Filter based methods for statistical linear inverse problems

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90 of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

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Solving Linear Inverse Problems Using Gan Priors: An Algorithm with Provable Guarantees

2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) ◽

10.1109/icassp.2018.8462233 ◽

2018 ◽

Cited By ~ 19

Author(s):

Viraj Shah ◽

Chinmay Hegde

Keyword(s):

Inverse Problems ◽

Linear Inverse Problems

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On some nonlinear inverse problems in elasticity

Theoretical and Applied Mechanics ◽

10.2298/tam1102125a ◽

2011 ◽

Vol 38 (2) ◽

pp. 125-154 ◽

Cited By ~ 1

Author(s):

S. Andrieux ◽

H.D. Bui

Keyword(s):

Inverse Problems ◽

Closed Form ◽

Variational Equation ◽

Material Constant ◽

Displacement Discontinuity ◽

Crack Plane ◽

Nonlinear Case ◽

Elastic Solids ◽

Linear Inverse Problems ◽

Statics And Dynamics

In this paper, we make a review of some inverse problems in elasticity, in statics and dynamics, in acoustics, thermoelasticity and viscoelasticity. Crack inverse problems have been solved in closed form, by considering a nonlinear variational equation provided by the reciprocity gap functional. This equation involves the unknown geometry of the crack and the boundary data. It results from the symmetry lost between current fields and adjoint fields which is related to their support. The nonlinear equation is solved step by step by considering linear inverse problems. The normal to the crack plane, then the crack plane and finally the geometry of the crack, defined by the support of the crack displacement discontinuity, are determined explicitly. We also consider the problem of a volumetric defect viewed as the perturbation of a material constant in elastic solids which satisfies the nonlinear Calderon?s equation. The nonlinear problem reduces to two successive ones: a source inverse problem and a Volterra integral equation of the first kind. The first problem provides information on the inclusion geometry. The second one provides the magnitude of the perturbation. The geometry of the defect in the nonlinear case is obtained in closed form and compared to the linearized Calderon?s solution. Both geometries, in linearized and nonlinear cases, are found to be the same.

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