Investigation Methods for Inverse Problems

Vladimir G. Romanov

• Publisher: De Gruyter
• Year: 2014
• Print ISBN: 9783110364194
• eBook ISBN: 9783110943849
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This monograph deals with some inverse problems of mathematical physics. It introduces new methods for studying inverse problems and gives obtained results, which are related to the conditional well posedness of the problems. The main focus lies on time-domain inverse problems for hyperbolic equations and the kinetic transport equation.

Table of contents

• Preface
• 1 Introduction
• 1.1 One-dimensional inverse kinematics problem
• 1.2 Inverse dynamical problem for a string
• 1.3 Inverse problems for a layered medium
• 2 Ray statements of inverse problems
• 2.1 Posing of the inverse problems
• 2.2 Asymptotic expansion
• 2.2.1 Asymptotic expansion of the solution
• 2.2.2 Reduction of the inverse problem
• 2.2.3 Construction of τ(x, y)
• 2.2.4 Proof of the expansion in the odd-dimensional case
• 2.2.5 Proof of the expansion in the even-dimensional case
• 2.2.6 Proof of the auxiliary lemma
• 2.3 Uniqueness theorems for the inverse problem
• 2.3.1 A proof of the stability estimate for the integral geometry problem
• 2.3.2 Uniqueness theorem for the integral geometry problem related to a vector field
• 2.3.3 Proof of the uniqueness theorem for inverse kinematics problem
• 2.3.4 The wave equation with an attenuation
• 2.3.5 Concluding remarks
• 2.4 Inverse problems related to a local heterogeneity
• 3 Local solvability of some inverse problems
• 3.1 Banach’s spaces of analytic functions
• 3.2 Determining coefficients of the lower terms
• 3.2.1 Determining a coefficient of the lower term
• 3.2.2 Determining an attenuation coefficient
• 3.3 Determining the speed of the sound
• 3.4 A regularization method for solving an inverse problem
• 3.4.1 Theorems related to the system of integro-differential equations
• 3.4.2 Estimates of a solution to the algebraic equations
• 3.4.3 Convergence the approximate solution to the exact one
• 4 Inverse problems with single measurements
• 4.1 Determining coefficient of the lowest term
• 4.1.1 Statement of the problem and stability estimates
• 4.1.2 Proof of the stability theorems
• 4.1.3 Proof of Lemma 4.1.3
• 4.1.4 Proof of Lemma 4.1.4
• 4.2 Determining coefficients under first derivatives
• 4.3 Determining the speed of sound in the wave equation
• 4.3.1 Formulation of the problem and a stability estimate of the solution
• 4.3.2 Proof of Theorem 4.3.1
• 4.3.3 Proof of Lemma 4.3.5
• 4.3.4 Proof of Lemma 4.3.6
• 4.3.5 Proof of the inequality (4.3.24)
• 4.4 Case of a point source
• 4.4.1 Formulations of the problem and results
• 4.4.2 Proofs of the stability theorems
• 4.4.3 Properties of a solution to problem (4.4.1)
• 4.4.4 Proof of Lemma 4.4.3
• 4.4.5 Proof of Lemma 4.4.4
• 5 Stability estimates related to inverse problems for the transport equation
• 5.1 The problem of determining the relaxation and a density of inner sources
• 5.1.1 Statement of basic and auxiliary problems
• 5.1.2 The basic results
• 5.1.3 Proof of Theorem 5.1.1
• 5.1.4 Proof of the auxiliary lemmas
• 5.2 A stability estimate in the problem of determining the dispersion index and relaxation in 2D
• 5.2.1 Statement of the problem and the basic results
• 5.2.2 Proof of Lemma 5.2.1
• 5.2.3 A priori estimates
• 5.2.4 Proof of Theorem 5.2.3
• 5.3 The problem of determining the dispersion index and relaxation in 3D
• 5.3.1 Statement of the problem and the main results
• 5.3.2 Proof of Lemma 5.3.1
• 5.3.3 A priori estimates for function ω(x, v)
• 5.3.4 Estimates for functions ώ(x, v) and σ̃(x)
• 5.3.5 A priori estimates and differential properties of function u¯(x,v,v°)
• 5.3.6 A priori estimates and properties of function v(x, v,v°)
• 5.3.7 Equations for the derivatives of function ṽ(x, u, v°)
• 5.3.8 Proof of inequality (5.3.24)
• 5.3.9 Proof of inequality (5.3.25)
• 5.3.10 Proof of inequality (5.3.26)
• 5.3.11 Auxiliary formulae
• Bibliography