Proximal point methods for inverse problems

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Title: Proximal point methods for inverse problems
Author: Paulhamus, Marc
Abstract: Numerous mathematical models in applied mathematics can be expressed as a partial differential equation involving certain coefficients. These coefficients are known and they describe some physical properties of the model. The direct problem in this context is to solve the partial differential equation. By contrast, an inverse problem asks for the identification of the variable coefficients when a certain measurement of a solution of the partial differential equation is available. One of the most commonly used approaches for solving this inverse problem is by posing a constrained minimization problem which can be written as a variational inequality. This paper investigates the inverse problem of identifying certain material parameters in the fourth-order partial differential equations representing the beam and plate models. This inverse problem has attracted a great deal of attention in recent years and has found numerous applications. Since the numerical treatment of the fourth-order problems is rather challenging, the first part of the paper describes in detail the finite element approach for solving the direct problem. The inverse problem is solved by posing an optimization problem whose solution is an approximation of the parameters sought. The optimization problem is solved by gradient based approaches, and in this setting, the most challenging aspect is the computations of the gradient of the objective function. We present a detailed treatment of three approaches to compute the gradient, namely, the so-called adjoint method, adjoint stiffness based approach, and the classical gradient computation. We also present a comparison among these different ways of gradient computation. We use different proximal point methods to solve the inverse problem. It is known that proximal point method is a regularization method which has a significantly different behavior than the well-known Tikhonov regularization method. We present a detailed comparative analysis of the numerical efficiency of several proximal point methods.
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Date: 2011-08-26

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