Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers
Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers
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具有不相交源和接收器的非线性双曲方程的反问题
DOI：
--
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
L. Oksanen
中科院分区：
文献类型：
--
作者：
A. Feizmohammadi;M. Lassas;L. Oksanen
Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension
$n+1\geqslant 3$
and with partial data. We consider the case when the set
$\Omega _{\mathrm{in}}$
, where the sources are supported, and the set
$\Omega _{\mathrm{out}}$
, where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point
$p_{in}\in \Omega _{\mathrm{in}}$
and the past of the point
$p_{out}\in \Omega _{\mathrm{out}}$
. In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in
$\Omega _{\mathrm{in}}$
and observations in
$\Omega _{\mathrm{out}}$
, determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.
Abstract The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension
$n+1\geqslant 3$
and with partial data. We consider the case when the set
$\Omega _{\mathrm{in}}$
, where the sources are supported, and the set
$\Omega _{\mathrm{out}}$
, where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point
$p_{in}\in \Omega _{\mathrm{in}}$
and the past of the point
$p_{out}\in \Omega _{\mathrm{out}}$
. In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in
$\Omega _{\mathrm{in}}$
and observations in
$\Omega _{\mathrm{out}}$
, determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.
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拉普拉斯连接的反演问题
DOI：
10.48550/arxiv.1509.02645
发表时间：
2015
期刊：
arXiv e-prints
影响因子：
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作者：
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