The motivation of this work is an inverse problem for the acoustic wave equation, where an array of sensors probes an unknown medium with pulses and measures the scattered waves. The goal of the inversion is to determine from these measurements the structure of the scattering medium, modeled by a spatially varying acoustic impedance function. Many inversion algorithms assume that the mapping from the unknown impedance to the scattered waves is approximately linear. The linearization, known as the Born approximation, is not accurate in strongly scattering media, where the waves undergo multiple reflections before they reach the sensors in the array. Thus, the reconstructions of the impedance have numerous artifacts. The main result of the paper is a novel, linear-algebraic algorithm that uses a reduced order model (ROM) to map the data to those corresponding to the single scattering (Born) model. The ROM construction is based only on the measurements at the sensors in the array. The ROM is a proxy for the wave propagator operator, that propagates the wave in the unknown medium over the duration of the time sampling interval. The output of the algorithm can be input into any off-the-shelf inversion software that incorporates state of the art linear inversion algorithms to reconstruct the unknown acoustic impedance.
The dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data
Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术院数学研究所）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems
Johannes Kepler University Linz（林茨约翰·开普勒大学）；Technical University of Denmark（丹麦技术大学）；University of Vienna（维也纳大学）；Johann Radon Institute Linz（林茨约翰·拉顿研究所）
