Filtering Sparse Regular Observed Linear and Nonlinear Turbulent System
|
|
- Randolf Murphy
- 5 years ago
- Views:
Transcription
1 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 Sciences, New York University July 22, 28
2 What is filtering (data assimilation)? A predictor-corrector method that includes observations (via Bayesian update) to improve the real time prediction.
3 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering?
4 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering? 2. Computational efficiency: how big of ensemble is needed in representing the uncertainty of billions of variables?
5 Difficulties in Real Time Filtering and Prediction of Turbulent Signals from Partial Observations 1. For turbulent signals from nature with many scales, even with mesh refinement the model has inaccuracies from parametrization, under-resolution, etc. How to overcome them? Can judicious model error help filtering? 2. Computational efficiency: how big of ensemble is needed in representing the uncertainty of billions of variables? 3. The most accurate ensemble filters is not immune from catastrophic filter divergence (beyond machine infinity).
6 Goal: Provide math guidelines and new numerical strategies thru modern applied math paradigm Modelling Turbulent Signals Filtering Stochastic Langevin Models Complex Nonlinear Dynamical Systems Extended Kalman Filter Classical Criteria: Observability Controllability Numerical Analysis Classical Von-Neumann stability analysis for frozen coef!cient linear systems
7 Filtering Linear Problem Real Space Fourier Space Simplest Tur6ule"t Model Constant Coef!cient Linear Stochastic PDE FT!"depe"de"t Fourier -oe. cie"t Langevin equation Classical Kalman Filter Innovative Strategy Fourier Domain Kalman Filter 8ois9 :6ser;atio"s FT Fourier -oe. cie"ts o. t=e "ois9 o6ser;atio"s
8 How to deal with Sparse Regularly Spaced Observations? ALIASING!! 1.8 m=25, l=3, k=2, N=5 sin(25x) sin(3x) sin(3x j )=sin(25x j ).6.4.2!.2!.4!.6!.8!
9 Example: 123 grid pts (61 modes) but only 41 observations (2 modes) available Physical Space sparse observations for P=3 Fourier Space aliasing set!(1) = {1,-4,42} for P=3 and M= aliasing set!(11) = {11,-3,52} for P=3 and M=2
10 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π
11 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l)
12 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l) Reduced Fourier Domain Kalman Filter (RFDKF) RFDKF : v l (t) = u l (t) + η o l (t), where η o l (t) N (, r o /2M + 1), k N, l M.
13 Example: Stochastically forced advection-diffusion equation u(x, t) t = 2 u(x, t) µ x x 2 u(x, t)+ F (x, t)+σ(x)ẇ (t), x 2π Fourier Domain Kalman Filter (FDKF) dû k (t) = [( µk 2 ik)û k (t) + ˆF k (t)]dt + σ k dw k (t), FDKF : v l (t) = u ki (t) + ηl o (t), k i A(l) Reduced Fourier Domain Kalman Filter (RFDKF) RFDKF : v l (t) = u l (t) + η o l (t), where η o l (t) N (, r o /2M + 1), k N, l M. Strongly Damped Approximate Filter (SDAF, VSDAF): Observation is modeled as in FDKF but we implement it with dynamic-less unresolved modes. e µk2 1 t = O(1), e µk2 i t = O(ɛ) 1, 2 i P.
14 Skill of cheaper approximate filters with model error depends on observation time at given wave number and how rough spectrum is for truth signal Decorrelation time vs observation time T corr = 1/! k =1/(µ k 2 ) "t 1 =.1 "t 2 =.1 Decorrelation time 1 1!1 1!2 1!3 1! wave number k
15 ETKF Filter Divergence (K = 15, r = 4%), observability is violated Extreme event, t 2 =.1, E k = k 5/3 RMS ETKF, <corr>= filtered unfiltered obs u(x,1!t) ETKF, at T=1!t, corr=.7692 true unfiltered filtered obs time!5! model space 25 ETKF, at T=5!t, corr= ETKF, at T=1!t, corr=.913 u(x,5!t) !5! u(x,1!t) 2 1!1!
16 SDAF high skill (observability is satisfied) Spontaneous development of extreme event for t 2 =.1 and E k = k 5/3 RMS SDAF, <corr>= filtered unfiltered obs u(x,1!t) SDAF, at T=1!t, corr= true unfiltered filtered obs time!5! model space 25 SDAF, at T=5!t, corr= SDAF, at T=1!t, corr= u(x,5!t) !5! model space u(x,1!t) 2 1!1! model space
17 Radical Filtering Strategy for Nonlinear System Stochastically forced linear PDE FT Uncoupled Langevin eqn Replace the Nonlinear terms with an Ornstein-Uhlenbeck process Nonlinear Chaotic Dynamical Systems FT Coupled nonlinear ODE through nonlinear terms
18 Filtering turbulent nonlinear dynamical systems L-96 model (Lorenz 1996), 4 modes. (absorbing ball property) du j dt = (u j+1 u j 2 )u j 1 u j + F, j =,..., J 1 Energy Rescaled Variables: F=6 weakly chaotic F=8 strongly chaotic F=16 fully turbulent 2 F=6 2 F=8 2 F= time
19 Climatological Variance and Correlation time x 1!3 Variance Spectrum F= F=5 F=6 F=8 F= Correlation times F= F=5 F=6 F=8 F=16 6 Variance Correlation time Wave Numbers Wave Numbers Climatological Stochastic Model (CSM): fit the damping coefficient and stochastic noise strength to these climatological statistical quantities.
20 Regularly spaced sparse observations: weakly chaotic regime F = 6, P = 2, r o = 1.96, t = !2!4 EAKF TRUE! space !2!4 EAKF C67! space !2!4 FDKF C67! space !2!4 RFDKF C67! space
21 Regularly spaced sparse observations: weakly chaotic regime F = 6, P = 2, r o = 1.96, t =.234 hrs Table: This is a regime where EAKF true is superior. scheme RMS corr. EAKF true EAKF CSM ETKF true - ETKF CSM FDKF CSM RFDKF CSM No Filter 2.8 -
22 Regularly spaced sparse observations: fully turbulent regime F = 16, P = 2, r o =.81, t =.78 hrs EAKF TRUE EAKF CSM !5!1! space !5!1! space 2 FDKF CSM 2 ETKF CSM !5!5!1!1! space! space
23 Regularly spaced sparse observations: fully turbulent regime F = 16, P = 2, r o =.81, t =.78 hrs Table: This is a regime where FDKF is superior. Scheme RMS corr. EAKF true - EAKF CSM ETKF true - ETKF CSM FDKF CSM No Filter 6.3 -
24 Summary:
25 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation.
26 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes.
27 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes. Practically, our radical strategy is independent of tunable parameters and ensemble size.
28 Summary: For M < N sparse regular observations, FDKF reduces (2N+1)-dim filtering problem to M decoupled P-dim filtering problems with a single scalar observation. The poor-man s CSM model degrades the filtering skill in the weakly chaotic regime but suggests encouraging results in the strongly chaotic and fully turbulent regimes. Practically, our radical strategy is independent of tunable parameters and ensemble size. Catastrophic filter divergence in a chaotic DS with absorbing ball property needs further mathematical theory.
29 References: MG Majda and Grote, Explicit off-line criteria for stable accurate time filtering of strongly unstable spatially extended systems, PNAS 14(4), , 27. CHM Castronovo, Harlim, and Majda, Mathematical test criteria for filtering complex systems: Plentiful observations, J. Comp. Phys., 227(7), , 28. HM1 Harlim and Majda, Filtering nonlinear dynamical systems with linear stochastic models, Nonlinearity 21(6), , 28. HM2 Harlim and Majda, Mathematical test criteria for filtering complex systems: Regularly sparse observations, J. Comp. Phys., 227(1), , 28. HM3 Harlim and Majda, Catastrophic filter divergence in filtering nonlinear dissipative systems, submitted to Comm. Math. Sci., 28.
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
More informationSuperparameterization 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
More informationSystematic*strategies*for*real*time*filtering*of* turbulent*signals*in*complex*systems*
Systematic*strategies*for*real*time*filtering*of* turbulent*signals*in*complex*systems* AndrewJ.Majda DepartmentofMathematicsand CenterofAtmosphereandOceanSciences CourantInstituteofMathematicalSciences
More informationMATHEMATICAL STRATEGIES FOR FILTERING TURBULENT DYNAMICAL SYSTEMS. Andrew J. Majda. John Harlim. Boris Gershgorin
DISCRETE AND CONTINUOUS doi:.94/dcds..7.44 DYNAMICAL SYSTEMS Volume 7, Number, June pp. 44 486 MATHEMATICAL STRATEGIES FOR FILTERING TURBULENT DYNAMICAL SYSTEMS Andrew J. Majda Department of Mathematics
More informationFiltering Nonlinear Dynamical Systems with Linear Stochastic Models
Filtering Nonlinear Dynamical Systems with Linear Stochastic Models J Harlim and A J Majda Department of Mathematics and Center for Atmosphere and Ocean Science Courant Institute of Mathematical Sciences,
More informationErgodicity 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
More informationData 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
More informationA 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,
More informationData 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
More informationEdinburgh Research Explorer
Edinburgh Research Explorer Dynamic Stochastic Superresolution of sparsely observed turbulent systems Citation for published version: Branicki, M & Majda, AJ 23, 'Dynamic Stochastic Superresolution of
More informationFILTERING COMPLEX TURBULENT SYSTEMS
FILTERING COMPLEX TURBULENT SYSTEMS by Andrew J. Majda Department of Mathematics and Center for Atmosphere and Ocean Sciences, Courant Institute for Mathematical Sciences, New York University and John
More informationWhat 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
More informationReal Time Filtering and Parameter Estimation for Dynamical Systems with Many Degrees of Freedom. NOOOl
REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions,
More informationMany contemporary problems in science ranging from protein
Blended Particle Filters for Large Dimensional Chaotic Dynamical Systems Andrew J. Majda, Di Qi, Themistoklis P. Sapsis Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute
More informationEnKF 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
More informationState 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
More informationSystematic 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
More informationSemiparametric 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,
More informationEnKF and Catastrophic filter divergence
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
More informationStochastic 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
More informationLagrangian data assimilation for point vortex systems
JOT J OURNAL OF TURBULENCE http://jot.iop.org/ Lagrangian data assimilation for point vortex systems Kayo Ide 1, Leonid Kuznetsov 2 and Christopher KRTJones 2 1 Department of Atmospheric Sciences and Institute
More informationPREDICTING EXTREME EVENTS FOR PASSIVE SCALAR TURBULENCE IN TWO-LAYER BAROCLINIC FLOWS THROUGH REDUCED-ORDER STOCHASTIC MODELS
PREDICTING EXTREME EVENTS FOR PASSIVE SCALAR TURBULENCE IN TWO-LAYER BAROCLINIC FLOWS THROUGH REDUCED-ORDER STOCHASTIC MODELS DI QI AND ANDREW J. MAJDA Abstract. The capability of using imperfect stochastic
More informationGaussian 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
More informationControlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter
Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter Georg A. Gottwald and Lewis Mitchell School of Mathematics and Statistics,
More informationGaussian 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
More informationRelative 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
More informationLocalization 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.
More informationAspects of the practical application of ensemble-based Kalman filters
Aspects of the practical application of ensemble-based Kalman filters Lars Nerger Alfred Wegener Institute for Polar and Marine Research Bremerhaven, Germany and Bremen Supercomputing Competence Center
More informationAdaptive 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
More informationA Global Atmospheric Model. Joe Tribbia NCAR Turbulence Summer School July 2008
A Global Atmospheric Model Joe Tribbia NCAR Turbulence Summer School July 2008 Outline Broad overview of what is in a global climate/weather model of the atmosphere Spectral dynamical core Some results-climate
More informationPDEs, part 3: Hyperbolic PDEs
PDEs, part 3: Hyperbolic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2011 Hyperbolic equations (Sections 6.4 and 6.5 of Strang). Consider the model problem (the
More informationBayesian 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
More informationForecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from Noisy Data
Forecasting Turbulent Modes with Nonparametric Diffusion Models: Learning from Noisy Data Tyrus Berry Department of Mathematics, The Pennsylvania State University, University Park, PA 682 John Harlim Department
More informationData 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
More informationKalman Filter and Ensemble Kalman Filter
Kalman Filter and Ensemble Kalman Filter 1 Motivation Ensemble forecasting : Provides flow-dependent estimate of uncertainty of the forecast. Data assimilation : requires information about uncertainty
More informationFiltering 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,
More informationPhysics Constrained Nonlinear Regression Models for Time Series
Physics Constrained Nonlinear Regression Models for Time Series Andrew J. Majda and John Harlim 2 Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute of Mathematical
More informationLocal Ensemble Transform Kalman Filter
Local Ensemble Transform Kalman Filter Brian Hunt 11 June 2013 Review of Notation Forecast model: a known function M on a vector space of model states. Truth: an unknown sequence {x n } of model states
More information6.435, System Identification
SET 6 System Identification 6.435 Parametrized model structures One-step predictor Identifiability Munther A. Dahleh 1 Models of LTI Systems A complete model u = input y = output e = noise (with PDF).
More informationA 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
More informationIntroduction to Turbulent Dynamical Systems in Complex Systems
Introduction to Turbulent Dynamical Systems in Complex Systems Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 216 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda
More informationIntroduction to asymptotic techniques for stochastic systems with multiple time-scales
Introduction to asymptotic techniques for stochastic systems with multiple time-scales Eric Vanden-Eijnden Courant Institute Motivating examples Consider the ODE {Ẋ = Y 3 + sin(πt) + cos( 2πt) X() = x
More informationState Space Representation of Gaussian Processes
State Space Representation of Gaussian Processes Simo Särkkä Department of Biomedical Engineering and Computational Science (BECS) Aalto University, Espoo, Finland June 12th, 2013 Simo Särkkä (Aalto University)
More informationChapter 6: Ensemble Forecasting and Atmospheric Predictability. Introduction
Chapter 6: Ensemble Forecasting and Atmospheric Predictability Introduction Deterministic Chaos (what!?) In 1951 Charney indicated that forecast skill would break down, but he attributed it to model errors
More informationSmoothers: Types and Benchmarks
Smoothers: Types and Benchmarks Patrick N. Raanes Oxford University, NERSC 8th International EnKF Workshop May 27, 2013 Chris Farmer, Irene Moroz Laurent Bertino NERSC Geir Evensen Abstract Talk builds
More informationEnsemble 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
More informationThe role of additive and multiplicative noise in filtering complex dynamical systems
The role of additive and multiplicative noise in filtering complex dynamical systems By Georg A. Gottwald 1 and John Harlim 2 1 School of Mathematics and Statistics, University of Sydney, NSW 26, Australia,
More informationMathematical test models for superparametrization in anisotropic turbulence
Mathematical test models for superparametrization in anisotropic turbulence Andrew J. Majda a,1 and Marcus J. Grote b a Department of Mathematics and Climate, Atmosphere, Ocean Science, Courant Institute
More informationData 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
More informationControlling Overestimation of Error Covariance in Ensemble Kalman Filters with Sparse Observations: A Variance-Limiting Kalman Filter
2650 M O N T H L Y W E A T H E R R E V I E W VOLUME 139 Controlling Overestimation of Error Covariance in Ensemble Kalman Filters with Sparse Observations: A Variance-Limiting Kalman Filter GEORG A. GOTTWALD
More informationSystematic stochastic modeling. of atmospheric variability. Christian Franzke
Systematic stochastic modeling of atmospheric variability Christian Franzke Meteorological Institute Center for Earth System Research and Sustainability University of Hamburg Daniel Peavoy and Gareth Roberts
More informationRobust 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
More informationLagrangian 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:
More informationBred vectors: theory and applications in operational forecasting. Eugenia Kalnay Lecture 3 Alghero, May 2008
Bred vectors: theory and applications in operational forecasting. Eugenia Kalnay Lecture 3 Alghero, May 2008 ca. 1974 Central theorem of chaos (Lorenz, 1960s): a) Unstable systems have finite predictability
More informationEnsembles and Particle Filters for Ocean Data Assimilation
DISTRIBUTION STATEMENT A. Approved for public release; distribution is unlimited. Ensembles and Particle Filters for Ocean Data Assimilation Robert N. Miller College of Oceanic and Atmospheric Sciences
More informationDART_LAB Tutorial Section 2: How should observations impact an unobserved state variable? Multivariate assimilation.
DART_LAB Tutorial Section 2: How should observations impact an unobserved state variable? Multivariate assimilation. UCAR 2014 The National Center for Atmospheric Research is sponsored by the National
More informationData Assimilation in Slow Fast Systems Using Homogenized Climate Models
APRIL 2012 M I T C H E L L A N D G O T T W A L D 1359 Data Assimilation in Slow Fast Systems Using Homogenized Climate Models LEWIS MITCHELL AND GEORG A. GOTTWALD School of Mathematics and Statistics,
More informationModel Uncertainty Quantification for Data Assimilation in partially observed Lorenz 96
Model Uncertainty Quantification for Data Assimilation in partially observed Lorenz 96 Sahani Pathiraja, Peter Jan Van Leeuwen Institut für Mathematik Universität Potsdam With thanks: Sebastian Reich,
More informationChapter 3. Finite Difference Methods for Hyperbolic Equations Introduction Linear convection 1-D wave equation
Chapter 3. Finite Difference Methods for Hyperbolic Equations 3.1. Introduction Most hyperbolic problems involve the transport of fluid properties. In the equations of motion, the term describing the transport
More informationLow-dimensional chaotic dynamics versus intrinsic stochastic noise: a paradigm model
Physica D 199 (2004) 339 368 Low-dimensional chaotic dynamics versus intrinsic stochastic noise: a paradigm model A. Majda a,b, I. Timofeyev c, a Courant Institute of Mathematical Sciences, New York University,
More informationLecture 17: Initial value problems
Lecture 17: Initial value problems Let s start with initial value problems, and consider numerical solution to the simplest PDE we can think of u/ t + c u/ x = 0 (with u a scalar) for which the solution
More informationAddressing 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
More informationThe 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
More informationChapter 4. Nonlinear Hyperbolic Problems
Chapter 4. Nonlinear Hyperbolic Problems 4.1. Introduction Reading: Durran sections 3.5-3.6. Mesinger and Arakawa (1976) Chapter 3 sections 6-7. Supplementary reading: Tannehill et al sections 4.4 and
More informationData Assimilation: Finding the Initial Conditions in Large Dynamical Systems. Eric Kostelich Data Mining Seminar, Feb. 6, 2006
Data Assimilation: Finding the Initial Conditions in Large Dynamical Systems Eric Kostelich Data Mining Seminar, Feb. 6, 2006 kostelich@asu.edu Co-Workers Istvan Szunyogh, Gyorgyi Gyarmati, Ed Ott, Brian
More informationLow-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification
Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification Andrew J Majda joint work with Di Qi Morse Professor of Arts and Science Department of Mathematics Founder of
More informationOn the (multi)scale nature of fluid turbulence
On the (multi)scale nature of fluid turbulence Kolmogorov axiomatics Laurent Chevillard Laboratoire de Physique, ENS Lyon, CNRS, France Laurent Chevillard, Laboratoire de Physique, ENS Lyon, CNRS, France
More informationCorrecting 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
More informationEnKF-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.
More informationAt A Glance. UQ16 Mobile App.
At A Glance UQ16 Mobile App Scan the QR code with any QR reader and download the TripBuilder EventMobile app to your iphone, ipad, itouch or Android mobile device. To access the app or the HTML 5 version,
More informationA review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
More informationZentrum für Technomathematik
Zentrum für Technomathematik Fachbereich 3 Mathematik und Informatik On the influence of model nonlinearity and localization on ensemble Kalman smoothing Lars Nerger Svenja Schulte Angelika Bunse-Gerstner
More informationPerformance 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
More informationP 1.86 A COMPARISON OF THE HYBRID ENSEMBLE TRANSFORM KALMAN FILTER (ETKF)- 3DVAR AND THE PURE ENSEMBLE SQUARE ROOT FILTER (EnSRF) ANALYSIS SCHEMES
P 1.86 A COMPARISON OF THE HYBRID ENSEMBLE TRANSFORM KALMAN FILTER (ETKF)- 3DVAR AND THE PURE ENSEMBLE SQUARE ROOT FILTER (EnSRF) ANALYSIS SCHEMES Xuguang Wang*, Thomas M. Hamill, Jeffrey S. Whitaker NOAA/CIRES
More information1. Introduction Systems evolving on widely separated time-scales represent a challenge for numerical simulations. A generic example is
COMM. MATH. SCI. Vol. 1, No. 2, pp. 385 391 c 23 International Press FAST COMMUNICATIONS NUMERICAL TECHNIQUES FOR MULTI-SCALE DYNAMICAL SYSTEMS WITH STOCHASTIC EFFECTS ERIC VANDEN-EIJNDEN Abstract. Numerical
More informationMathematical Theories of Turbulent Dynamical Systems
Mathematical Theories of Turbulent Dynamical Systems Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 2016 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Mathematical
More informationConvective-scale data assimilation in the Weather Research and Forecasting model using a nonlinear ensemble filter
Convective-scale data assimilation in the Weather Research and Forecasting model using a nonlinear ensemble filter Jon Poterjoy, Ryan Sobash, and Jeffrey Anderson National Center for Atmospheric Research
More informationIII Subjective probabilities. III.4. Adaptive Kalman filtering. Probability Course III:4 Bologna 9-13 February 2015
III Subjective probabilities III.4. Adaptive Kalman filtering III.4.1 A 2-dimensional Kalman filter system Obs-Tfc= correction Bias? Corr = A(t) Tfc It appears as if the old bias has abruptly changed into
More informationThe Planetary Boundary Layer and Uncertainty in Lower Boundary Conditions
The Planetary Boundary Layer and Uncertainty in Lower Boundary Conditions Joshua Hacker National Center for Atmospheric Research hacker@ucar.edu Topics The closure problem and physical parameterizations
More informationReducing the Impact of Sampling Errors in Ensemble Filters
Reducing the Impact of Sampling Errors in Ensemble Filters Jeffrey Anderson NCAR Data Assimilation Research Section The National Center for Atmospheric Research is sponsored by the National Science Foundation.
More informationIntroduction to ensemble forecasting. Eric J. Kostelich
Introduction to ensemble forecasting Eric J. Kostelich SCHOOL OF MATHEMATICS AND STATISTICS MSRI Climate Change Summer School July 21, 2008 Co-workers: Istvan Szunyogh, Brian Hunt, Edward Ott, Eugenia
More informationEnhancing information transfer from observations to unobserved state variables for mesoscale radar data assimilation
Enhancing information transfer from observations to unobserved state variables for mesoscale radar data assimilation Weiguang Chang and Isztar Zawadzki Department of Atmospheric and Oceanic Sciences Faculty
More informationA scaling limit from Euler to Navier-Stokes equations with random perturbation
A scaling limit from Euler to Navier-Stokes equations with random perturbation Franco Flandoli, Scuola Normale Superiore of Pisa Newton Institute, October 208 Newton Institute, October 208 / Subject of
More informationMOX EXPONENTIAL INTEGRATORS FOR MULTIPLE TIME SCALE PROBLEMS OF ENVIRONMENTAL FLUID DYNAMICS. Innsbruck Workshop October
Innsbruck Workshop October 29 21 EXPONENTIAL INTEGRATORS FOR MULTIPLE TIME SCALE PROBLEMS OF ENVIRONMENTAL FLUID DYNAMICS Luca Bonaventura - Modellistica e Calcolo Scientifico Dipartimento di Matematica
More informationEnsemble 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
More informationModel error in coupled atmosphereocean data assimilation. Alison Fowler and Amos Lawless (and Keith Haines)
Model error in coupled atmosphereocean data assimilation Alison Fowler and Amos Lawless (and Keith Haines) Informal DARC seminar 11 th February 2015 Introduction Coupled atmosphere-ocean DA schemes have
More informationNonparametric Uncertainty Quantification for Stochastic Gradient Flows
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows Tyrus Berry and John Harlim Abstract. This paper presents a nonparametric statistical modeling method for quantifying uncertainty
More informationObservability, a Problem in Data Assimilation
Observability, Data Assimilation with the Extended Kalman Filter 1 Observability, a Problem in Data Assimilation Chris Danforth Department of Applied Mathematics and Scientific Computation, UMD March 10,
More informationWe honor Ed Lorenz ( ) who started the whole new science of predictability
Atmospheric Predictability: From Basic Theory to Forecasting Practice. Eugenia Kalnay Alghero, May 2008, Lecture 1 We honor Ed Lorenz (1917-2008) who started the whole new science of predictability Ed
More informationA Simple Dynamical Model Capturing the Key Features of Central Pacific El Niño (Supporting Information Appendix)
A Simple Dynamical Model Capturing the Key Features of Central Pacific El Niño (Supporting Information Appendix) Nan Chen and Andrew J. Majda Department of Mathematics, and Center for Atmosphere Ocean
More informationSome Basic Statistical Modeling Issues in Molecular and Ocean Dynamics
Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics p. 1/2 Some Basic Statistical Modeling Issues in Molecular and Ocean Dynamics Peter R. Kramer Department of Mathematical Sciences
More informationOn the influence of model nonlinearity and localization on ensemble Kalman smoothing
QuarterlyJournalof theroyalmeteorologicalsociety Q. J. R. Meteorol. Soc. 140: 2249 2259, October 2014 A DOI:10.1002/qj.2293 On the influence of model nonlinearity and localization on ensemble Kalman smoothing
More informationA State Space Model for Wind Forecast Correction
A State Space Model for Wind Forecast Correction Valrie Monbe, Pierre Ailliot 2, and Anne Cuzol 1 1 Lab-STICC, Université Européenne de Bretagne, France (e-mail: valerie.monbet@univ-ubs.fr, anne.cuzol@univ-ubs.fr)
More informationA 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
More informationEnsemble square-root filters
Ensemble square-root filters MICHAEL K. TIPPETT International Research Institute for climate prediction, Palisades, New Yor JEFFREY L. ANDERSON GFDL, Princeton, New Jersy CRAIG H. BISHOP Naval Research
More informationStochastic 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
More informationStochastic 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
More informationDATA-DRIVEN TECHNIQUES FOR ESTIMATION AND STOCHASTIC REDUCTION OF MULTI-SCALE SYSTEMS
DATA-DRIVEN TECHNIQUES FOR ESTIMATION AND STOCHASTIC REDUCTION OF MULTI-SCALE SYSTEMS A Dissertation Presented to the Faculty of the Department of Mathematics University of Houston In Partial Fulfillment
More informationEarth System Modeling Domain decomposition
Earth System Modeling Domain decomposition Graziano Giuliani International Centre for Theorethical Physics Earth System Physics Section Advanced School on Regional Climate Modeling over South America February
More informationAdaptive Inflation for Ensemble Assimilation
Adaptive Inflation for Ensemble Assimilation Jeffrey Anderson IMAGe Data Assimilation Research Section (DAReS) Thanks to Ryan Torn, Nancy Collins, Tim Hoar, Hui Liu, Kevin Raeder, Xuguang Wang Anderson:
More information