A fast subspace optimization method for nonlinear inverse problems in Banach spaces with an application in parameter identification

Wald, Anne

doi:10.1088/1361-6420/aac8f3

Cited by 21 publications

(31 citation statements)

References 22 publications

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“…The latter properties are required for many regularisation techniques that are used to find a stable solution of the usually ill-posed parameter identification problems. Examples are the classical Landweber method [16], Tikhonov regularisation [14], Gauss-Newton methods [18,25], or sequential subspace optimisation techniques [30,31]. An overview of suitable techniques can be found in [10,13,20,27].…”

Section: Introductionmentioning

confidence: 99%

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Hoffmann¹,

Wald²

2020

Preprint

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Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography.

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Section: Introductionmentioning

confidence: 99%

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Hoffmann¹,

Wald²

2020

Preprint

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“…An overview of different methods for solving inverse problems and deeper insights to regularization theory can be found in the standard textbooks [9,23,25,30]. For nonlinear problems, such as the one presented in this article, these methods have to be adapted correspondingly (see [22,30,39,40]).…”

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confidence: 99%

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

Rothermel¹,

Panfilenko²,

Sharma³

et al. 2021

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Mathematical models describing the behavior of viscoelastic materials are often based on evolution equations that measure the change in stress depending on its material parameters such as stiffness, viscosity or relaxation time. In this article, we introduce a Maxwell-based rheological model, define the associated forward operator and the inverse problem in order to determine the number of Maxwell elements and the material parameters of the underlying viscoelastic material. We perform a relaxation experiment by applying a strain to the material and measure the generated stress. Since the measured data varies with the number of Maxwell elements, the forward operator of the underlying inverse problem depends on parts of the solution. By introducing assumptions on the relaxation times, we propose a clustering algorithm to resolve this problem. We provide the calculations that are necessary for the minimization process and conclude with numerical results by investigating unperturbed as well as noisy data. We present different reconstruction approaches based on minimizing a least squares functional. Furthermore, we look at individual stress components to analyze different displacement rates. Finally, we study reconstructions with shortened data sets to obtain assertions on how long experiments have to be performed to identify conclusive material parameters.

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“…Standard methods such as the Landweber iteration scheme prove to be tremendously slow when applied to such a high-dimensional nonlinear inverse problem. To increase numerical efficiency Sequential Subspace Optimization (SESOP) techniques have been developed and analyzed for various settings, see [18,22,23,30,32]. The general idea is to reduce the number of iterations until the stopping criterion is fulfilled.…”

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confidence: 99%

“…The calculation of the projection yields a regulation of the step widths. This technique has been successfully applied, for example in parameter identification [24,31,32], demonstrating a significant increase in efficiency. In addition, they have been used and analyzed in combination with, e.g., sparsity constraints, total variation or Nesterov methods [10,16,29].…”

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confidence: 99%

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Sequential subspace optimization for recovering stored energy functions in hyperelastic materials from time-dependent data

Klein¹,

Schuster²,

Wald³

2019

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Monitoring structures of elastic materials for defect detection by means of ultrasound waves (Structural Health Monitoring, SHM) demands for an efficient computation of parameters which characterize their mechanical behavior. Hyperelasticity describes a nonlinear elastic behavior where the second Piola-Kirchhoff stress tensor is given as a derivative of a scalar function representing the stored (strain) energy. Since the stored energy encodes all mechanical properties of the underlying material, the inverse problem of computing this energy from measurements of the displacement field is very important regarding SHM. The mathematical model is represented by a high-dimensional parameter identification problem for a nonlinear, hyperbolic system with given initial and boundary values. Iterative methods for solving this problem, such as the Landweber iteration, are very timeconsuming. The reason is the fact that such methods demand for several numerical solutions of the hyperbolic system in each iteration step. In this contribution we present an iterative method based on sequential subspace optimization (SESOP) which in general uses more than only one search direction per iteration and explicitly determines the step size. This leads to a significant acceleration compared to the Landweber method, even with only one search direction and an optimized step size. This is demonstrated by means of several numerical tests.

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A fast subspace optimization method for nonlinear inverse problems in Banach spaces with an application in parameter identification

Cited by 21 publications

References 22 publications

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

A method for determining the parameters in a rheological model for viscoelastic materials by minimizing Tikhonov functionals

Sequential subspace optimization for recovering stored energy functions in hyperelastic materials from time-dependent data

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