A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem

Benjamin Crestel, Alen Alexanderian, Georg Stadler, Omar Ghattas

Research output: Contribution to journalArticle

Abstract

The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.

Original languageEnglish (US)
Article number074008
JournalInverse Problems
Volume33
Issue number7
DOIs
StatePublished - Jun 21 2017

Fingerprint

Nonlinear Inverse Problems
Hermann Von Helmholtz
Inverse problems
Inverse Problem
Encoding
Experiment
Computational Cost
Costs
Linearly
Discretization
Experiments
Infinite Dimensions
Linear Combination
Numerical methods
Inversion
Objective function
Numerical Methods
Optimise
Vary
Monitoring

Keywords

  • A-optimal experimental design
  • Bayesian nonlinear inverse problem
  • Helmholtz equation
  • randomized trace estimator
  • source encoding

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics

Cite this

A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem. / Crestel, Benjamin; Alexanderian, Alen; Stadler, Georg; Ghattas, Omar.

In: Inverse Problems, Vol. 33, No. 7, 074008, 21.06.2017.

Research output: Contribution to journalArticle

@article{3c61255305524cd4a2dce597ae935b25,
title = "A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem",
abstract = "The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.",
keywords = "A-optimal experimental design, Bayesian nonlinear inverse problem, Helmholtz equation, randomized trace estimator, source encoding",
author = "Benjamin Crestel and Alen Alexanderian and Georg Stadler and Omar Ghattas",
year = "2017",
month = "6",
day = "21",
doi = "10.1088/1361-6420/aa6d8e",
language = "English (US)",
volume = "33",
journal = "Inverse Problems",
issn = "0266-5611",
publisher = "IOP Publishing Ltd.",
number = "7",

}

TY - JOUR

T1 - A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem

AU - Crestel, Benjamin

AU - Alexanderian, Alen

AU - Stadler, Georg

AU - Ghattas, Omar

PY - 2017/6/21

Y1 - 2017/6/21

N2 - The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.

AB - The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.

KW - A-optimal experimental design

KW - Bayesian nonlinear inverse problem

KW - Helmholtz equation

KW - randomized trace estimator

KW - source encoding

UR - http://www.scopus.com/inward/record.url?scp=85021739555&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021739555&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/aa6d8e

DO - 10.1088/1361-6420/aa6d8e

M3 - Article

AN - SCOPUS:85021739555

VL - 33

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 7

M1 - 074008

ER -