Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A^∗A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.     

Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C¹ finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's fixed point theorem for a continuous map on L^p-space in a computer. We present detailed verification procedures and give some numerical examples.     

Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients

One of the basic inverse problems in an anisotropic media is the determination of coefficients in a bounded domain with a single measurement. We consider the problem of finding the coefficient of the second derivatives in a second-order hyperbolic equation with variable coefficients.

Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement. Moreover we show that our uniqueness results yield the Lipschitz stability estimate in L ² space for solution to the inverse problem under consideration.     

The generalized weighted Moore-Penrose inverse

In this paper, we definite a generalized weighted Moore-Penrose inverse A _M,N ⁺ of a given matrixA, and give the necessary and sufficient conditions for its existence. We also prove its uniqueness and give a representation of it. In the end we point out this generalized inverse is also a prescribed rangT and null spaceS of {2}-(or outer) inverse ofA.     

Uniqueness Theorems for Sturm–Liouville Operators with Boundary Conditions Polynomially Dependent on the Eigenparameter from Spectral Data

In this paper, we discuss the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter. We establish some uniqueness theorems on the potential q(x) for the half inverse problem and the interior inverse problem from spectral data, respectively.     

On inverse problems in electromagnetic field in classical mechanics at fixed energy

In this article, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic and nonrelativistic Newton equations in a static external electromagnetic field (V, B), V∈C², B∈C¹ in classical mechanics. Developing the approach going back to Gerver-Nadinashvili 1983's work on an inverse problem of mechanics, we obtain, in particular, theorems of uniqueness.     

Uniqueness of solutions to inverse Sturm–Liouville problems with potential using three spectra

The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L²(0,a).     

A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse problem with memory effect

We consider an inverse problem for a one-dimensional integrodifferential hyperbolic system, which comes from a simplified model of thermoelasticity. This inverse problem aims to identify the displacement u, the temperature η and the memory kernel k simultaneously from the weighted measurement data of temperature. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. Moreover, we prove that the solution to this inverse problem depends continuously on the noisy data in suitable Sobolev spaces. For this nonlinear inverse problem, our theoretical results guarantee the solvability for the proposed physical model and the well-posedness for small measurement time τ, which is quite different from general inverse problems.     

Existence and uniqueness of solutions to the GRID macroscopic growth equation

In this paper we prove the existence and uniqueness of solutions to the initial value problems associated with the GRID integro-differential equation describing macroscopic growth of an organism. We consider the general form of the macroscopic growth operator Φ and study the set of conditions on Φ that are sufficient to guarantee existence and uniqueness of solutions in Rⁿ,n=1,2,3.     

The homogeneous dirichlet problem for quasilinear anisotropic degenerate parabolic-hyperbolic equation with L initial value

The aim of this paper is to prove the well-posedness (existence and uniqueness) of the L^p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L^p initial value. We use the device of doubling variables and some technical analysis to prove the uniqueness result. Moreover we can prove that the L^p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Weighted inverse maximum perfect matching problems under the Hamming distance

Given an undirected network G(V, E, c) and a perfect matching M ⁰, the inverse maximum perfect matching problem is to modify the cost vector as little as possible such that the given perfect matching M ⁰ can form a maximum perfect matching. The modification can be measured by different norms. In this paper, we consider the weighted inverse maximum perfect matching problems under the Hamming distance, where we use the weighted Hamming distance to measure the modification of the edges. We consider both of the sum-type and the bottleneck-type problems. For the general case of the sum-type case, we show it is NP-hard. For the bottleneck-type, we present a strongly polynomial algorithm which can be done in O(m · n ³).     

Uniqueness and stability of the minimizer for a binary functional arising in an inverse heat conduction problem

The local well-posedness of the minimizer of an optimal control problem is studied in this paper. The optimization problem concerns an inverse problem of simultaneously reconstructing the initial temperature and heat radiative coefficient in a heat conduction equation. Being different from other ordinary optimization problems, the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters. Particularly, since the status of the two unknown coefficients in the cost functional are different, the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem. The necessary condition which must be satisfied by the minimizer is deduced. By assuming the terminal time T is relatively small, an L² estimate regarding the minimizer is obtained, from which the uniqueness and stability of the minimizer can be deduced immediately.     

Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the Stroock-Varadhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for Ld-almost every initial condition x whenever existence and uniqueness is known at the PDE level in the L^∞-setting (and, conversely, if existence and uniqueness of martingale solutions is known for Ld-a.e. initial condition, then existence and uniqueness for the PDE holds). In the second part of the paper we consider situations where, on the one hand, no pointwise uniqueness result for the martingale problem is known and, on the other hand, well-posedness for the Fokker-Planck equation can be proved. Thus, the theory developed in the first part of the paper is applicable. In particular, we will study the Fokker-Planck equation in two somehow extreme situations: in the first one, assuming uniform ellipticity of the diffusion coefficients and Lipschitz regularity in time, we are able to prove existence and uniqueness in the L²-setting; in the second one we consider an additive noise and, assuming the drift b to have BV regularity and allowing the diffusion matrix a to be degenerate (also identically 0), we prove existence and uniqueness in the L^∞-setting. Therefore, in these two situations, our theory yields existence, uniqueness and stability results for martingale solutions.     

Perturbations and expressions for generalized inverses in Banach spaces and Moore-Penrose inverses in Hilbert spaces of closed linear operators

The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore-Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under T-bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression T⁺(I+δTT⁺)^-1. Utilizing these results, we investigate the expression for the Moore-Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore-Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore-Penrose inverse to have the simplest expression T^†(I+δTT^†)^-1. The results obtained in this paper extend and improve many recent results in this area.     

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for the Schrdinger equation in Sobolev spaces of negative orders, and use these estimates to prove the uniqueness in the inverse problem of determining Lp-potentials. An L2-level observability inequality and unique continuation results for the Schrdinger equation are also obtained.     

Inverse Sturm–Liouville problems with the potential known on an interior subinterval

The uniqueness problem of inverse Sturm–Liouville problems with the potential known on an interior subinterval is considered. We prove that the potential on the entire interval and boundary conditions are uniquely determined in terms of the known eigenvalues and some information on the eigenfunctions at some interior point (interior spectral data). Moreover, we also concern with the situation where the potential q is C^2k-smoothness at some given points.     

Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.