Sample-based UQ via Computational Linear Algebra (not MCMC) Colin Fox Al Parker, John Bardsley March 2012
|
|
- Logan Garrett
- 5 years ago
- Views:
Transcription
1 Sample-based UQ via Computational Linear Algebra (not MCMC) Colin Fox Al Parker, John Bardsley March 2012
2 Contents Example: sample-based inference in a large-scale problem Sampling built on computational linear algebra Gibbs sampling a Gauss-Seidel poloynomial acceleration of GS... and of MCMC sampling based on (P)CG, computing A 1/2, BFGS, PGCG a for Gaussians
3 Ocean circulation :: 2 samples from the posterior Data on traces, assert physics and observational models, infer abyssal advection 3 Oxygen, run 1 3 Oxygen, run McKeague Nicholls Speer Herbei 2005 Statistical Inversion of South Atlantic Circulation in an Abyssal Neutral Density Layer
4 Gibbs sampling from normal distributions Gibbs sampling a repeatedly samples from (block) conditional distributions Forward and reverse sweep simulates a reversible kernel (self-adjoint wrt π) Normal distributions π (x) = det (A) 2π n exp { 1 } 2 xt Ax + b T x precision matrix A, covariance matrix Σ = A 1 (both SPD) wlog treat zero mean Particularly interested in case where A is sparse (GMRF) and n large When is π (0) is also normal, then so is the n-step distribution A (n) A Σ (n) Σ In what sense is stochastic relaxation related to relaxation? What decomposition of A is this performing? a Glauber 1963 (heat-bath algorithm), Turcin 1971, Geman and Geman 1984
5 Matrix formulation of Gibbs sampling from N(0, A 1 ) Let y = (y 1, y 2,..., y n ) T Component-wise Gibbs updates each component in sequence from the (normal) conditional distributions One sweep over all n components can be written y (k+1) = D 1 Ly (k+1) D 1 L T y (k) + D 1/2 z (k) where: D = diag(a), L is the strictly lower triangular part of A, z (k 1) N(0, I) y (k+1) = Gy (k) + c (k) c (k) is iid noise with zero mean, finite covariance Spot the similarity to Gauss-Seidel iteration for solving Ax = b x (k+1) = D 1 Lx (k+1) D 1 L T x (k) + D 1 b Goodman & Sokal 1989; Amit & Grenander 1991
6 Matrix formulation of Gibbs sampling from N(0, A 1 ) Let y = (y 1, y 2,..., y n ) T Component-wise Gibbs updates each component in sequence from the (normal) conditional distributions One sweep over all n components can be written y (k+1) = D 1 Ly (k+1) D 1 L T y (k) + D 1/2 z (k) where: D = diag(a), L is the strictly lower triangular part of A, z (k 1) N(0, I) y (k+1) = Gy (k) + c (k) c (k) is iid noise with zero mean, finite covariance Spot the similarity to Gauss-Seidel iteration for solving Ax = b x (k+1) = D 1 Lx (k+1) D 1 L T x (k) + D 1 b Goodman & Sokal 1989; Amit & Grenander 1991
7 Gibbs converges solver converges Theorem 1 Let A = M N, M invertible. The stationary linear solver x (k+1) = M 1 Nx (k) + M 1 b = Gx (k) + M 1 b converges, if and only if the random iteration y (k+1) = M 1 Ny (k) + M 1 c (k) = Gy (k) + M 1 c (k) converges in distribution. Here c (k) iid π n has zero mean and finite variance Proof. Both converge iff ϱ(g) < 1 Convergent splittings generate convergent (generalized) Gibbs samplers Mean converges with asymptotic convergence factor ϱ(g), covariance with ϱ(g) 2 Young 1971 Thm 3-5.1, Duflo 1997 Thm , Goodman & Sokal, 1989, Galli & Gao 2001 F Parker 2012
8 Average contractive iterated random functions Diaconis Freedman SIAM Review 1999, Stenflo JDEA 2012
9 Some not so common Gibbs samplers for N(0, A 1 ) splitting/sampler M Var ( c (k)) = M T + N converge if 1 Richardson ω I 2 ω I A 0 < ω < 2 ϱ(a) Jacobi D 2D A A SDD GS/Gibbs D + L D always SOR/B&F 1 ω D + L SSOR/REGS ω 2 ω M SORD 1 M T SOR ω 2 ω 2 ω ω ( D 0 < ω < 2 MSOR D 1 M T SOR 0 < ω < 2 +N T ) SOR D 1 N SOR Want: convenient to solve Mu = r and sample from N(0, M T + N) Relaxation parameter ω can accelerate Gibbs SSOR is a forwards and backwards sweep of SOR to give a symmetric splitting SOR: Adler 1981; Barone & Frigessi 1990, Amit & Grenander 1991, SSOR: Roberts & Sahu 1997
10 Some not so common Gibbs samplers for N(0, A 1 ) splitting/sampler M Var ( c (k)) = M T + N converge if 1 Richardson ω I 2 ω I A 0 < ω < 2 ϱ(a) Jacobi D 2D A A SDD GS/Gibbs D + L D always SOR/B&F 1 ω D + L SSOR/REGS ω 2 ω M SORD 1 M T SOR ω 2 ω 2 ω ω ( D 0 < ω < 2 MSOR D 1 M T SOR 0 < ω < 2 +N T ) SOR D 1 N SOR Want: convenient to solve Mu = r and sample from N(0, M T + N) Relaxation parameter ω can accelerate Gibbs SSOR is a forwards and backwards sweep of SOR to give a symmetric splitting SOR: Adler 1981; Barone & Frigessi 1990, Amit & Grenander 1991, SSOR: Roberts & Sahu 1997
11 A closer look at convergence To sample from N(µ, A 1 ) where Aµ = b Split A = M N, M invertible. G = M 1 N, and c (k) iid N(0, M T + N) The iteration ( ) y (k+1) = Gy (k) + M 1 (c (k) + b implies and Var E ( y (m)) [ ( µ = G m E y (0)) ] µ ( y (m)) [ ( A 1 = G m Var y (0)) A 1] G m (Hence asymptotic average convergence factors ϱ(g) and ϱ(g) 2 ) Errors go down as the polynomial P m (I G) = (I (I G)) m = ( I M 1 A ) m P m (λ) = (1 λ) m solver and sampler have exactly the same error polynomial
12 A closer look at convergence To sample from N(µ, A 1 ) where Aµ = b Split A = M N, M invertible. G = M 1 N, and c (k) iid N(0, M T + N) The iteration ( ) y (k+1) = Gy (k) + M 1 (c (k) + b implies and Var E ( y (m)) [ ( µ = G m E y (0)) ] µ ( y (m)) [ ( A 1 = G m Var y (0)) A 1] G m (Hence asymptotic average convergence factors ϱ(g) and ϱ(g) 2 ) Errors go down as the polynomial P m (I G) = (I (I G)) m = ( I M 1 A ) m P m (λ) = (1 λ) m solver and sampler have exactly the same error polynomial
13 Controlling the error polynomial The splitting gives the iteration operator A = 1 ( τ M ) M N τ G τ = ( I τm 1 A ) and error polynomial P m (λ) = (1 τλ) m The sequence of parameters τ 1, τ 2,..., τ m gives the error polynomial P m (λ) = m (1 τ l λ) l=1... so we can choose the zeros of P m! This gives a non-stationary solver non-homogeneous Markov chain Golub & Varga 1961, Golub & van Loan 1989, Axelsson 1996, Saad 2003, F & Parker 2012
14 The best (Chebyshev) polynomial 10 iterations, factor of 300 improvement Choose 1 = λ n + λ 1 τ l 2 + λ n λ 1 2 cos where λ 1 λ n are extreme eigenvalues of M 1 A ( π 2l + 1 ) 2p l = 0, 1, 2,..., p 1
15 Second-order accelerated sampler First-order accelerated iteration turns out to be unstable Numerical stability, and optimality at each step, is given by the second-order iteration y (k+1) = (1 α k )y (k 1) + α k y (k) + α k τ k M 1 (c (k) Ay (k) ) with α k and τ k chosen so error polynomial satisfies Chebyshev recursion. Theorem 2 2 nd -order solver converges 2 nd -order sampler converges (given correct noise distribution) Error polynomial is optimal, at each step, for both mean and covariance Asymptotic average reduction factor (Axelsson 1996) is σ = 1 λ 1 /λ n 1 + λ 1 /λ n Axelsson 1996, F & Parker 2012
16 input : The SSOR splitting M, N of A; smallest eigenvalue λ min of M 1 A; largest eigenvalue λ max of M 1 A; relaxation parameter ω; initial state y (0) ; k max output: y (k) approximately distributed as N(0, A 1 ) set γ = ( ( 2 ω 1) 1/2, δ = λmax λ min 4 set α = 1, β = 2/θ, τ = 1/θ, τ c = 2 τ 1; for k = 1,..., k max do if k = 0 then b = 2 α 1, a = τ cb, κ = τ; end ) 2, θ = λ max +λ min 2 ; else b = 2(1 α)/β + 1, a = τ c + (b 1) (1/τ + 1/κ 1), κ = β + (1 α)κ; end sample z N(0, I); c = γb 1/2 D 1/2 z; x = y (k) + M 1 (c Ay (k) ); sample z N(0, I); c = γa 1/2 D 1/2 z; w = x y (k) + M T (c Ax); y (k+1) = α(y (k) y (k 1) + τw) + y (k 1) ; β = (θ βδ) 1 ; α = θβ; Algorithm 1: Chebyshev accelerated SSOR sampling from N(0, A 1 )
17 10 10 lattice (d = 100) sparse precision matrix [A] ij = 10 4 δ ij + n i if i = j 1 if i j and s i s j otherwise 1 relative error SSOR, ω=1 SSOR, ω= Cheby SSOR, ω=1 Cheby SSOR, ω= Cholesky flops x times faster
18 lattice (d = 10 6 ) sparse precision matrix only used sparsity, no other special structure
19 Accelerating MCMC (cartoon) MH-MCMC repeatedly simulates fixed transition kernel P with πp = π via detailed balance (self-adjoint in π) n-step distribution: π (n) = π (n 1) P = π (0) P n n-step error in distribution: π (n) π = ( π (0) π ) P n error polynomial: P n = (I (I P)) n = (1 λ) n Hence convergence is geometric Polynomial acceleration: (modify iteration to give desired error polynomial) (choose x (i) w.p. α i ) π (0) n α i P i error poly: Q n = i=1 n α i (1 λ) i i=1
20 Accelerating MCMC (cartoon) MH-MCMC repeatedly simulates fixed transition kernel P with πp = π via detailed balance (self-adjoint in π) n-step distribution: π (n) = π (n 1) P = π (0) P n n-step error in distribution: π (n) π = ( π (0) π ) P n error polynomial: P n = (I (I P)) n = (1 λ) n Hence convergence is geometric Polynomial acceleration: (modify iteration to give desired error polynomial) (choose x (i) w.p. α i ) π (0) n α i P i error poly: Q n = i=1 n α i (1 λ) i i=1
21 CG (Krylov space) sampling x 2 p (0) x 2 5 =p (0) (0) x (2) p (1) x (1) x (2) x x p (1) x (1) x x 5 (1) x (0) x 1 x (0) x 1 Gibbs sampling & Gauss Seidel CG sampling & optimization Mutually conjugate vectors (wrt A) are independent directions V T AV = D A 1 = VD 1 V T so if z N (0, I) then x = V D 1 z N(0, A 1 ) Schneider & Willsky 2003, F 2008, Parker & F SISC 2012
22 CG sampling algorithm Apart from step 3, this is exactly (linear) CG optimization Var(y k ) is the CG polynomial Parker & F SISC 2012
23 CG sampling example So CG (by itself) over-smooths : initialize Gibbs with CG Parker & F SISC 2012, Schneider & Willsky 2003
24 Evaluating f(a) = A 1/2 If z N(0, I) then x = A 1/2 z N(0, A 1 ) Compute a rational approximation x = A 1/2 z Nα j (A σ j I) 1 x j=1 on spectrum {σ j } of A, based on Lanczos approximation: x = x 0 + β 0 VT 1/2 e 1 Contour integration Continuous deformation ODE all using CG Simpson Turner & Pettitt 2008, Aune Eidsvik & Pokern STCO 2012
25 Sampling by BFGS BFGS optimization over log π builds an approximation to the inverse Hessian A 1 A so if z 0 N(0, A 1 0 ) and z i N(0, 1) then N ρ i s i s T i i=1 z 0 + N ρi s i z i N(0, A 1 ) i=1 Can propagate sample covariance in large-scale EnKF... though CG sampler does a better job (don t know why) and seeding proposal in AM Veersé 1999, Auvinen Bardsley Haario & Kauranne 2010, Bardsley Solonen Parker Haario & Howard IJUQ 2011
26 Non-negativity constrained sampling by PGCG Linear inverse problem b = Ax + η η N(0, σ 2 I) ( p(x, λ, δ b) λ n/2+αλ 1 δ n/2+αδ 1 exp λ 2 Ax b 2 δ ) 2 xt Cx β λ λ β δ δ with probability mass projected to constraint boundary in metric of Hessian of log π { } 1 x k = arg min x 0 2 xt (λ k A T A + δ k C)x x T (λ k A T b + w) where w N(0, λ k A T A + δ k C). Compute by projected-gradient CG Bardsley F IPSE 2011
27 Some observations In the Gaussian setting stochastic relaxation is fundamentally equivalent to classical relaxation existing solvers (mutligrid, fast multipole, parallel tools) can be used as samplers More generally Accelerated smplers are not homogeneous chains not standard MCMC need a convergence theory for non-stationary random iterations Fast sampling looks like a good way to do UQ in large-scale problems error = Cn o o = 1/2 for Monte Carlo (not good)... but we are never in the asymptotic regime Further... C is much smaller than lattice methods especially when using importance sampling and scales weakly with problem dimension
28 Conclusion
Polynomial accelerated MCMC... and other sampling algorithms inspired by computational optimization
Polynomial accelerated MCMC... and other sampling algorithms inspired by computational optimization Colin Fox fox@physics.otago.ac.nz Al Parker, John Bardsley Fore-words Motivation: an inverse oceanography
More informationMonte Carlo simulation inspired by computational optimization. Colin Fox Al Parker, John Bardsley MCQMC Feb 2012, Sydney
Monte Carlo simulation inspired by computational optimization Colin Fox fox@physics.otago.ac.nz Al Parker, John Bardsley MCQMC Feb 2012, Sydney Sampling from π(x) Consider : x is high-dimensional (10 4
More informationMonitoring milk-powder dryers via Bayesian inference in an FPGA. Colin Fox Markus Neumayer, Al Parker, Pat Suggate
Monitoring milk-powder dryers via Bayesian inference in an FPGA Colin Fox fox@physics.otago.ac.nz Markus Neumayer, Al Parker, Pat Suggate Three newish technologies Gibbs sampling for capacitance tomography
More information9.1 Preconditioned Krylov Subspace Methods
Chapter 9 PRECONDITIONING 9.1 Preconditioned Krylov Subspace Methods 9.2 Preconditioned Conjugate Gradient 9.3 Preconditioned Generalized Minimal Residual 9.4 Relaxation Method Preconditioners 9.5 Incomplete
More informationDeblurring Jupiter (sampling in GLIP faster than regularized inversion) Colin Fox Richard A. Norton, J.
Deblurring Jupiter (sampling in GLIP faster than regularized inversion) Colin Fox fox@physics.otago.ac.nz Richard A. Norton, J. Andrés Christen Topics... Backstory (?) Sampling in linear-gaussian hierarchical
More informationPolynomial accelerated solutions to a LARGE Gaussian model for imaging biofilms: in theory and finite precision
Journal of the American Statistical Association ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: http://www.tandfonline.com/loi/uasa20 Polynomial accelerated solutions to a LARGE Gaussian model
More informationScientific Computing WS 2018/2019. Lecture 9. Jürgen Fuhrmann Lecture 9 Slide 1
Scientific Computing WS 2018/2019 Lecture 9 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 9 Slide 1 Lecture 9 Slide 2 Simple iteration with preconditioning Idea: Aû = b iterative scheme û = û
More informationCLASSICAL ITERATIVE METHODS
CLASSICAL ITERATIVE METHODS LONG CHEN In this notes we discuss classic iterative methods on solving the linear operator equation (1) Au = f, posed on a finite dimensional Hilbert space V = R N equipped
More informationAlgebraic Multigrid as Solvers and as Preconditioner
Ò Algebraic Multigrid as Solvers and as Preconditioner Domenico Lahaye domenico.lahaye@cs.kuleuven.ac.be http://www.cs.kuleuven.ac.be/ domenico/ Department of Computer Science Katholieke Universiteit Leuven
More informationMath Introduction to Numerical Analysis - Class Notes. Fernando Guevara Vasquez. Version Date: January 17, 2012.
Math 5620 - Introduction to Numerical Analysis - Class Notes Fernando Guevara Vasquez Version 1990. Date: January 17, 2012. 3 Contents 1. Disclaimer 4 Chapter 1. Iterative methods for solving linear systems
More informationSome definitions. Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization. A-inner product. Important facts
Some definitions Math 1080: Numerical Linear Algebra Chapter 5, Solving Ax = b by Optimization M. M. Sussman sussmanm@math.pitt.edu Office Hours: MW 1:45PM-2:45PM, Thack 622 A matrix A is SPD (Symmetric
More informationFEM and Sparse Linear System Solving
FEM & sparse system solving, Lecture 7, Nov 3, 2017 1/46 Lecture 7, Nov 3, 2015: Introduction to Iterative Solvers: Stationary Methods http://people.inf.ethz.ch/arbenz/fem16 Peter Arbenz Computer Science
More informationLecture 18 Classical Iterative Methods
Lecture 18 Classical Iterative Methods MIT 18.335J / 6.337J Introduction to Numerical Methods Per-Olof Persson November 14, 2006 1 Iterative Methods for Linear Systems Direct methods for solving Ax = b,
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationChapter 7 Iterative Techniques in Matrix Algebra
Chapter 7 Iterative Techniques in Matrix Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128B Numerical Analysis Vector Norms Definition
More information7.3 The Jacobi and Gauss-Siedel Iterative Techniques. Problem: To solve Ax = b for A R n n. Methodology: Iteratively approximate solution x. No GEPP.
7.3 The Jacobi and Gauss-Siedel Iterative Techniques Problem: To solve Ax = b for A R n n. Methodology: Iteratively approximate solution x. No GEPP. 7.3 The Jacobi and Gauss-Siedel Iterative Techniques
More informationIntroduction to Bayesian methods in inverse problems
Introduction to Bayesian methods in inverse problems Ville Kolehmainen 1 1 Department of Applied Physics, University of Eastern Finland, Kuopio, Finland March 4 2013 Manchester, UK. Contents Introduction
More informationClassical iterative methods for linear systems
Classical iterative methods for linear systems Ed Bueler MATH 615 Numerical Analysis of Differential Equations 27 February 1 March, 2017 Ed Bueler (MATH 615 NADEs) Classical iterative methods for linear
More informationIterative Methods for Smooth Objective Functions
Optimization Iterative Methods for Smooth Objective Functions Quadratic Objective Functions Stationary Iterative Methods (first/second order) Steepest Descent Method Landweber/Projected Landweber Methods
More informationBindel, Fall 2016 Matrix Computations (CS 6210) Notes for
1 Iteration basics Notes for 2016-11-07 An iterative solver for Ax = b is produces a sequence of approximations x (k) x. We always stop after finitely many steps, based on some convergence criterion, e.g.
More informationLINEAR SYSTEMS (11) Intensive Computation
LINEAR SYSTEMS () Intensive Computation 27-8 prof. Annalisa Massini Viviana Arrigoni EXACT METHODS:. GAUSSIAN ELIMINATION. 2. CHOLESKY DECOMPOSITION. ITERATIVE METHODS:. JACOBI. 2. GAUSS-SEIDEL 2 CHOLESKY
More informationClone MCMC: Parallel High-Dimensional Gaussian Gibbs Sampling
Clone MCMC: Parallel High-Dimensional Gaussian Gibbs Sampling Andrei-Cristian Bărbos IMS Laboratory Univ. Bordeaux - CNRS - BINP andbarbos@u-bordeaux.fr Jean-François Giovannelli IMS Laboratory Univ. Bordeaux
More informationIterative Methods for Solving A x = b
Iterative Methods for Solving A x = b A good (free) online source for iterative methods for solving A x = b is given in the description of a set of iterative solvers called templates found at netlib: http
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 3: Iterative Methods PD
More informationThe Conjugate Gradient Method
The Conjugate Gradient Method Classical Iterations We have a problem, We assume that the matrix comes from a discretization of a PDE. The best and most popular model problem is, The matrix will be as large
More informationLecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector
More informationFrom Stationary Methods to Krylov Subspaces
Week 6: Wednesday, Mar 7 From Stationary Methods to Krylov Subspaces Last time, we discussed stationary methods for the iterative solution of linear systems of equations, which can generally be written
More informationIterative Methods. Splitting Methods
Iterative Methods Splitting Methods 1 Direct Methods Solving Ax = b using direct methods. Gaussian elimination (using LU decomposition) Variants of LU, including Crout and Doolittle Other decomposition
More informationPreconditioning Techniques Analysis for CG Method
Preconditioning Techniques Analysis for CG Method Huaguang Song Department of Computer Science University of California, Davis hso@ucdavis.edu Abstract Matrix computation issue for solve linear system
More informationComputer Vision Group Prof. Daniel Cremers. 11. Sampling Methods: Markov Chain Monte Carlo
Group Prof. Daniel Cremers 11. Sampling Methods: Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationComputer Vision Group Prof. Daniel Cremers. 10a. Markov Chain Monte Carlo
Group Prof. Daniel Cremers 10a. Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative is Markov Chain
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationParallel Numerics, WT 2016/ Iterative Methods for Sparse Linear Systems of Equations. page 1 of 1
Parallel Numerics, WT 2016/2017 5 Iterative Methods for Sparse Linear Systems of Equations page 1 of 1 Contents 1 Introduction 1.1 Computer Science Aspects 1.2 Numerical Problems 1.3 Graphs 1.4 Loop Manipulations
More information4.6 Iterative Solvers for Linear Systems
4.6 Iterative Solvers for Linear Systems Why use iterative methods? Virtually all direct methods for solving Ax = b require O(n 3 ) floating point operations. In practical applications the matrix A often
More informationHere is an example of a block diagonal matrix with Jordan Blocks on the diagonal: J
Class Notes 4: THE SPECTRAL RADIUS, NORM CONVERGENCE AND SOR. Math 639d Due Date: Feb. 7 (updated: February 5, 2018) In the first part of this week s reading, we will prove Theorem 2 of the previous class.
More informationIntroduction to Scientific Computing
(Lecture 5: Linear system of equations / Matrix Splitting) Bojana Rosić, Thilo Moshagen Institute of Scientific Computing Motivation Let us resolve the problem scheme by using Kirchhoff s laws: the algebraic
More informationMCMC Sampling for Bayesian Inference using L1-type Priors
MÜNSTER MCMC Sampling for Bayesian Inference using L1-type Priors (what I do whenever the ill-posedness of EEG/MEG is just not frustrating enough!) AG Imaging Seminar Felix Lucka 26.06.2012 , MÜNSTER Sampling
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationIterative Solution methods
p. 1/28 TDB NLA Parallel Algorithms for Scientific Computing Iterative Solution methods p. 2/28 TDB NLA Parallel Algorithms for Scientific Computing Basic Iterative Solution methods The ideas to use iterative
More informationGeneral Construction of Irreversible Kernel in Markov Chain Monte Carlo
General Construction of Irreversible Kernel in Markov Chain Monte Carlo Metropolis heat bath Suwa Todo Department of Applied Physics, The University of Tokyo Department of Physics, Boston University (from
More informationIterative Methods and Multigrid
Iterative Methods and Multigrid Part 3: Preconditioning 2 Eric de Sturler Preconditioning The general idea behind preconditioning is that convergence of some method for the linear system Ax = b can be
More informationGoal: to construct some general-purpose algorithms for solving systems of linear Equations
Chapter IV Solving Systems of Linear Equations Goal: to construct some general-purpose algorithms for solving systems of linear Equations 4.6 Solution of Equations by Iterative Methods 4.6 Solution of
More informationIterative Methods and Multigrid
Iterative Methods and Multigrid Part 1: Introduction to Multigrid 2000 Eric de Sturler 1 12/02/09 MG01.prz Basic Iterative Methods (1) Nonlinear equation: f(x) = 0 Rewrite as x = F(x), and iterate x i+1
More informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More information9. Iterative Methods for Large Linear Systems
EE507 - Computational Techniques for EE Jitkomut Songsiri 9. Iterative Methods for Large Linear Systems introduction splitting method Jacobi method Gauss-Seidel method successive overrelaxation (SOR) 9-1
More informationLecture 16 Methods for System of Linear Equations (Linear Systems) Songting Luo. Department of Mathematics Iowa State University
Lecture 16 Methods for System of Linear Equations (Linear Systems) Songting Luo Department of Mathematics Iowa State University MATH 481 Numerical Methods for Differential Equations Songting Luo ( Department
More informationEfficient Variational Inference in Large-Scale Bayesian Compressed Sensing
Efficient Variational Inference in Large-Scale Bayesian Compressed Sensing George Papandreou and Alan Yuille Department of Statistics University of California, Los Angeles ICCV Workshop on Information
More informationComputational statistics
Computational statistics Markov Chain Monte Carlo methods Thierry Denœux March 2017 Thierry Denœux Computational statistics March 2017 1 / 71 Contents of this chapter When a target density f can be evaluated
More informationBayesian Methods and Uncertainty Quantification for Nonlinear Inverse Problems
Bayesian Methods and Uncertainty Quantification for Nonlinear Inverse Problems John Bardsley, University of Montana Collaborators: H. Haario, J. Kaipio, M. Laine, Y. Marzouk, A. Seppänen, A. Solonen, Z.
More informationNumerical Linear Algebra Primer. Ryan Tibshirani Convex Optimization
Numerical Linear Algebra Primer Ryan Tibshirani Convex Optimization 10-725 Consider Last time: proximal Newton method min x g(x) + h(x) where g, h convex, g twice differentiable, and h simple. Proximal
More informationStabilization and Acceleration of Algebraic Multigrid Method
Stabilization and Acceleration of Algebraic Multigrid Method Recursive Projection Algorithm A. Jemcov J.P. Maruszewski Fluent Inc. October 24, 2006 Outline 1 Need for Algorithm Stabilization and Acceleration
More informationCHAPTER 5. Basic Iterative Methods
Basic Iterative Methods CHAPTER 5 Solve Ax = f where A is large and sparse (and nonsingular. Let A be split as A = M N in which M is nonsingular, and solving systems of the form Mz = r is much easier than
More informationEXAMPLES OF CLASSICAL ITERATIVE METHODS
EXAMPLES OF CLASSICAL ITERATIVE METHODS In these lecture notes we revisit a few classical fixpoint iterations for the solution of the linear systems of equations. We focus on the algebraic and algorithmic
More informationComparison of Fixed Point Methods and Krylov Subspace Methods Solving Convection-Diffusion Equations
American Journal of Computational Mathematics, 5, 5, 3-6 Published Online June 5 in SciRes. http://www.scirp.org/journal/ajcm http://dx.doi.org/.436/ajcm.5.5 Comparison of Fixed Point Methods and Krylov
More informationNumerical Solution Techniques in Mechanical and Aerospace Engineering
Numerical Solution Techniques in Mechanical and Aerospace Engineering Chunlei Liang LECTURE 3 Solvers of linear algebraic equations 3.1. Outline of Lecture Finite-difference method for a 2D elliptic PDE
More informationIntroduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos & Aarti Singh Contents Markov Chain Monte Carlo Methods Goal & Motivation Sampling Rejection Importance Markov
More informationGenerative Models and Stochastic Algorithms for Population Average Estimation and Image Analysis
Generative Models and Stochastic Algorithms for Population Average Estimation and Image Analysis Stéphanie Allassonnière CIS, JHU July, 15th 28 Context : Computational Anatomy Context and motivations :
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationOn Reparametrization and the Gibbs Sampler
On Reparametrization and the Gibbs Sampler Jorge Carlos Román Department of Mathematics Vanderbilt University James P. Hobert Department of Statistics University of Florida March 2014 Brett Presnell Department
More informationMobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti
Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More informationTopics. The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems
Topics The CG Algorithm Algorithmic Options CG s Two Main Convergence Theorems What about non-spd systems? Methods requiring small history Methods requiring large history Summary of solvers 1 / 52 Conjugate
More informationIntroduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Bastian Steder
Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Bastian Steder Reference Book Thrun, Burgard, and Fox: Probabilistic Robotics Vectors Arrays of numbers Vectors represent
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 24: Preconditioning and Multigrid Solver Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 5 Preconditioning Motivation:
More informationIntroduction to Iterative Solvers of Linear Systems
Introduction to Iterative Solvers of Linear Systems SFB Training Event January 2012 Prof. Dr. Andreas Frommer Typeset by Lukas Krämer, Simon-Wolfgang Mages and Rudolf Rödl 1 Classes of Matrices and their
More informationIterative techniques in matrix algebra
Iterative techniques in matrix algebra Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan September 12, 2015 Outline 1 Norms of vectors and matrices 2 Eigenvalues and
More informationComputer Vision Group Prof. Daniel Cremers. 14. Sampling Methods
Prof. Daniel Cremers 14. Sampling Methods Sampling Methods Sampling Methods are widely used in Computer Science as an approximation of a deterministic algorithm to represent uncertainty without a parametric
More informationScalable kernel methods and their use in black-box optimization
with derivatives Scalable kernel methods and their use in black-box optimization David Eriksson Center for Applied Mathematics Cornell University dme65@cornell.edu November 9, 2018 1 2 3 4 1/37 with derivatives
More information1 Conjugate gradients
Notes for 2016-11-18 1 Conjugate gradients We now turn to the method of conjugate gradients (CG), perhaps the best known of the Krylov subspace solvers. The CG iteration can be characterized as the iteration
More informationConjugate gradient method. Descent method. Conjugate search direction. Conjugate Gradient Algorithm (294)
Conjugate gradient method Descent method Hestenes, Stiefel 1952 For A N N SPD In exact arithmetic, solves in N steps In real arithmetic No guaranteed stopping Often converges in many fewer than N steps
More informationSpatial Statistics with Image Analysis. Outline. A Statistical Approach. Johan Lindström 1. Lund October 6, 2016
Spatial Statistics Spatial Examples More Spatial Statistics with Image Analysis Johan Lindström 1 1 Mathematical Statistics Centre for Mathematical Sciences Lund University Lund October 6, 2016 Johan Lindström
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.
More informationComputational Economics and Finance
Computational Economics and Finance Part II: Linear Equations Spring 2016 Outline Back Substitution, LU and other decomposi- Direct methods: tions Error analysis and condition numbers Iterative methods:
More informationLab 1: Iterative Methods for Solving Linear Systems
Lab 1: Iterative Methods for Solving Linear Systems January 22, 2017 Introduction Many real world applications require the solution to very large and sparse linear systems where direct methods such as
More informationM.A. Botchev. September 5, 2014
Rome-Moscow school of Matrix Methods and Applied Linear Algebra 2014 A short introduction to Krylov subspaces for linear systems, matrix functions and inexact Newton methods. Plan and exercises. M.A. Botchev
More informationStability 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
More informationNumerical Methods I Non-Square and Sparse Linear Systems
Numerical Methods I Non-Square and Sparse Linear Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 25th, 2014 A. Donev (Courant
More informationOUTLINE ffl CFD: elliptic pde's! Ax = b ffl Basic iterative methods ffl Krylov subspace methods ffl Preconditioning techniques: Iterative methods ILU
Preconditioning Techniques for Solving Large Sparse Linear Systems Arnold Reusken Institut für Geometrie und Praktische Mathematik RWTH-Aachen OUTLINE ffl CFD: elliptic pde's! Ax = b ffl Basic iterative
More informationA 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
More informationScientific Computing
Scientific Computing Direct solution methods Martin van Gijzen Delft University of Technology October 3, 2018 1 Program October 3 Matrix norms LU decomposition Basic algorithm Cost Stability Pivoting Pivoting
More informationStat. 758: Computation and Programming
Stat. 758: Computation and Programming Eric B. Laber Department of Statistics, North Carolina State University Lecture 4a Sept. 10, 2015 Ambition is like a frog sitting on a Venus flytrap. The flytrap
More informationAn introduction to adaptive MCMC
An introduction to adaptive MCMC Gareth Roberts MIRAW Day on Monte Carlo methods March 2011 Mainly joint work with Jeff Rosenthal. http://www2.warwick.ac.uk/fac/sci/statistics/crism/ Conferences and workshops
More informationSTA 294: Stochastic Processes & Bayesian Nonparametrics
MARKOV CHAINS AND CONVERGENCE CONCEPTS Markov chains are among the simplest stochastic processes, just one step beyond iid sequences of random variables. Traditionally they ve been used in modelling a
More informationMaster Thesis Literature Study Presentation
Master Thesis Literature Study Presentation Delft University of Technology The Faculty of Electrical Engineering, Mathematics and Computer Science January 29, 2010 Plaxis Introduction Plaxis Finite Element
More informationThe convergence of stationary iterations with indefinite splitting
The convergence of stationary iterations with indefinite splitting Michael C. Ferris Joint work with: Tom Rutherford and Andy Wathen University of Wisconsin, Madison 6th International Conference on Complementarity
More informationAPPLIED NUMERICAL LINEAR ALGEBRA
APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface 1 Introduction 1 1.1 Basic Notation
More informationJACOBI S ITERATION METHOD
ITERATION METHODS These are methods which compute a sequence of progressively accurate iterates to approximate the solution of Ax = b. We need such methods for solving many large linear systems. Sometimes
More informationThe Particle Filter. PD Dr. Rudolph Triebel Computer Vision Group. Machine Learning for Computer Vision
The Particle Filter Non-parametric implementation of Bayes filter Represents the belief (posterior) random state samples. by a set of This representation is approximate. Can represent distributions that
More information6. Iterative Methods for Linear Systems. The stepwise approach to the solution...
6 Iterative Methods for Linear Systems The stepwise approach to the solution Miriam Mehl: 6 Iterative Methods for Linear Systems The stepwise approach to the solution, January 18, 2013 1 61 Large Sparse
More informationAnalysis of Iterative Methods for Solving Sparse Linear Systems C. David Levermore 9 May 2013
Analysis of Iterative Methods for Solving Sparse Linear Systems C. David Levermore 9 May 2013 1. General Iterative Methods 1.1. Introduction. Many applications lead to N N linear algebraic systems of the
More informationJae Heon Yun and Yu Du Han
Bull. Korean Math. Soc. 39 (2002), No. 3, pp. 495 509 MODIFIED INCOMPLETE CHOLESKY FACTORIZATION PRECONDITIONERS FOR A SYMMETRIC POSITIVE DEFINITE MATRIX Jae Heon Yun and Yu Du Han Abstract. We propose
More informationLinear Dynamical Systems (Kalman filter)
Linear Dynamical Systems (Kalman filter) (a) Overview of HMMs (b) From HMMs to Linear Dynamical Systems (LDS) 1 Markov Chains with Discrete Random Variables x 1 x 2 x 3 x T Let s assume we have discrete
More informationLecture 11. Fast Linear Solvers: Iterative Methods. J. Chaudhry. Department of Mathematics and Statistics University of New Mexico
Lecture 11 Fast Linear Solvers: Iterative Methods J. Chaudhry Department of Mathematics and Statistics University of New Mexico J. Chaudhry (UNM) Math/CS 375 1 / 23 Summary: Complexity of Linear Solves
More informationSAMPLING ALGORITHMS. In general. Inference in Bayesian models
SAMPLING ALGORITHMS SAMPLING ALGORITHMS In general A sampling algorithm is an algorithm that outputs samples x 1, x 2,... from a given distribution P or density p. Sampling algorithms can for example be
More informationNumerical Linear Algebra Primer. Ryan Tibshirani Convex Optimization /36-725
Numerical Linear Algebra Primer Ryan Tibshirani Convex Optimization 10-725/36-725 Last time: proximal gradient descent Consider the problem min g(x) + h(x) with g, h convex, g differentiable, and h simple
More informationReversible Markov chains
Reversible Markov chains Variational representations and ordering Chris Sherlock Abstract This pedagogical document explains three variational representations that are useful when comparing the efficiencies
More informationTheory of Iterative Methods
Based on Strang s Introduction to Applied Mathematics Theory of Iterative Methods The Iterative Idea To solve Ax = b, write Mx (k+1) = (M A)x (k) + b, k = 0, 1,,... Then the error e (k) x (k) x satisfies
More informationUniversity of Toronto Department of Statistics
Norm Comparisons for Data Augmentation by James P. Hobert Department of Statistics University of Florida and Jeffrey S. Rosenthal Department of Statistics University of Toronto Technical Report No. 0704
More informationA Search and Jump Algorithm for Markov Chain Monte Carlo Sampling. Christopher Jennison. Adriana Ibrahim. Seminar at University of Kuwait
A Search and Jump Algorithm for Markov Chain Monte Carlo Sampling Christopher Jennison Department of Mathematical Sciences, University of Bath, UK http://people.bath.ac.uk/mascj Adriana Ibrahim Institute
More information