Numerical methods for experimental design of large-scale linear ill-posed inverse problems

Haber, Eldad; Horesh, Lior; Tenorio, Luis

doi:10.1088/0266-5611/24/5/055012

Cited by 131 publications

(195 citation statements)

References 27 publications

(43 reference statements)

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“…For underdetermined problems, the convex relaxed l 1 -norm problem provably finds a near-optimal solution also with respect to l 0 , i.e., the sparsest solution [4,11,12]; we refer to [29] for an overview of numerical methods employed in this context. In a different interpretation of the sparsity approach, the authors in [15] find optimal experimental designs for linear inverse problems. Here, a sparsity term is used to rule out experiments that do not contribute significant information for the inversion.…”

Section: Introduction Optimal Control Problems With Control Costs Of Lmentioning

confidence: 99%

Directional Sparsity in Optimal Control of Partial Differential Equations

Herzog¹,

Stadler²,

Wachsmuth³

2012

SIAM J. Control Optim.

110

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Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For time-dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses a nondifferentiable cost functional to implement the sparsity requirements; additionally, bound constraints for the optimal controls can be included. We study the resulting problem in appropriate function spaces and present two solution methods of Newton type, based on different formulations of the optimality system. Using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms.

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Section: Introduction Optimal Control Problems With Control Costs Of Lmentioning

confidence: 99%

Directional Sparsity in Optimal Control of Partial Differential Equations

Herzog¹,

Stadler²,

Wachsmuth³

2012

SIAM J. Control Optim.

110

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“…It is also worth remarking that our approach of improving the prior based on the statistics of the ground-truth is different from recent approaches that optimise the prior based on the denoising result [9,16,15,4,6]. These approaches can provide improved results in practise, but no longer have the simple MAP interpretation.…”

Section: Image Statistics and The Jump Partmentioning

confidence: 90%

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

Hintermüller

Valkonen

2015

SIAM J. Imaging Sci.

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Recently, non-convex regularisation models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies ϕ in the total variation type functional TV ϕ (u) := ϕ(|∇u(x)|) dx. In this paper, it is demonstrated that for typical choices of ϕ, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, BV(Ω). In particular, if ϕ(t) = t q for q ∈ (0, 1) and TV ϕ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to BV(Ω), then still TV ϕ (u) = ∞ for u not piecewise constant. If, on the other hand, TV ϕ is defined analogously via continuously differentiable functions, then TV ϕ ≡ 0, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy ϕ(t) = t q is linearised for high values. The fact, that this kind of energies actually better matches reality and improves reconstructions, is demonstrated by statistics and numerical experiments.Mathematics subject classification: 26B30, 49Q20, 65J20.

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“…Other than recent numerous exceptions (see [8,5,12,15,9,14] and references therein), optimal experimental design of ill-posed problems has been somewhat an under-researched topic. In the case of ill-posed problems, the selection of optimal weights for (8) is more difficult because this estimate is biased and its bias depends on the unknown m: bias( m) = −λ 2 C(w) −1 M m, where the inverse Fisher matrix is given by…”

Section: The Ill-posed Casementioning

confidence: 99%

“…For example, let us return to the well-posed parameter estimation example with Aj = SL −1 Qj. The penalized least-squares estimate (8…”

Section: Introductionmentioning

confidence: 99%

Optimal design of simultaneous source encoding

Haber

Doel

Horesh

2014

Inverse Problems in Science and Engineering

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A broad range of parameter estimation problems involve the collection of an excessively large number of observations N . Typically each such observation involves excitation of the domain through injection of energy at some pre-defined sites and recording of the response of the domain at another set of locations. It has been observed that similar results can often be obtained by considering a far smaller number K of multiple linear superpositions of experiments with K << N . This allows the construction of the solution to the inverse problem in time O(K) instead of O(N ). Given these considerations it should not be necessary to perform all the N experiments but only a much smaller number of K experiments with simultaneous sources in superpositions with certain weights. Devising such procedure would results in a drastic reduction in acquisition time.The question we attempt to rigorously investigate in this work is: what are the optimal weights ? We formulate the problem as an optimal experimental design problem and show that by leveraging techniques from this field an answer is readily available. Designing optimal experiments requires some statistical framework and therefore the statistical framework that one chooses to work with plays a major role in the selection of the weights.

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Numerical methods for experimental design of large-scale linear ill-posed inverse problems

Cited by 131 publications

References 27 publications

Directional Sparsity in Optimal Control of Partial Differential Equations

Directional Sparsity in Optimal Control of Partial Differential Equations

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

Optimal design of simultaneous source encoding

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