Determination of the Robin coefficient in a nonlinear boundary condition for a steady‐state problem

In this paper, a constant heat transfer coefficient present in a nonlinear Robin‐type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one‐dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.     

Approximation of finite-section equations by piecewise constant functions

The problem is studied of reducing the amount of discrete information required for achieving a prescribed accuracy of solving Fredholm integral equations of the first kind on a half-line. The equations are solved by the finite-section method combined with piecewise constant interpolation of the kernel and the right-hand side at uniform grid points. The approximating properties of the discretization schemes are examined, and the corresponding computational costs are analyzed.     

An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

We consider an inverse problem for identifying a leading coefficient α(x) in −(α(x)y′(x))′ + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm-Liouville operator. We transform y(x) to u(x, t) =  (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function α(x). When α(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y′ and α. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of α(x).     

The determination of radially dependent conductivity coefficient

In this paper we discuss the question of identifying the radially dependent coefficient a(r) in the elliptic equation div(a(r) ▽u)=0 in the unit disk by Dirichlet and Neumann data.We establish a condition to guarantee the uniqueness of this determination. One of the applications of this study is the determination of the radially dependent conductivity coefficient of layered medium.     

Asymptotics of the solution to Robin problem

Convergence of the solution to the exterior Robin problem to the solution of the Dirichlet problem, as the impedance tends to infinity, is proved. The rate of convergence is established. A method for deriving higher order terms of the asymptotics of the solution is given.     

Stability estimate for an inverse boundary coefficient problem in thermal imaging

We establish a stability estimate for an inverse boundary coefficient problem in thermal imaging. The inverse problem under consideration consists in the determination of a boundary coefficient appearing in a boundary value problem for the heat equation with Robin boundary condition (we note here that the initial condition is assumed to be a priori unknown). Our stability estimate is of logarithmic type and it is essentially based on a logarithmic estimate for a Cauchy problem for the Laplace equation.     

Logarithmic convergence rates for the identification of a nonlinear Robin coefficient

We consider the identification of a nonlinear corrosion profile from single voltage boundary data and show injectivity of the parameter-to-output map. We demonstrate that Tikhonov regularization can be applied in order to solve the inverse problem in a stable manner despite the presence of noisy data. In combination with a logarithmic stability estimate for the underlying Cauchy problem, rates for the convergence of the regularized solutions are proven using a source condition that does not involve the Fréchet derivative of the parameter-to-output map. We present sufficient conditions for the existence of a source function and illustrate our approach by means of numerical examples.     

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

ASYMPTOTIC ESTIMATION FOR SOLUTION OF A CLASS OF SEMI-LINEAR ROBIN PROBLEMS

A class of semi-linear Robin problem is considered. Under appropriate assumptions, the existence and asymptotic behavior of its solution are studied more carefully. Using stretched variables, the formal asymptotic expansion of solution for the problem is constructed and the uniform validity of the solution is obtained by using the method of upper and lower solution.     

Solution of the Robin problem for the Laplace equation

For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Evaluation of diffusion coefficient based on multiple forward solutions

This note is concerned with the identification of the unknown diffusion coefficient for a parabolic equation. It introduces an iterative algorithm that can be used to recover the unknown function. The algorithm assumes an initial guess for the unknown function and obtains a background field. It obtains an equation for the error field. It then formulates three forward problems for the error field. These three formulations share the same unknown function which is the correction to the assumed value of the unknown diffusion coefficient. By equating the responses of these three formulations, the algorithm obtains two working equations for the unknown function. A number of numerical examples are also used to study the performance of the algorithm.     

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave-convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.     

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two‐sided estimates for this term in a variety of situations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Critical points of the regular part of the harmonic Green function with Robin boundary condition

In this paper we consider the Green function for the Laplacian in a smooth bounded domain with Robin boundary condition

and its regular part S_λ(x,y), where b>0 is smooth. We show that in general, as λ→∞, the Robin function R_λ(x)=S_λ(x,x) has at least 3 critical points. Moreover, in the case b≡const we prove that R_λ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when bconst there are critical points of R_λ near non-degenerate critical points of b.     

Identifying the radiative coefficient of an evolutional type heat conduction equation by optimization method

This paper studies an evolutional type inverse problem of identifying the radiative coefficient of heat conduction equation when the over-specified data is given. Problems of this type have important applications in several fields of applied science. Being different from other ordinary inverse coefficient problems, the unknown coefficient in this paper depends on both the space variable x and the time t. Based on the optimal control framework, the inverse problem is transformed into an optimization problem and a new cost functional is constructed in the paper. The existence, uniqueness and stability of the minimizer of the cost functional are proved, and the necessary conditions which must be satisfied by the minimizer are also given. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations.     

‐solution of the Neumann,Robin and transmission problem for the scalar Oseen equation

We find necessary and sufficient conditions for the existence of an ‐solution of the Neumann problem, the Robin problem and the transmission problem for the scalar Oseen equation in three‐dimensional open sets. As a consequence we study solutions of the generalized jump problem.     

A new refinement indicator for adaptive parameterization: Application to the estimation of the diffusion coefficient in an elliptic problem

We propose a new refinement indicator (NRI) for adaptive parameterization to determine the diffusion coefficient in an elliptic equation in two-dimensional space. The diffusion coefficient is assumed to be a piecewise constant space function. The unknowns are both the parameter values and the zonation. Refinement indicators are used to localize parameter discontinuities in order to construct iteratively the zonation (parameterization). The refinement indicator is obtained usually by using the first-order effect on the objective function of removing degrees of freedom for a current set of parameters. In this work, in order to reduce the computation costs, we propose a new refinement indicator based on the second-order effect on the objective function. This new refinement indicator depends on the objective function, and its first and second derivatives with respect to the parameter constraints. Numerical experiments show the high efficiency of the new refinement indicator compared to the standard one.