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Register a new userTheoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction
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In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Holder stability with both partial boundary and interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.
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The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that makes it possible to study the inverse problem in all dimensions $n+1geq 3$. We consider the case when the set $Omega_{textrm{in}}$, where the sources are supported, and the set $Omega_{textrm{out}}$, where the observations are made, are separated. As model problems we study both a quasi-linear and also 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 first half we study multiple-fold linearization of the non-linear wave equation near real parts of Gaussian beams that results in a three-wave interaction. We show that the three-wave interaction can produce a three-to-one scattering data. In the second half of the paper, we study an abstract formulation of the three-to-one scattering relation showing that it recovers the topological, differential and conformal structures of the manifold in a causal diamond set that is the intersection of the future of the point $p_{in}in Omega_{textrm{in}}$ and the past of the point $p_{out}in Omega_{textrm{out}}$. The results do not require any assumptions on the conjugate or cut points.
In this article, we investigate inverse source problems for a wide range of PDEs of parabolic and hyperbolic types as well as time-fractional evolution equations by partial interior observation. Restricting the source terms to the form of separated variables, we establish uniqueness results for simultaneously determining both temporal and spatial components without non-vanishing assumptions at $t=0$, which seems novel to the best of our knowledge. Remarkably, mostly we allow a rather flexible choice of the observation time not necessarily starting from $t=0$, which fits into various situations in practice. Our main approach is based on the combination of the Titchmarsh convolution theorem with unique continuation properties and time-analyticity of the PDEs under consideration.
We develop a version of Haar and Holmgren methods which applies to discontinuous solutions of nonlinear hyperbolic systems and allows us to control the L1 distance between two entropy solutions. The main difficulty is to cope with linear hyperbolic systems with discontinuous coefficients. Our main observation is that, while entropy solutions contain compressive shocks only, the averaged matrix associated with two such solutions has compressive or undercompressive shocks, but no rarefaction-shocks -- which are recognized as a source for non-uniqueness and instability. Our Haar-Holmgren-type method rests on the geometry associated with the averaged matrix and takes into account adjoint problems and wave cancellations along generalized characteristics. It extends the method proposed earlier by LeFloch et al. for genuinely nonlinear systems. In the present paper, we cover solutions with small total variation and a class of systems with general flux that need not be genuinely nonlinear and includes for instance fluid dynamics equations. We prove that solutions generated by Glimm or front tracking schemes depend continuously in the L1 norm upon their initial data, by exhibiting an L1 functional controling the distance between two solutions.
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.
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