We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter λ near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of λ and log λ. The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.
When solving large scale semidefinite programs that admit a low-rank solution, a very efficient heuristic is the Burer-Monteiro factorization: Instead of optimizing over the full matrix, one optimizes over its low-rank factors. This strongly reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points which can prevent local optimization algorithms from finding the solution. Boumal, Voroninski, and Bandeira [2018] have recently shown that, when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: For almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: For smaller values of the size, second-order critical points are not generically optimal, even when considering only semidefinite programs with a rank 1 solution.
We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term q(t, x), we can recover the X-ray transform of time dependent potentials q(t, x) from the dynamical Dirichlet-to-Neumann map in a stable way. We derive conditional Hölder stability estimates for the X-ray transform of q(t, x). The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.Whenever f is a C 1 function on M, the gradient of f is defined as the vector field ∇ g f so that ∀X on M we haveIn local coordinates, we can writeThe metric tensor induces a Riemannian volume form which as in [9] we denote by,the dynamical Dirichlet-to-Neumann map. We allow work in integral geometry to tell us which class for which class of manifolds and admissible potentials we have stability and uniqueness results. We direct the reader to the preprint Uhlmann and Vasy [43] for recent injectivity and stability results on the X-ray transform. Future works by integral geometers regarding X-ray transforms will produce better stability results of the potentials.In Eskin [17], [16], [18], uses the boundary control method first introduced by Belishev [7], and Belishev and Kurylev [8]. Eskin's methods in [18] require the potential to be analytic in time. For a survey on the literature of the boundary control method, and explanation of the techniques, one should see the monograph by Lassas et. al, [1]. For some stability and uniqueness results for other partial differential equations with time-independent coefficients, see for example [4], [5], [30], [3], [25] and [11].Here the author has chosen to focus on references which are related to work on the X-ray transform for uniqueness and stability estimates for the hyperbolic problem.Using Green's theorem as in Alessandrini and Sylvester, [2] and Sylvester and Uhlmann, [42] and complex geometric optics to produce the X-ray transform, we derive stability results for the X-ray transform of potentials q(t, x). The author uses only the minimal amount of assumptions on the geometry for the Gaussian beam Ansatz to be well-defined. In particular, the metric must be at least three times differentiable, an assumption which was also used in [6]. It seems likely that using the same techniques developed by the author in [44], wave equations with C 1,1 (t, x) coefficients could also be examined.The study of the initial boundary value problem (1.2) has a long history, and these results are formulated building on the results of others. For references in this direction, we direct the reader to the articles by Isakov [21], [22] and Sun [41], and Isakov and Sun [23]. Using X-ray transform methods, the first uniqueness result for time dependent potentials for wave equations was established by Stefanov in [37] using the scattering relation when the geometry is Euclidean. Later, Sjöstrand and Ramm in R n in [35] establish...
We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These estimates imply convergence of a numerical scheme known as the Landweber iteration scheme for reconstructing the shear modulus distributions.
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