On the Relation Between an Inverse Problem for a System of Ordinary Differential Equations and an Initial Boundary Value Problem for a Hyperbolic System

This paper deals with a coefficient inverse problem of a system of ODEs whose coefficient matrix is the so-called generalized negative definite matrix. To solve the problem, an initial-boundary value problem of a hyperbolic system of PDEs is constructed. The existence and uniqueness of its solution and its asymptotic convergence with respect to one of the variables to the original inverse problem are proved. As a result, the solution of the inverse problem is reduced to the solution of the direct problem. A few numerical examples were solved to show the effectiveness of the method.     

A Stability Estimate for a Solution of the Problem of Determining the Dielectric Permittivity

We consider the problem of determining the dielectric permittivity for a nonconducting and nonmagnetic medium. As information we take the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindrical domain. These traces correspond to a solution to some direct problem for the Maxwell system. The impulse source of the current flux lies outside the domain in which the coefficient is sought. The main result of the article is a stability estimate for a solution to the inverse problem in question.     

Application of a Numerical-Asymptotic Approach to the Problem of Restoring the Parameters of a Local Stationary Source of Anthropogenic Pollution

Doklady Mathematics - A numerical-asymptotic approach is used to solve some coefficient inverse problems of tracer diffusion in the atmosphere. An asymptotic solution of the direct problem for an...     

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.     

有理函数积分的公式解法

被积函数为有理函数的不定积分求解通常是采用待定系数法。本文提出了这类积分的非待定系数公式解法，较完美地解决了这类积分问题。在实际应用中显示了这种新型方法是简捷的和有效的。它的优点还在于不仅对一些常规方法极为困难或无法解决的问题给以简明的解，而且借助于电子计算机可解决更复杂的问题。     

A new approach for numerical identification of conductivity

This paper studies the parameter estimation problem for a steady state flow in an inhomogeneous medium. For identifying the spatially varying diffusion coefficient from an observed solution to the forward problem, we propose a direct method using local Green's function technique. This could be used when the diffusion coefficient is discontinuous. The convergence order is calculated and numerical simulations are performed.     

Coefficient identification in Euler-Bernoulli equation from over-posed data

A method for solving the inverse problem for coefficient identification in the Euler-Bernoulli equation from over-posed data is presented. The original inverse problem is replaced by a minimization problem. The method is applied to the problem for identifying the coefficient in the case when it is a piece-wise polynomial function. Several examples are elaborated and the numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the second order of approximation.     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

Problem of determining the permittivity in the stationary system of Maxwell equations

The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity ε from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant ε₀ only inside a compact domain Ω ? R ³ with a smooth boundary S. To find ε inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω₀ and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution.     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

Numerical Modeling of Pollutant Dispersion and Oil Spreading by the Stochastic Discrete Particles Method

We consider two applications of the stochastic discrete particles method. The first one is concerned with the dispersion of a passive pollutant by a turbulent stream with a scale dependent diffusion coefficient. The second application deals with the problem of an oil spill spreading on the water surface described by transport–diffusion equation with a nonlinear diffusion coefficient. For the first problem we develop a discrete particles algorithm provided the diffusion coefficient obeys Richardson's "4/3" law and show good correspondence with the numerical and analytical results. The second problem is more involved and we develop a heuristic procedure based on the standard discrete particles random walk algorithm updating the dependence of each particle step variance on the dependent function. The obtained solution coincides well with analytical and direct one-dimensional finite-difference solutions both for instantaneous and continuous oil release.     

线性互补问题均衡解的存在形式与识别方法

研究了线性互补问题均衡解的存在形式与判定方法,给出了线性互补问题有解的充要条件,得到了带有几类特殊系数矩阵的线性互补问题的解的性质.在此基础上设计了求解线性互补问题均衡解的直接算法.     

The problem of recovering the permittivity coefficient from the modulus of the scattered electromagnetic field

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ? R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω ₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.     

The transmutation operator method for efficient solution of the inverse Sturm‐Liouville problem on a half‐line

The inverse Sturm‐Liouville problem on a half‐line is considered. With the aid of a Fourier‐Legendre series representation of the transmutation integral kernel and the Gel'fand‐Levitan equation, the numerical solution of the problem is reduced to a system of linear algebraic equations. The potential q is recovered from the first coefficient of the Fourier‐Legendre series. The resulting numerical method is direct and simple. The results of the numerical experiments are presented.     

INVERSE COEFFICIENT PROBLEMS FOR PARABOLIC HEMIVARIATIONAL INEQUALITIES

This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality. The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients. It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence. Based on this result the existence of a quasisolution of the inverse problem is obtained.     

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

In this article, we consider coefficient identification problems in heat transfer concerned with the determination of the space‐dependent perfusion coefficient and/or thermal conductivity from interior temperature measurements using the conjugate gradient method (CGM). We establish the direct, sensitivity and adjoint problems and the iterative CGM algorithm which has to be stopped according to the discrepancy principle in order to reconstruct a stable solution for the inverse problem. The Sobolev gradient concept is introduced in the CGM iterative algorithm in order to improve the reconstructions. The numerical results illustrated for both exact and noisy data, in one‐ and two‐dimensions for single or double coefficient identifications show that the CGM is an efficient and stable method of inversion.     

On the global existence and small dissipation limit for generalized dissipative Zakharov system

This paper deals with the existence and uniqueness of the global solutions to the initial boundary value problem for a generalized Zakharov system with direct self‐interaction of the dispersive waves and weak dissipation in the nondispersive subsystem. We prove the global existence of the generalized solution to the problem by a priori estimates and Galerkin method. We also establish the regularity of the global generalized solution and the existence and uniqueness of the global classical solution. Moreover, we obtain the convergence of the solutions of the generalized Zakharov system with dissipation as the dissipative coefficient approaches zero.     

Computational identification of the lowest space-wise dependent coefficient of a parabolic equation

In the present work, we consider a nonlinear inverse problem of identifying the lowest coefficient of a parabolic equation. The desired coefficient depends on spatial variables only. Additional information about the solution is given at the final time moment, i.e., we consider the final redefinition. An iterative process is used to evaluate the lowest coefficient, where at each iteration we solve the standard initial-boundary value problem for the parabolic equation. On the basis of the maximum principle for the solution of the differential problem, the monotonicity of the iterative process is established along with the fact that the coefficient is approached from above. The possibilities of the proposed computational algorithm are illustrated by numerical examples for a model two-dimensional problem.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

Source and coefficient inverse problems for an elliptic equation in a rectangle

The inverse problems of determining the source and coefficient of an elliptic equation in a rectangle are studied. Additional information on the solution to the direct problem (overdetermination) is the trace of its solution on an interval inside the rectangle. Sufficient existence and uniqueness conditions (global) are derived for the inverse problems. The study is performed in the class of continuously differentiable functions whose derivatives satisfy a Hölder condition.