EnKF and Catastrophic filter divergence
|
|
|
- Tracy Lindsey
- 5 years ago
- Views:
Transcription
1 EnKF and Catastrophic filter divergence David Kelly Kody Law Andrew Stuart Mathematics Department University of North Carolina Chapel Hill NC bkelly.com June 4, 2014 SIAM UQ 2014 Savannah. David Kelly (UNC Catastrophic EnKF June 4, / 19
2 Talk outline 1. What is EnKF? 2. What is known about EnKF? 3. How can we use stochastic analysis to better understand EnKF? David Kelly (UNC Catastrophic EnKF June 4, / 19
3 The filtering problem We have a deterministic model dv = F (v with v 0 N(m 0, C 0. We will denote v(t = Ψ t (v 0. Think of this as very high dimensional and nonlinear. We want to estimate v = v(h for some h > 0 and = 0, 1,..., J given the observations y = Hv + ξ for ξ iid N(0, Γ. David Kelly (UNC Catastrophic EnKF June 4, / 19
4 We can use EnKF to draw approximate samples from the posterior. David Kelly (UNC Catastrophic EnKF June 4, / 19
5 The set-up for EnKF Suppose we are given the ensemble {u (1,..., u (K } at time. For each ensemble member, we create an artificial observation y (k +1 = y +1 + ξ (k +1, ξ (k +1 iid N(0, Γ. We update each member using the Kalman update ( u (k +1 = Ψ h(u (k + G(u y (k +1 HΨ h(u (k, where G(u is the Kalman gain computed using the forecasted ensemble covariance Ĉ +1 = 1 K K (Ψ h (u (k Ψ h (u T (Ψ h (u (k Ψ h (u. k=1 David Kelly (UNC Catastrophic EnKF June 4, / 19
6 There aren t many theorems about EnKF. Ideally, we would like a theorem about long time behaviour of the filter for a finite ensemble size. David Kelly (UNC Catastrophic EnKF June 4, / 19
7 Filter divergence In certain situations, it has been observed ( that the ensemble can blow-up (ie. reach machine-infinity in finite time, even when the model has nice bounded solutions. This is known as catastrophic filter divergence. We would like to investigate whether this has a dynamical ustification or if it is simply a numerical artefact. Harlim, Mada (2010, Gottwald (2011, Gottwald, Mada (2013. David Kelly (UNC Catastrophic EnKF June 4, / 19
8 Assumptions on the dynamics We make a dissipativity assumption on the model. Namely that dv + Av + B(v, v = f with A linear elliptic and B bilinear, satisfying certain estimates and symmetries. This guarantees uniformly bounded solutions. Eg. 2d-Navier-Stokes, Lorenz-63, Lorenz-96. David Kelly (UNC Catastrophic EnKF June 4, / 19
9 Discrete time results Let e (k = u (k v. For a fixed observation frequency h > 0 Theorem (AS,DK,KL If H = Id and Γ = γ 2 Id then there exists constant β > 0 such that ( E e (k 2 e 2βh E e (k e Kγ 2 2βh 1 e 2βh 1 So no finite time blow-up. David Kelly (UNC Catastrophic EnKF June 4, / 19
10 Discrete time results with variance inflation Suppose we replace Ĉ +1 α 2 I + Ĉ+1 at each update step. This is known as additive variance inflation. Theorem (AS,DK,KL If H = Id and Γ = γ 2 Id then there exists constant β > 0 such that ( E e (k 2 θ E e (k 1 θ Kγ 2 1 θ where θ = γ2 e 2βh α 2 +γ 2. In particular, if we pick α large enough (so that θ < 1 then lim E e(k 2 2Kγ2 1 θ David Kelly (UNC Catastrophic EnKF June 4, / 19
11 For observations with h 1, we need another approach. David Kelly (UNC Catastrophic EnKF June 4, / 19
12 The EnKF equations look like a discretization Recall the ensemble update equation ( u (k +1 = Ψ h(u (k + G(u y (k +1 HΨ h(u (k ( = Ψ h (u (k + Ĉ+1H T (H T Ĉ +1 H + Γ 1 y (k +1 HΨ h(u (k Subtract u (k from both sides and divide by h u (k +1 u(k h = Ψ h(u (k u (k h ( + Ĉ+1H T (hh T Ĉ +1 H + hγ 1 y (k +1 HΨ h(u (k Clearly we need to rescale the noise (ie. Γ. David Kelly (UNC Catastrophic EnKF June 4, / 19
13 Continuous-time limit If we set Γ = h 1 Γ 0 and substitute y (k +1, we obtain u (k +1 u(k h But we know that and Ĉ +1 = 1 K = 1 K = Ψ h(u (k u (k h + Ĉ+1H T (hh T Ĉ +1 H + Γ 0 1 ( Hv + h 1/2 Γ 1/2 0 ξ +1 + h 1/2 Γ 1/2 0 ξ (k +1 HΨ h(u (k Ψ h (u (k = u (k + O(h K (Ψ h (u (k Ψ h (u T (Ψ h (u (k Ψ h (u k=1 K (u (k k=1 u T (u (k u + O(h = C(u + O(h David Kelly (UNC Catastrophic EnKF June 4, / 19
14 Continuous-time limit We end up with u (k +1 u(k h = Ψ h(u (k u (k h + C(u H T Γ 1 0 C(u H T Γ 1 ( h 1/2 ξ +1 + h 1/2 ξ (k +1 This looks like a numerical scheme for Itô S(PDE 0 H(u(k v + O(h du (k = F (u (k C(uH T Γ 1 0 H(u(k v ( ( dw (k + C(uH T Γ 1/2 0 + db. David Kelly (UNC Catastrophic EnKF June 4, / 19
15 Nudging du (k = F (u (k C(uH T Γ 1 0 H(u(k v ( ( dw (k + C(uH T Γ 1/2 0 + db. 1 - Extra dissipation term only sees differences in observed space 2 - Extra dissipation only occurs in the space spanned by ensemble David Kelly (UNC Catastrophic EnKF June 4, / 19
16 Kalman-Bucy limit If F were linear and we write m(t = 1 K K k=1 u(k (t then dm = F (m C(uHT Γ 1 0 H(m v + C(uH T Γ 1/2 db 0 + O(K 1/2. This is the equation for the Kalman-Bucy filter, with empirical covariance C(u. The remainder O(K 1/2 can be thought of as a sampling error. David Kelly (UNC Catastrophic EnKF June 4, / 19
17 Continuous-time results Theorem (AS,DK,KL Suppose that {u (k } K k=1 satisfy ( with H = Id and Γ = γ2 Id. Let e (k = u (k v. Then there exists constant β > 0 such that 1 K K E e (k (t 2 k=1 ( 1 K K E e (k (0 2 exp (βt. k=1 David Kelly (UNC Catastrophic EnKF June 4, / 19
18 Why do we need H = Id and Γ = γ 2 Id? In the equation du (k = F (u (k C(uH T Γ 1 0 H(u(k v ( dw (k + C(uH T Γ 1/2 0 + db. The energy pumped in by the noise must be balanced by contraction of (u (k v. So the operator must be positive-definite. C(uH T Γ 1 0 H Both C(u and H T Γ 1 0 H are pos-def, but this doesn t guarantee the same for the product! David Kelly (UNC Catastrophic EnKF June 4, / 19
19 Summary + Future Work (1 Writing down an SDE/SPDE allows us to see the important quantities in the algorithm. (2 Does not prove that catastrophic filter divergence is a numerical phenomenon, but is a decent starting point. (1 Improve the condition on H. (2 If we can measure the important quantities, then we can test the performance during the algorithm. Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time. D. Kelly, K.Law, A. Stuart. arxiv David Kelly (UNC Catastrophic EnKF June 4, / 19
EnKF and filter divergence
EnKF and filter divergence David Kelly Andrew Stuart Kody Law Courant Institute New York University New York, NY dtbkelly.com December 12, 2014 Applied and computational mathematics seminar, NIST. David
What do we know about EnKF?
What do we know about EnKF? David Kelly Kody Law Andrew Stuart Andrew Majda Xin Tong Courant Institute New York University New York, NY April 10, 2015 CAOS seminar, Courant. David Kelly (NYU) EnKF April
Data assimilation in high dimensions
Data assimilation in high dimensions David Kelly Kody Law Andy Majda Andrew Stuart Xin Tong Courant Institute New York University New York NY www.dtbkelly.com February 3, 2016 DPMMS, University of Cambridge
Data assimilation in high dimensions
Data assimilation in high dimensions David Kelly Courant Institute New York University New York NY www.dtbkelly.com February 12, 2015 Graduate seminar, CIMS David Kelly (CIMS) Data assimilation February
Ergodicity in data assimilation methods
Ergodicity in data assimilation methods David Kelly Andy Majda Xin Tong Courant Institute New York University New York NY www.dtbkelly.com April 15, 2016 ETH Zurich David Kelly (CIMS) Data assimilation
Homogenization for chaotic dynamical systems
Homogenization for chaotic dynamical systems David Kelly Ian Melbourne Department of Mathematics / Renci UNC Chapel Hill Mathematics Institute University of Warwick November 3, 2013 Duke/UNC Probability
Filtering the Navier-Stokes Equation
Filtering the Navier-Stokes Equation Andrew M Stuart1 1 Mathematics Institute and Centre for Scientific Computing University of Warwick Geometric Methods Brown, November 4th 11 Collaboration with C. Brett,
Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications
Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications Dongbin Xiu Department of Mathematics, Purdue University Support: AFOSR FA955-8-1-353 (Computational Math) SF CAREER DMS-64535
Performance of ensemble Kalman filters with small ensembles
Performance of ensemble Kalman filters with small ensembles Xin T Tong Joint work with Andrew J. Majda National University of Singapore Sunday 28 th May, 2017 X.Tong EnKF performance 1 / 31 Content Ensemble
A data-driven method for improving the correlation estimation in serial ensemble Kalman filter
A data-driven method for improving the correlation estimation in serial ensemble Kalman filter Michèle De La Chevrotière, 1 John Harlim 2 1 Department of Mathematics, Penn State University, 2 Department
Gaussian Filtering Strategies for Nonlinear Systems
Gaussian Filtering Strategies for Nonlinear Systems Canonical Nonlinear Filtering Problem ~u m+1 = ~ f (~u m )+~ m+1 ~v m+1 = ~g(~u m+1 )+~ o m+1 I ~ f and ~g are nonlinear & deterministic I Noise/Errors
The Kalman Filter. Data Assimilation & Inverse Problems from Weather Forecasting to Neuroscience. Sarah Dance
The Kalman Filter Data Assimilation & Inverse Problems from Weather Forecasting to Neuroscience Sarah Dance School of Mathematical and Physical Sciences, University of Reading s.l.dance@reading.ac.uk July
Adaptive ensemble Kalman filtering of nonlinear systems
Adaptive ensemble Kalman filtering of nonlinear systems Tyrus Berry George Mason University June 12, 213 : Problem Setup We consider a system of the form: x k+1 = f (x k ) + ω k+1 ω N (, Q) y k+1 = h(x
Filtering Sparse Regular Observed Linear and Nonlinear Turbulent System
Filtering Sparse Regular Observed Linear and Nonlinear Turbulent System John Harlim, Andrew J. Majda Department of Mathematics and Center for Atmosphere Ocean Science Courant Institute of Mathematical
A variance limiting Kalman filter for data assimilation: I. Sparse observational grids II. Model error
A variance limiting Kalman filter for data assimilation: I. Sparse observational grids II. Model error Georg Gottwald, Lewis Mitchell, Sebastian Reich* University of Sydney, *Universität Potsdam Durham,
Addressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF
Addressing the nonlinear problem of low order clustering in deterministic filters by using mean-preserving non-symmetric solutions of the ETKF Javier Amezcua, Dr. Kayo Ide, Dr. Eugenia Kalnay 1 Outline
Ensemble Kalman Filter
Ensemble Kalman Filter Geir Evensen and Laurent Bertino Hydro Research Centre, Bergen, Norway, Nansen Environmental and Remote Sensing Center, Bergen, Norway The Ensemble Kalman Filter (EnKF) Represents
Data assimilation with and without a model
Data assimilation with and without a model Tim Sauer George Mason University Parameter estimation and UQ U. Pittsburgh Mar. 5, 2017 Partially supported by NSF Most of this work is due to: Tyrus Berry,
Data assimilation as an optimal control problem and applications to UQ
Data assimilation as an optimal control problem and applications to UQ Walter Acevedo, Angwenyi David, Jana de Wiljes & Sebastian Reich Universität Potsdam/ University of Reading IPAM, November 13th 2017
State and Parameter Estimation in Stochastic Dynamical Models
State and Parameter Estimation in Stochastic Dynamical Models Timothy DelSole George Mason University, Fairfax, Va and Center for Ocean-Land-Atmosphere Studies, Calverton, MD June 21, 2011 1 1 collaboration
EnKF-based particle filters
EnKF-based particle filters Jana de Wiljes, Sebastian Reich, Wilhelm Stannat, Walter Acevedo June 20, 2017 Filtering Problem Signal dx t = f (X t )dt + 2CdW t Observations dy t = h(x t )dt + R 1/2 dv t.
Stochastic Modelling in Climate Science
Stochastic Modelling in Climate Science David Kelly Mathematics Department UNC Chapel Hill dtbkelly@gmail.com November 16, 2013 David Kelly (UNC) Stochastic Climate November 16, 2013 1 / 36 Why use stochastic
On the convergence of (ensemble) Kalman filters and smoothers onto the unstable subspace
On the convergence of (ensemble) Kalman filters and smoothers onto the unstable subspace Marc Bocquet CEREA, joint lab École des Ponts ParisTech and EdF R&D, Université Paris-Est, France Institut Pierre-Simon
Ensemble Data Assimilation and Uncertainty Quantification
Ensemble Data Assimilation and Uncertainty Quantification Jeff Anderson National Center for Atmospheric Research pg 1 What is Data Assimilation? Observations combined with a Model forecast + to produce
Sequential Monte Carlo Samplers for Applications in High Dimensions
Sequential Monte Carlo Samplers for Applications in High Dimensions Alexandros Beskos National University of Singapore KAUST, 26th February 2014 Joint work with: Dan Crisan, Ajay Jasra, Nik Kantas, Alex
Superparameterization and Dynamic Stochastic Superresolution (DSS) for Filtering Sparse Geophysical Flows
Superparameterization and Dynamic Stochastic Superresolution (DSS) for Filtering Sparse Geophysical Flows June 2013 Outline 1 Filtering Filtering: obtaining the best statistical estimation of a nature
Organization. I MCMC discussion. I project talks. I Lecture.
Organization I MCMC discussion I project talks. I Lecture. Content I Uncertainty Propagation Overview I Forward-Backward with an Ensemble I Model Reduction (Intro) Uncertainty Propagation in Causal Systems
A Note on the Particle Filter with Posterior Gaussian Resampling
Tellus (6), 8A, 46 46 Copyright C Blackwell Munksgaard, 6 Printed in Singapore. All rights reserved TELLUS A Note on the Particle Filter with Posterior Gaussian Resampling By X. XIONG 1,I.M.NAVON 1,2 and
arxiv: v2 [math.ds] 5 Apr 2013
![arxiv: v2 [math.ds] 5 Apr 2013](/thumbs/92/109705949.jpg "arxiv: v2 [math.ds] 5 Apr 2013")
Analysis of the 3DVAR Filter for the Partially Observed Lorenz 63 Model K. J. H. Law, A. Shukla and A. M. Stuart Warwick Mathematics Institute University of Warwick CV4 7AL, UK e-mail: k.j.h.law@warwick.ac.uk;
Convergence of Square Root Ensemble Kalman Filters in the Large Ensemble Limit
Convergence of Square Root Ensemble Kalman Filters in the Large Ensemble Limit Evan Kwiatkowski, Jan Mandel University of Colorado Denver December 11, 2014 OUTLINE 2 Data Assimilation Bayesian Estimation
Fast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 12, 215 Averaging and homogenization workshop, Luminy. Fast-slow systems
Bayesian Statistics and Data Assimilation. Jonathan Stroud. Department of Statistics The George Washington University
Bayesian Statistics and Data Assimilation Jonathan Stroud Department of Statistics The George Washington University 1 Outline Motivation Bayesian Statistics Parameter Estimation in Data Assimilation Combined
Bayesian Inverse problem, Data assimilation and Localization
Bayesian Inverse problem, Data assimilation and Localization Xin T Tong National University of Singapore ICIP, Singapore 2018 X.Tong Localization 1 / 37 Content What is Bayesian inverse problem? What is
Correcting biased observation model error in data assimilation
Correcting biased observation model error in data assimilation Tyrus Berry Dept. of Mathematical Sciences, GMU PSU-UMD DA Workshop June 27, 217 Joint work with John Harlim, PSU BIAS IN OBSERVATION MODELS
Rough Burgers-like equations with multiplicative noise
Rough Burgers-like equations with multiplicative noise Martin Hairer Hendrik Weber Mathematics Institute University of Warwick Bielefeld, 3.11.21 Burgers-like equation Aim: Existence/Uniqueness for du
Notes on Noise Contrastive Estimation (NCE)
Notes on Noise Contrastive Estimation NCE) David Meyer dmm@{-4-5.net,uoregon.edu,...} March 0, 207 Introduction In this note we follow the notation used in [2]. Suppose X x, x 2,, x Td ) is a sample of
CIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions
CIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions December 14, 2016 Questions Throughout the following questions we will assume that x t is the state vector at time t, z t is the
Sensitivity to Model Parameters
Sensitivity to Model Parameters C. David Levermore Department of Mathematics and Institute for Physical Science and Technology University of Maryland, College Park lvrmr@math.umd.edu Math 420: Mathematical
Lecture 16: State Space Model and Kalman Filter Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 6: State Space Model and Kalman Filter Bus 490, Time Series Analysis, Mr R Tsay A state space model consists of two equations: S t+ F S t + Ge t+, () Z t HS t + ɛ t (2) where S t is a state vector
Numerical Solutions of ODEs by Gaussian (Kalman) Filtering
Numerical Solutions of ODEs by Gaussian (Kalman) Filtering Hans Kersting joint work with Michael Schober, Philipp Hennig, Tim Sullivan and Han C. Lie SIAM CSE, Atlanta March 1, 2017 Emmy Noether Group
Lagrangian Data Assimilation and Manifold Detection for a Point-Vortex Model. David Darmon, AMSC Kayo Ide, AOSC, IPST, CSCAMM, ESSIC
Lagrangian Data Assimilation and Manifold Detection for a Point-Vortex Model David Darmon, AMSC Kayo Ide, AOSC, IPST, CSCAMM, ESSIC Background Data Assimilation Iterative process Forecast Analysis Background
Gaussian Process Approximations of Stochastic Differential Equations
Gaussian Process Approximations of Stochastic Differential Equations Cédric Archambeau Dan Cawford Manfred Opper John Shawe-Taylor May, 2006 1 Introduction Some of the most complex models routinely run
Data assimilation with and without a model
Data assimilation with and without a model Tyrus Berry George Mason University NJIT Feb. 28, 2017 Postdoc supported by NSF This work is in collaboration with: Tim Sauer, GMU Franz Hamilton, Postdoc, NCSU
A mechanism for catastrophic filter divergence in data assimilation for sparse observation networks
Manuscript prepared for Nonlin. Processes Geophys. with version 5. of the L A TEX class copernicus.cls. Date: 5 August 23 A mechanism for catastrophic filter divergence in data assimilation for sparse
Fast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
Lagrangian Data Assimilation and Its Application to Geophysical Fluid Flows
Lagrangian Data Assimilation and Its Application to Geophysical Fluid Flows Laura Slivinski June, 3 Laura Slivinski (Brown University) Lagrangian Data Assimilation June, 3 / 3 Data Assimilation Setup:
Data assimilation Schrödinger s perspective
Data assimilation Schrödinger s perspective Sebastian Reich (www.sfb1294.de) Universität Potsdam/ University of Reading IMS NUS, August 3, 218 Universität Potsdam/ University of Reading 1 Core components
GAMM-workshop in UQ, TU Dortmund. Characterization of fluctuations in stochastic homogenization. Mitia Duerinckx, Antoine Gloria, Felix Otto
GAMM-workshop in UQ, TU Dortmund Characterization of fluctuations in stochastic homogenization Mitia Duerinckx, Antoine Gloria, Felix Otto Max Planck Institut für Mathematik in den Naturwissenschaften,
Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods
Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods March 4, 29 M. Hairer 1,2, A. Stuart 1, J. Voss 1,3 1 Mathematics Institute, Warwick University,
A Stochastic Collocation based. for Data Assimilation
A Stochastic Collocation based Kalman Filter (SCKF) for Data Assimilation Lingzao Zeng and Dongxiao Zhang University of Southern California August 11, 2009 Los Angeles Outline Introduction SCKF Algorithm
ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
Systematic strategies for real time filtering of turbulent signals in complex systems
Systematic strategies for real time filtering of turbulent signals in complex systems Statistical inversion theory for Gaussian random variables The Kalman Filter for Vector Systems: Reduced Filters and
Semiparametric modeling: correcting low-dimensional model error in parametric models
Semiparametric modeling: correcting low-dimensional model error in parametric models Tyrus Berry a,, John Harlim a,b a Department of Mathematics, the Pennsylvania State University, 09 McAllister Building,
Dimension-Independent likelihood-informed (DILI) MCMC
Dimension-Independent likelihood-informed (DILI) MCMC Tiangang Cui, Kody Law 2, Youssef Marzouk Massachusetts Institute of Technology 2 Oak Ridge National Laboratory 2 August 25 TC, KL, YM DILI MCMC USC
Machine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 11 Maximum Likelihood as Bayesian Inference Maximum A Posteriori Bayesian Gaussian Estimation Why Maximum Likelihood? So far, assumed max (log) likelihood
Finite-size ensemble Kalman filters (EnKF-N)
Finite-size ensemble Kalman filters (EnKF-N) Marc Bocquet (1,2), Pavel Sakov (3,4) (1) Université Paris-Est, CEREA, joint lab École des Ponts ParisTech and EdF R&D, France (2) INRIA, Paris-Rocquencourt
State Estimation of Linear and Nonlinear Dynamic Systems
State Estimation of Linear and Nonlinear Dynamic Systems Part I: Linear Systems with Gaussian Noise James B. Rawlings and Fernando V. Lima Department of Chemical and Biological Engineering University of
Balanced multivariate model errors of an intermediate coupled model for ensemble Kalman filter data assimilation
Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi:10.1029/2007jc004621, 2008 Balanced multivariate model errors of an intermediate coupled model for ensemble Kalman filter data
LARGE-SCALE TRAFFIC STATE ESTIMATION
Hans van Lint, Yufei Yuan & Friso Scholten A localized deterministic Ensemble Kalman Filter LARGE-SCALE TRAFFIC STATE ESTIMATION CONTENTS Intro: need for large-scale traffic state estimation Some Kalman
The Ensemble Kalman Filter:
p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen 23, Ocean Dynamics, Vol 53, No 4 p.2 The Ensemble
Data Assimilation Research Testbed Tutorial
Data Assimilation Research Testbed Tutorial Section 2: How should observations of a state variable impact an unobserved state variable? Multivariate assimilation. Single observed variable, single unobserved
Methods of Data Assimilation and Comparisons for Lagrangian Data
Methods of Data Assimilation and Comparisons for Lagrangian Data Chris Jones, Warwick and UNC-CH Kayo Ide, UCLA Andrew Stuart, Jochen Voss, Warwick Guillaume Vernieres, UNC-CH Amarjit Budiraja, UNC-CH
Stability of Ensemble Kalman Filters
Stability of Ensemble Kalman Filters Idrissa S. Amour, Zubeda Mussa, Alexander Bibov, Antti Solonen, John Bardsley, Heikki Haario and Tuomo Kauranne Lappeenranta University of Technology University of
Localization and Correlation in Ensemble Kalman Filters
Localization and Correlation in Ensemble Kalman Filters Jeff Anderson NCAR Data Assimilation Research Section The National Center for Atmospheric Research is sponsored by the National Science Foundation.
Application of the Ensemble Kalman Filter to History Matching
Application of the Ensemble Kalman Filter to History Matching Presented at Texas A&M, November 16,2010 Outline Philosophy EnKF for Data Assimilation Field History Match Using EnKF with Covariance Localization
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
A stochastic particle system for the Burgers equation.
A stochastic particle system for the Burgers equation. Alexei Novikov Department of Mathematics Penn State University with Gautam Iyer (Carnegie Mellon) supported by NSF Burgers equation t u t + u x u
STA414/2104 Statistical Methods for Machine Learning II
STA414/2104 Statistical Methods for Machine Learning II Murat A. Erdogdu & David Duvenaud Department of Computer Science Department of Statistical Sciences Lecture 3 Slide credits: Russ Salakhutdinov Announcements
Convergence of the Ensemble Kalman Filter in Hilbert Space
Convergence of the Ensemble Kalman Filter in Hilbert Space Jan Mandel Center for Computational Mathematics Department of Mathematical and Statistical Sciences University of Colorado Denver Parts based
Stochastic contraction BACS Workshop Chamonix, January 14, 2008
Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Q.-C. Pham N. Tabareau J.-J. Slotine Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 1 / 19 Why stochastic contraction?
Algebraic Geometry and Model Selection
Algebraic Geometry and Model Selection American Institute of Mathematics 2011/Dec/12-16 I would like to thank Prof. Russell Steele, Prof. Bernd Sturmfels, and all participants. Thank you very much. Sumio
Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations.
Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations. May 6, 2009 Motivation Constitutive Equations EnKF algorithm Some results Method Navier Stokes equations
A Spectral Approach to Linear Bayesian Updating
A Spectral Approach to Linear Bayesian Updating Oliver Pajonk 1,2, Bojana V. Rosic 1, Alexander Litvinenko 1, and Hermann G. Matthies 1 1 Institute of Scientific Computing, TU Braunschweig, Germany 2 SPT
Sampling the posterior: An approach to non-gaussian data assimilation
Physica D 230 (2007) 50 64 www.elsevier.com/locate/physd Sampling the posterior: An approach to non-gaussian data assimilation A. Apte a, M. Hairer b, A.M. Stuart b,, J. Voss b a Department of Mathematics,
Hierarchical Bayesian Inversion
Hierarchical Bayesian Inversion Andrew M Stuart Computing and Mathematical Sciences, Caltech cw/ S. Agapiou, J. Bardsley and O. Papaspiliopoulos SIAM/ASA JUQ 2(2014), pp. 511--544 cw/ M. Dunlop and M.
EnKF Review. P.L. Houtekamer 7th EnKF workshop Introduction to the EnKF. Challenges. The ultimate global EnKF algorithm
Overview 1 2 3 Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation 6th EnKF Purpose EnKF equations localization After the 6th EnKF (2014), I decided with Prof. Zhang to summarize progress
A new Hierarchical Bayes approach to ensemble-variational data assimilation
A new Hierarchical Bayes approach to ensemble-variational data assimilation Michael Tsyrulnikov and Alexander Rakitko HydroMetCenter of Russia College Park, 20 Oct 2014 Michael Tsyrulnikov and Alexander
DATA ASSIMILATION FOR FLOOD FORECASTING
DATA ASSIMILATION FOR FLOOD FORECASTING Arnold Heemin Delft University of Technology 09/16/14 1 Data assimilation is the incorporation of measurement into a numerical model to improve the model results
Robust Ensemble Filtering With Improved Storm Surge Forecasting
Robust Ensemble Filtering With Improved Storm Surge Forecasting U. Altaf, T. Buttler, X. Luo, C. Dawson, T. Mao, I.Hoteit Meteo France, Toulouse, Nov 13, 2012 Project Ensemble data assimilation for storm
A diamagnetic inequality for semigroup differences
A diamagnetic inequality for semigroup differences (Irvine, November 10 11, 2001) Barry Simon and 100DM 1 The integrated density of states (IDS) Schrödinger operator: H := H(V ) := 1 2 + V ω =: H(0, V
Regularization by noise in infinite dimensions
Regularization by noise in infinite dimensions Franco Flandoli, University of Pisa King s College 2017 Franco Flandoli, University of Pisa () Regularization by noise King s College 2017 1 / 33 Plan of
Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation
Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation arxiv:1507.08319v1 [math.pr] 29 Jul 2015 Xin T Tong, Andrew J Majda and David Kelly July 31, 2015 Abstract The Ensemble
Localization in the ensemble Kalman Filter
Department of Meteorology Localization in the ensemble Kalman Filter Ruth Elizabeth Petrie A dissertation submitted in partial fulfilment of the requirement for the degree of MSc. Atmosphere, Ocean and
Superparameterization and Dynamic Stochastic Superresolution (DSS) for Filtering Sparse Geophysical Flows
SP and DSS for Filtering Sparse Geophysical Flows Superparameterization and Dynamic Stochastic Superresolution (DSS) for Filtering Sparse Geophysical Flows Presented by Nan Chen, Michal Branicki and Chenyue
Stochastic Analogues to Deterministic Optimizers
Stochastic Analogues to Deterministic Optimizers ISMP 2018 Bordeaux, France Vivak Patel Presented by: Mihai Anitescu July 6, 2018 1 Apology I apologize for not being here to give this talk myself. I injured
Revision of TR-09-25: A Hybrid Variational/Ensemble Filter Approach to Data Assimilation
Revision of TR-9-25: A Hybrid Variational/Ensemble ilter Approach to Data Assimilation Adrian Sandu 1 and Haiyan Cheng 1 Computational Science Laboratory Department of Computer Science Virginia Polytechnic
EKF and SLAM. McGill COMP 765 Sept 18 th, 2017
EKF and SLAM McGill COMP 765 Sept 18 th, 2017 Outline News and information Instructions for paper presentations Continue on Kalman filter: EKF and extension to mapping Example of a real mapping system:
A Matrix Theoretic Derivation of the Kalman Filter
A Matrix Theoretic Derivation of the Kalman Filter 4 September 2008 Abstract This paper presents a matrix-theoretic derivation of the Kalman filter that is accessible to students with a strong grounding
Bayesian Calibration of Simulators with Structured Discretization Uncertainty
Bayesian Calibration of Simulators with Structured Discretization Uncertainty Oksana A. Chkrebtii Department of Statistics, The Ohio State University Joint work with Matthew T. Pratola (Statistics, The
Hairer /Gubinelli-Imkeller-Perkowski
Hairer /Gubinelli-Imkeller-Perkowski Φ 4 3 I The 3D dynamic Φ 4 -model driven by space-time white noise Let us study the following real-valued stochastic PDE on (0, ) T 3, where ξ is the space-time white
From Boltzmann Equations to Gas Dynamics: From DiPerna-Lions to Leray
From Boltzmann Equations to Gas Dynamics: From DiPerna-Lions to Leray C. David Levermore Department of Mathematics and Institute for Physical Science and Technology University of Maryland, College Park
On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection-diffusion equations
On the relationship of local projection stabilization to other stabilized methods for one-dimensional advection-diffusion equations Lutz Tobiska Institut für Analysis und Numerik Otto-von-Guericke-Universität
The Ensemble Kalman Filter:
p.1 The Ensemble Kalman Filter: Theoretical formulation and practical implementation Geir Evensen Norsk Hydro Research Centre, Bergen, Norway Based on Evensen, Ocean Dynamics, Vol 5, No p. The Ensemble
Relative Merits of 4D-Var and Ensemble Kalman Filter
Relative Merits of 4D-Var and Ensemble Kalman Filter Andrew Lorenc Met Office, Exeter International summer school on Atmospheric and Oceanic Sciences (ISSAOS) "Atmospheric Data Assimilation". August 29
Implicit sampling for particle filters. Alexandre Chorin, Mathias Morzfeld, Xuemin Tu, Ethan Atkins
0/20 Implicit sampling for particle filters Alexandre Chorin, Mathias Morzfeld, Xuemin Tu, Ethan Atkins University of California at Berkeley 2/20 Example: Try to find people in a boat in the middle of
Sequential Monte Carlo Methods in High Dimensions
Sequential Monte Carlo Methods in High Dimensions Alexandros Beskos Statistical Science, UCL Oxford, 24th September 2012 Joint work with: Dan Crisan, Ajay Jasra, Nik Kantas, Andrew Stuart Imperial College,
Soliton-like Solutions to NLS on Compact Manifolds
Soliton-like Solutions to NLS on Compact Manifolds H. Christianson joint works with J. Marzuola (UNC), and with P. Albin (Jussieu and UIUC), J. Marzuola, and L. Thomann (Nantes) Department of Mathematics
Nonlinear error dynamics for cycled data assimilation methods
Nonlinear error dynamics for cycled data assimilation methods A J F Moodey 1, A S Lawless 1,2, P J van Leeuwen 2, R W E Potthast 1,3 1 Department of Mathematics and Statistics, University of Reading, UK.
ENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 19: Random Processes Jie Liang School of Engineering Science Simon Fraser University 1 Outline Random processes Stationary random processes Autocorrelation of random processes
Stochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.