Inverse Problems for Partial Differential Equations

By Y. Belov. 2002

789 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/fc0a2961ca4623530222c698419f88fcf160d7ae

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In Mathematical Physics

Citations in the corpus

Sensitivity analysis of an inverse problem for the wave equation with caustics (2012, result): “Moreover, our result is fundamentally different from those in the literature where Lipschitz, Hölder or logarithmic estimates are derived; see, for example, [6], [1], [33], [25], [26], [27], [28], [19] and [14].”
Fourier regularization for a backward heat equation (2007, background): “The BHCP is a typical ill-posed problem [2,3].”
Optimal Transport for Seismic Full Waveform Inversion (2016, background): “For additional details on uniqueness and stability, we refer to [10, 15, 16].”
Global uniqueness of a multidimensional inverse problem for a nonlinear parabolic equation by a Carleman estimate (2004, background): “In the rest of publications about this method, it has been used so far for proofs of uniqueness and stability results for multidimensional inverse problems for linear PDEs only (including systems of PDEs), see, e.g., Imanuvilov and Yamamoto [5], Imanuvilov, Isakov and Yamamoto [6], Klibanov [11,13,1”
Cloaking via change of variables in electric impedance tomography (2008, neutral): “We have given just a few of the most important references; for more complete surveys, see [11, 21, 43, 44].”
Parameter estimation for the stochastically perturbed Navier-Stokes equations (2010, methodology): “We note that different types of methods and approaches are used to study inverse problems for deterministic PDEs, and we refer the reader to [20, 21] and references therein.”
Electrical impedance tomography (2002, neutral): “This problem is much more difficult than EIT, and there are few theoretical results (see for example [93]).”
Why is the Cauchy problem severely ill-posed? (2007, background): “When Cauchy’s problem has a solution (it is unique by Holmgren’s theorem, see [17]), consistency is to be understood trivially: the solution of the variational equation also satisfies the Cauchy problem and the uniqueness is preserved.”
Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping (2007, methodology): “Typical tools such as Carleman’s estimates [27] do not apply (due to non-locality of von Karman brackets).”
Uniqueness and stability in an inverse problem for the Schrödinger equation (2007, positive): “ISAKOV in [4] uses local Carleman estimates for the Schrödinger equation to prove uniqueness of the solution.”
An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem (2000, neutral): “For generalizations of the above results to more general spaces and other related work see the references given in [15, 23, 27].”
CONTROLLABILITY OF PARTIAL DIFFERENTIAL EQUATIONS AND ITS SEMI-DISCRETE APPROXIMATIONS (2006, positive): “Note however that controllability is not the only application of the observability inequalities, which are also of systematic use in the context of Inverse Problems (Isakov, [116]).”
Stability of High-Order Perturbative Methods for the Computation of Dirichlet-Neumann Operators (2001, neutral): “In this manner, the DNO has been brought to bear on problems (direct and inverse) relating to a wide variety of applications that include electromagnetic and acoustic scattering, nondestructive evaluation, and boundary value and free boundary problems from solid and fluid mechanics; see (13, 21), (1”
Inverse transport theory and applications (2009, positive): “This behavior is quite similar to what is observed in the reconstruction of a diffusion coefficient from Cauchy data in the inverse conductivity problem [59] or in the diffusion approximation to the transport equation (34).”
The stability for the Cauchy problem for elliptic equations (2009, background): “In this introduction we do not intend to present a complete discussion on the historical development on this subject since Hadamard, because various monographs, Lavrent′ev [64], Payne [83], Lavrent′ev, Romanov and Šǐsatskĭı [65], Isakov [47, 48], Hörmander [43, 45], already contain abundant info”
Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation (2009, positive): “In the case of α = β = 1, our inverse problem is concerned with the one-dimensional diffusion equation, and we can refer to Isakov and Kindermann [14], Murayama [28], Pierce [30], Suzuki [37, 38], Suzuki and Murayama [39].”
Inverse and Optimization Problems in Heat Transfer (2006, positive): “Although the solution techniques for inverse problems do not necessarily make use of optimizati on techniques, many popular methods are nowadays based on them [1-21].”
Inverse source problem for the hyperbolic equation with a time-dependent principal part (2015, positive): “The Carleman estimate (2.8) deﬁnitely relies on some conditions on d and the principal coeﬃcients a jk (see Amirov and Yamamoto [1], Isakov [15], Kha˘ıdarov [19]).”