ideas and experiments 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 TUM eaccess 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
Motivation Dynamical systems In the lecture Numerical Methods for ODEs we considered a discrete-in-time dynamical system on X := R n, v t = Φ(v t 1 ), t N for some evolution map Φ : X X and some initial value v 0 X. In this seminar, we consider such a dynamical system under uncertainties. Uncertainties are modelled using randomness. Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 8 / 28
Motivation Adding uncertainties (A) Uncertain initial value and deterministic dynamics v t = Φ(v t 1 ), v 0 N(m 0, C 0 ) t N corresponds to a discretised ODE with uncertain initial value Example: periodic motion of a pendulum with uncertain initial position states (v t ) t N are now uncertain as well Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 9 / 28
Motivation Adding uncertainties (B) Uncertain initial value and stochastic dynamics v t = Φ(v t 1 ) + ξ t, ξ t N(0, C t ), t N v 0 N(m 0, C 0 ) corresponds to a discretised stochastic differential equation (SDE) with uncertain initial value Example: motion of a pendulum with uncertain initial position and uncertain time-dependent friction states (v t ) t N are now uncertain as well Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 10 / 28
Motivation Adding observations Assumption: true underlying trajectory (vt true ) t N0 Observation: we observe the true trajectory in terms of a noisy signal (Y t ) t N : Y t := H(v true t ) + η t, η t N(0, Γ), t N Data assimilation: Identify the true trajectory (vt true ) t N0 based on (Y t ) t N Forecast future states with current data Correct past states with current data Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 11 / 28
Motivation: Weather forecasting Triangulation of the globe, actually much finer and more irregular ( c Deutscher Wetterdienst) True trajectory (vt true ( )) t N : weather averaged over 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 (system of 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 forecasting Challenges: X is very high-dimensional hundreds of millions of spatial grid points high memory requirement Φ requires a supercomputer cannot be solved on a regular fine grid ( 2km grid size) one solve takes 8 minutes Y t is high-dimensional, sparse high memory requirement not always informative Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 15 / 28
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 Kalman Smoother 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] Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 20 / 28
(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] Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 25 / 28
Supervision Supervisor Ullmann Ullmann Latz Wagner Wagner Ullmann Wagner Latz Latz Topic B1 B2 B3 B4 B5 B6 B7 M1 M2 Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 26 / 28
Tentative schedule Date Topic B1, B2 B3, B4 B5, B6 B7, M1 M2 Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 27 / 28
References [DJ11] A. Doucet, A. Johansen: A tutorial on particle filtering and smoothing: fifteen years 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. SIAM Review, 54, pp. 801 823, 2012 [LSZ15] K. Law, A. M. Stuart, K. Zygalakis: Data Assimilation. A Mathematical Introduction. Springer-Verlag, 2015. [MCB11] J. Mandel, L. Cobb, J. Beezley: On the convergence of the Ensemble Kalman filter. Applications of Mathematics, 6, pp. 533 541, 2011. [SS17] C. Schillings, A.M. Stuart: Analysis of the Ensemble Kalman Filter for Inverse Problems. SIAM J. Numer. Anal., 55, pp. 1264 1290, 2017. Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 28 / 28