Lipschitz Stability Research Articles

Global Lipschitz stability for an inverse coefficient problem for a mean field game system

For an inverse coefficient problem of determining a state‐varying factor in the corresponding Hamiltonian for a mean field game system, we prove the global Lipschitz stability by spatial data of one component and interior data in an arbitrarily chosen subdomain over a time interval. The proof is based on Carleman estimates with different norms.

Analysis and computation of an inverse source problem for the biharmonic wave equation

Abstract In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on the Fourier series expansion by multi-wavenumber boundary measurements. The stability of the proposed method is analyzed and numerical experiments are presented to verify the accuracy and efficiency.

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

Locally Lipschitz Stability of Solutions to a Parametric Parabolic Optimal Control Problem with Mixed Pointwise Constraints

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Abstract In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. We then apply these Carleman estimates to investigate two inverse problems for the space semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the H"{o}lder stability for the discrete Cauchy problem.

Variation of parameters and initial time difference Lipschitz stability of impulsive differential equations

In this paper, we investigate the Lipschitz stability of a perturbed impulsive differential system concerning the unperturbed system. We employ the variation of parameters or the constant of variation for impulsive differential systems with an initial time difference.

Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs

Abstract In 1996 Seo proved that two appropriate pairs of current and voltage data measured on the surface of a planar homogeneous object are sufficient to determine a conductive polygonal inclusion with known deviating conductivity. Here we show that the corresponding linearized forward map is injective, and from this we deduce Lipschitz stability of the solution of the original nonlinear inverse problem. We also treat the case of an insulating polygonal inclusion, in which case a single pair of Cauchy data is already sufficient for the same purpose.

Determining a time‐varying potential in time‐fractional diffusion from observation at a single point

AbstractWe discuss the identification of a time‐dependent potential in a time‐fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem. Numerically, we develop an easily implementable iterative algorithm to recover the unknown coefficient, and also derive rigorous error bounds for the discrete reconstruction. These results are attained by leveraging the (discrete) solution theory of direct problems, and applying error estimates that are optimal with respect to problem data regularity. Numerical simulations are provided to demonstrate the theoretical results.

Finite element approximation of unique continuation of functions with finite dimensional trace

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

Square Root LASSO: Well-Posedness, Lipschitz Stability, and the Tuning Trade-Off

Square Root LASSO: Well-Posedness, Lipschitz Stability, and the Tuning Trade-Off

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We explore the Lipschitz stability of solutions to the Hunter–Saxton equation with respect to the initial data. In particular, we study the stability of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\alpha $$\\end{document}-dissipative solutions constructed using a generalised method of characteristics approach, where α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a function determining the energy loss at each position in space.

Carleman estimate and applications for a heat equation with multiple singularities

The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: ∑ j = 1 m μ j | x − x j | 2 , where x 1 , x 2 , … , x m are different points, defined in a bounded C 2 domain Ω ⊂ R n , and μ 1 , … , μ m are real constants. More precisely, we show that for any μ j ≤ ( n − 2 ) 2 4 , j = 1 , … , m , we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function.

Inverse stable reconstruction of 3 coefficients for the heterogeneous Maxwell equations by finite number of partial interior observations

We consider an inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell’s equations in a bounded domain of R3 by means of a finite number of interior data of as less as possible components of the solutions. Our main result is a Lipschitz stability estimate for the inverse problem and our proof relies on a Carleman estimate for the heterogeneous Maxwell’s equations.

Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations

Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

Optimality conditions and Lipschitz stability for non-smooth semilinear elliptic optimal control problems with sparse controls

Optimality conditions and Lipschitz stability for non-smooth semilinear elliptic optimal control problems with sparse controls

Lipschitz stability of some canonical Jordan bases of real H-selfadjoint matrices under small perturbations

In this paper we prove that the new Jordan bases called γ-flipped orthogonal conjugate symmetric (γ-FOCS) are Lipschitz stable under small perturbations as well. We also establish the Lipschitz stability for the classical real canonical Jordan bases.

Lipschitz Stability for Impulsive Riemann–Liouville Fractional Differential Equations

Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann–Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singularity at the initial time and at any point of the impulses. This leads to the need to appropriately generalize the classical concept of Lipschitz stability. Two derivative types of Lyapunov functions are utilized in order to deduce sufficient conditions for the new stability concept. Three examples are provided for illustration purpose of the theoretical results.

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

Lipschitz stability for determination of states and inverse source problem for the mean field game equations

Numerical recovery of a time-dependent potential in subdiffusion *

In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted Lp(0,T) norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.

Convexification for the viscocity solution for a coefficient inverse problem for the radiative transfer equation

A coefficient inverse problem for the radiative transport equation is considered. The globally convergent numerical method, the so-called convexification, is developed. For the first time, the viscosity solution is considered for a boundary value problem for the resulting system of two coupled partial differential equations. A Lipschitz stability estimate is proved for this boundary value problem using a Carleman estimate for the Laplace operator. Next, the global convergence analysis is provided via that Carleman estimate. Results of numerical experiments demonstrate a high computational efficiency of this approach.