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

N1 - Funding Information:
This research was partially supported by DOE grants DE-SC0009286 and DE-SC0010518, and NSF grants CBET-1508713 and CBET-150700.
Publisher Copyright:
© 2017 IOP Publishing Ltd.

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

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U2 - 10.1088/1361-6420/aa6d8e

DO - 10.1088/1361-6420/aa6d8e

M3 - Article

AN - SCOPUS:85021739555

SN - 0266-5611

VL - 33

JO - Inverse Problems

JF - Inverse Problems

IS - 7

M1 - 074008

ER -