Finite-dimensional parametric problems in inverse problems of astrophysics

1990 ◽  
Vol 1 (2) ◽  
pp. 169-179
Author(s):  
A. V. Goncharskii ◽  
S. Yu. Romanov ◽  
V. V. Stepanov ◽  
A. M. Cherepashchuk
Keyword(s):  
Inverse Problems ◽  
Finite Dimensional ◽  
Parametric Problems

scholarly journals Exact solutions of infinite dimensional total-variation regularized problems

2018 ◽  
Vol 8 (3) ◽  
pp. 407-443 ◽  
Author(s):  
Axel Flinth ◽  
Pierre Weiss
Keyword(s):  
Inverse Problems ◽  
Exact Solutions ◽  
Total Variation ◽  
Linear Mapping ◽  
Scattered Data Approximation ◽  
Convex Programs ◽  
Linear Measurements ◽  
Data Approximation ◽  
Infinite Dimensional ◽  
Finite Dimensional

Abstract We study the solutions of infinite dimensional inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary function. The first contribution describes the solution’s structure: we show that under mild assumptions, there always exists an $m$-sparse solution, where $m$ is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exact solutions of the infinite dimensional problem can be obtained by solving one or two consecutive finite dimensional convex programs depending on the measurement functions structures. We finish by showing an application on scattered data approximation. These results extend recent advances in the understanding of total-variation regularized inverse problems.

scholarly journals Convergence of a series associated with the convexification method for coefficient inverse problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Michael V. Klibanov ◽  
Dinh-Liem Nguyen
Keyword(s):  
Fourier Series ◽  
Inverse Problems ◽  
Convergence Property ◽  
Convergence Result ◽  
Orthogonal Projections ◽  
Finite Dimensional ◽  
Coefficient Inverse Problems ◽  
Ill Posed ◽  
Fourier Series Approximation ◽  
Truncated Fourier Series

AbstractThis paper is concerned with the convergence of a series associated with a certain version of the convexification method. That version has been recently developed by the research group of the first author for solving coefficient inverse problems. The convexification method aims to construct a globally convex Tikhonov-like functional with a Carleman weight function in it. In the previous works, the construction of the strictly convex weighted Tikhonov-like functional assumes a truncated Fourier series (i.e. a finite series instead of an infinite one) for a function generated by the total wave field. In this paper, we prove a convergence property for this truncated Fourier series approximation. More precisely, we show that the residual of the approximate PDE obtained by using the truncated Fourier series tends to zero in L^{2} as the truncation index in the truncated Fourier series tends to infinity. The proof relies on a convergence result in the H^{1}-norm for a sequence of L^{2}-orthogonal projections on finite-dimensional subspaces spanned by elements of a special Fourier basis. However, due to the ill-posed nature of coefficient inverse problems, we cannot prove that the solution of that approximate PDE, which results from the minimization of that Tikhonov-like functional, converges to the correct solution.

scholarly journals On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

2021 ◽  
pp. 1-23
Author(s):  
Noè Angelo Caruso ◽  
Alessandro Michelangeli ◽  
Paolo Novati
Keyword(s):  
Inverse Problems ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Projection Methods ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Galerkin Truncation ◽  
General Projection ◽  
Theoretical Results

In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov–Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

scholarly journals Effective Solution of Ill-Posed Inverse Problems with Stabilized Forward Solver

2021 ◽  
pp. 343-357
Author(s):  
Marcin Łoś ◽  
Robert Schaefer ◽  
Maciej Smołka
Keyword(s):  
Inverse Problems ◽  
Galerkin Method ◽  
Computational Cost ◽  
Forward Problem ◽  
Effective Solution ◽  
Numerical Example ◽  
Parametric Problems ◽  
Ill Posed ◽  
The Common ◽  
Forward Error

AbstractWe consider inverse parametric problems for elliptic variational PDEs. They are solved through the minimization of misfit functionals. Main difficulties encountered consist in the misfit multimodality and insensitivity as well as in the weak conditioning of the direct (forward) problem, that therefore requires stabilization. A complex multi-population memetic strategy hp-HMS combined with the Petrov-Galerkin method stabilized by the Demkowicz operator is proposed to overcome obstacles mentioned above. This paper delivers the theoretical motivation for the common inverse/forward error scaling, that can reduce significantly the computational cost of the whole strategy. A short illustrative numerical example is attached at the end of the paper.

The stability of finite-dimensional inverse problems

1995 ◽  
Vol 11 (4) ◽  
pp. 889-911 ◽  
Author(s):  
H J S Dorren ◽  
R K Snieder
Keyword(s):  
Inverse Problems ◽  
Finite Dimensional ◽  
The Stability

On the clustering of stationary points of Tikhonov’s functional for conditionally well-posed inverse problems

2020 ◽  
Vol 28 (5) ◽  
pp. 713-725
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Inverse Problems ◽  
Small Neighborhood ◽  
Stationary Points ◽  
Conditional Stability ◽  
Finite Dimensional ◽  
Tikhonov’S Scheme ◽  
Dimensional Version ◽  
Dimensional Section ◽  
Well Posed ◽  
Solution Estimates

AbstractIn a Hilbert space, we consider a class of conditionally well-posed inverse problems for which the Hölder type estimate of conditional stability on a bounded closed and convex subset holds. We investigate a finite-dimensional version of Tikhonov’s scheme in which the discretized Tikhonov’s functional is minimized over the finite-dimensional section of the set of conditional stability. For this optimization problem, we prove that each its stationary point that is located not too far from the desired solution of the original inverse problem in reality belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of discretization errors and error level in input data are also given.

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