Local Misfit Approximation in Memetic Solving of Ill-Posed Inverse Problems

Łoś, Marcin; Schaefer, Robert; Sawicki, Jakub; Smółka, Maciej

doi:10.1007/978-3-319-55849-3_20

Cited by 3 publications

(6 citation statements)

References 13 publications

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“…The above theory supports the stochastic strategies of multimodal analysis gathered in [14,19]. It may be also used to study the features of a Clustered Genetic Search (CGS) [21,27] and Hierarchic Memetic Strategy (HMS) dedicated for approximating lowlands and their basins of attraction [15,16]. The solving of real-world engineering ill-posed inverse problems from Classes C.6 -C.9 by the methods mentioned above is presented in papers [2,3,11,22,23].…”

Section: Conclusion and Perspective Of Applicationsmentioning

confidence: 63%

“…Proof. First, from the Picard-Lindelöf theorem, we know that, in our case, every point from D belongs to exactly one trajectory of (16). Of course, this trajectory can converge to one point at most.…”

Section: Separability Of Attraction Sets and Basins Of Attractionmentioning

confidence: 85%

“…If there was a solution of (16) starting from a point different than y, then this solution would have common points with A; however, this is of course impossible since all elements of A are stationary points for (16) and any solution would get stuck in any of them. Therefore, R y does not contain points other than y.…”

Section: Sets and Basins Of Attractionmentioning

confidence: 99%

“…In the latter case, there are points in a neighborhood of y with a value of f less than f (y). But those points cannot belong to R y because the solutions of (16) have decreasing values of f for increasing times. The other elements of the neighborhood are stationary points; hence, also in this case,…”

Section: Sets and Basins Of Attractionmentioning

confidence: 99%

“…We suggest to avoid the term "plateau", as it is less concrete than the proposed terms of lowland, upland, and shelf. Even though the term is used in the literature (see [16,17,19]), it introduces some ambiguity.…”

Section: Introductionmentioning

confidence: 99%

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Misfit Landforms Imposed by Ill-Conditioned Inverse Parametric Problems

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In this paper, we put forward a new topological taxonomy that allows us to distinguish and separate multiple solutions to ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters called the landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We have proven their important separability and completeness conditions. In particular, lowlands, uplands, and shelves are pairwise disjoint, and there are no other subsets of the positive measure in the admissible domain on which the misfit function takes a constant value. The topological taxonomy is related to the second, "local" one, which characterizes the types of ill-conditioning of the particular solutions. We hope that the proposed results will be helpful for a better and more precise formulation of ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used in solving these problems.

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Section: Conclusion and Perspective Of Applicationsmentioning

confidence: 63%

Section: Separability Of Attraction Sets and Basins Of Attractionmentioning

confidence: 85%

Section: Sets and Basins Of Attractionmentioning

confidence: 99%

Section: Sets and Basins Of Attractionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Misfit Landforms Imposed by Ill-Conditioned Inverse Parametric Problems

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et al. 2018

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Two-Phase Strategy Managing Insensitivity in Global Optimization

Sawicki¹,

Smółka²,

Łoś³

et al. 2017

Applications of Evolutionary Computation

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Approximating landscape insensitivity regions in solving ill-conditioned inverse problems

et al. 2018

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Solving ill-posed continuous, global optimization problems is challenging. No well-established methods are available to handle the objective intensity that appears when studying the inversion of non-invasive tumor tissue diagnosis or geophysical applications. The paper presents a complex metaheuristic method that identifies regions of objective function's insensitivity (plateaus). It is composed of a multi-deme hierarchic memetic strategy coupled with random sample clustering, cluster integration, and a special kind of local evolution processes using the multiwinner selection that allows to breed the demes to cover each plateau separately. The final phase consists in a smooth local objective approximation which determines the shape of the plateaus by analyzing the objective level sets. We test the method on benchmarks with multiple non-convex plateaus and in an actual geophysical application of magnetotelluric data inversion.

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Local Misfit Approximation in Memetic Solving of Ill-Posed Inverse Problems

Cited by 3 publications

References 13 publications

Misfit Landforms Imposed by Ill-Conditioned Inverse Parametric Problems

Misfit Landforms Imposed by Ill-Conditioned Inverse Parametric Problems

Two-Phase Strategy Managing Insensitivity in Global Optimization

Approximating landscape insensitivity regions in solving ill-conditioned inverse problems

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