A parametric level set based collage method for an inverse problem in elliptic partial differential equations
| Autor: | Xiaoliang Cheng, Ye Zhang, Guangliang Lin |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Work (thermodynamics)
Collage theorem Beräkningsmatematik 02 engineering and technology 01 natural sciences Level set Regularization 0202 electrical engineering electronic engineering information engineering Applied mathematics 0101 mathematics Parametric level set Parametric statistics Mathematics Total variation Applied Mathematics Computational mathematics Inverse problem Partial differential equations 010101 applied mathematics Discontinuity (linguistics) Computational Mathematics Elliptic partial differential equation 020201 artificial intelligence & image processing |
| Popis: | In this work, based on the collage theorem, we develop a new numerical approach to reconstruct the locations of discontinuity of the conduction coefficient in elliptic partial differential equations (PDEs) with inaccurate measurement data and coefficient value. For a given conductivity coefficient, one can construct a contraction mapping such that its fixed point is just the gradient of a solution to the elliptic system. Therefore, the problem of reconstructing a conductivity coefficient in PDEs can be considered as an approximation of the observation data by the fixed point of a contraction mapping. By collage theorem, we translate it to seek a contraction mapping that keeps the observation data as close as possible to itself, which avoids solving adjoint problems when applying the gradient descent method to the corresponding optimization problem. Moreover, the total variation regularizing strategy is applied to tackle the ill-posedness and the parametric level set technique is adopted to represent the discontinuity of the conductivity coefficient. Various numerical simulations are given to show the efficiency of the proposed method. Funding Agencies:Alexander von Humboldt-StiftungNational Natural Science Foundation of China (NSFC) 11571311 |
| Databáze: | OpenAIRE |
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