Numerical Methods for Large-Scale Ill-Posed Inverse Problems - PowerPoint PPT Presentation

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Numerical Methods for Large-Scale Ill-Posed Inverse Problems Julianne Chung University of Maryland Collaborators: James G. Nagy (Emory) Eldad Haber (Emory) Dianne O’Leary (University of Maryland) Ioannis Sechopoulos (Emory) Chao Yang (Lawrence Berkeley National Laboratory) Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an inverse problem? Forward Model Physical System Input Signal Output Signal Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an inverse problem? Forward Model Physical System Input Signal Output Signal Inverse Problem Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Image Deblurring Given: Blurred image and some information about the blurring Goal: Compute approximation of true image Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Super-Resolution Imaging 1−th low resolution image Given: LR images and some information about the motion parameters Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Super-Resolution Imaging Given: LR images and some information about the motion parameters Goal: Improve parameters and approximate HR image Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Application: Tomographic Imaging Given: 2D projection images Goal: Approximate 3D volume Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an Ill-Posed Inverse Problem? Hadamard (1923): A problem is ill-posed if the solution does not exist, is not unique, or does not depend continuously on the data. Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks What is an Ill-Posed Inverse Problem? Hadamard (1923): A problem is ill-posed if the solution does not exist, is not unique, or does not depend continuously on the data. True image: Blurred & noisy image: Forward Problem Invers e Problem Inverse Solution: Naive inverse solution is corrupted with noise! Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Outline Regularization for Least Squares Systems 1 High Performance Implementation 2 Polyenergetic Tomosynthesis 3 Concluding Remarks 4 Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Outline Regularization for Least Squares Systems 1 High Performance Implementation 2 Polyenergetic Tomosynthesis 3 Concluding Remarks 4 Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks The Linear Problem b = Ax + ε where x ∈ R n - true data A ∈ R m × n - large, ill-conditioned matrix ε ∈ R m - noise, statistical properties may be known b ∈ R m - known, observed data Goal: Given b and A , compute approximation of x Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Regularization Tikhonov Regularization � b � A � � � � � � || b − Ax || 2 2 + λ 2 || Lx || 2 � � min ⇔ min − x � � 2 0 λ L x x � � 2 Selecting a good regularization parameter, λ , is difficult Discrepancy Principle Generalized Cross-Validation L-curve Difficult for large-scale problems Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 Relative Error 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 20 40 60 80 100 120 140 Iteration Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 1 Relative Error 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 0.3 20 40 60 80 100 120 140 Iteration Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 1 Relative Error 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 0.3 20 40 60 80 100 120 140 Iteration Iteration 28 Solution gets better Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Regularization for Least Squares Systems High Performance Implementation The Linear Problem: b = Ax + ε Polyenergetic Tomosynthesis The Nonlinear Problem: b = A ( y ) x + ε Concluding Remarks Illustration of Semi-convergence Behavior min x � b − Ax � 2 Iteration 0 Typical Behavior for Ill-Posed Problems C G 1.1 Relative Error 1 0.9 0.8 0.7 0.6 0.5 0.4 Iteration 10 Iteration 85 0.3 20 40 60 80 100 120 140 Iteration Iteration 28 Solution gets better Julianne Chung Numerical Methods for Large-Scale Ill-Posed Inverse Problems

Numerical Methods for Large-Scale Ill-Posed Inverse Problems - PowerPoint PPT Presentation

Regularization for Least Squares Systems High Performance Implementation Polyenergetic Tomosynthesis Concluding Remarks Numerical Methods for Large-Scale Ill-Posed Inverse Problems Julianne Chung University of Maryland Collaborators: James

Cascadic Multilevel Methods for Cascadic Multilevel Methods for Large-Scale Ill-Posed Problems

Numerical Methods for Ill-Posed Problems III Lothar Reichel Summer School on Applied Analysis TU

Numerical Methods for Ill-Posed Problems I Lothar Reichel Summer School on Applied Analysis TU

Numerical Methods for Linear Discrete Ill-Posed Problems Lothar Reichel Como, May 2018 Part 1:

Numerical Methods for Solving Large Scale Eigenvalue Problems Lecture 2, February 28, 2018:

Linear and Nonlinear SP 2 Methods for Large Scale Eigenvalue Calculations Zhaojun Bai

On iterative regularization methods for nonlinear inverse problems Antonio Leito

C++ Concepts for C++ Concepts for Ill-posed Inverse Problems Ill-posed Inverse Problems Or: How

A Randomized Likelihood Method for Data Reduction in Large-scale Inverse Problems Ellen B. Le

Optimal Experimental Design for Large-Scale Bayesian Inverse Problems via Multi-PDE-Constrained

Subspace Regularization for Large Linear Discrete Ill-Posed Problems 8th GAMM Workshop Applied

Survey of Fast Methods for large-scale tree estimation J E S U S S A N D O V A L Introduction

Time-space adaptive numerical methods for multi-scale reaction waves simulation M. Duarte 1 M.

PARALLEL LARGE-SCALE SIMULATION OF VISCOELASTIC FLUID FLOW INSTABILITIES Mehmet SAHIN 17 th

Open tools and methods for large scale segmentation of Very High Resolution satellite images

SDEs in large dimension and numerical methods Part 1: Sampling the canonical distribution T.

Large-Scale Numerical Simulation of Fluid Structure Interactions in Low Reynolds Number Flows APS

Numerical algorithms for large-scale Hamiltonian eigenproblems Peter Benner Professur Mathematik

Compressive sensing principles and iterative sparse recovery for inverse and ill-posed problems

Inverse Problems Recovering x 0 R N from noisy observations y = x 0 + w R P Inverse

Numerical qualitative analysis of a large-scale model for measles spread Hossein Zivari-Piran

SDEs in large dimension and numerical methods Part 2: Sampling metastable dynamics T. Lelivre

INDAM intensive period Computational Methods for Inverse Problems in Imaging Conclusive

FINANCING LARGE SCALE SOLAR Large Scale Solar Conference - Sydney Gloria Chan Director, Large