Numerical implementations of dynamical probe method for non-stationary heat equation

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Abstract

The dynamical probe method for non-stationary heat equation is developed recently, which aims to detect an unknown inclusion of conductive material from the boundary measurement data. The Runge approximation is used to define some indicator function for this method, which is a mathematical testing machine to detect the inclusion. A numerical realization of the Runge approximation is the key to this method. By using a regularizing method, a realization scheme is given for the Runge approximation, and numerical examples are given to show the validity of the dynamical probe method.

Introduction

In the recent 15 years, several reconstruction schemes for identifying an unknown inclusion for steady state heat conductor have been proposed. They are the linear sampling method, probe method, singular sources method, no response test, range test, factorization method, enclosure method etc, see [2], [6], [7], [9], [11], [12], [13] and the references therein. None of these schemes is data fitting scheme using Tikhonov’s regularization. Each of them gave an exact scheme theoretically to recover the unknown inclusion directly from given measurement data.

On the other hand, for non steady state conductors, Nakamura and his collaborators have recently developed a probe typed reconstruction scheme for the one space dimensional case [3] for heat conduction process, which aims to detect the unknown inclusion of heat conductive material from the boundary measurement data. We refer to this scheme by dynamical probe method. Compared with the standard optimization method which considers the solution to this inverse problem as an minimizer of some cost functional, the advantage of this new scheme is that it is a theoretically exact method and no iteration process is required. The basic idea of this scheme is to construct an indicator function J(y,s) which can detect the boundary of the inclusion by using the boundary measurement data of heat field. For this function, the variable is the parameter (y,s). When (y,s) approaches to the boundary of inclusion from outside the inclusion, this indicator blows up. The indicator function is defined via the pre-indicator function I(y,s;y,s) by taking some limiting procedure. For the numerical realization of this scheme, it is more convenient to use the pre-indicator function than the indicator function in order to avoid the limiting procedure.

Due to the diffusion of heat conduction with respect to time t, the information about the inclusion contained in the measurement data is relatively weak. Therefore, an efficient numerical realization scheme for this new dynamical probe method should be studied, which is the purpose of this paper.

On the other hand, it also should be noticed that the dynamical probe method is not the only one method to detect the inclusion in heat conductive media. A single one sided measurement version of the enclosure method has been applied in one dimensional case by Ikehata recently, see [8].

The inverse problem for identifying an unknown inclusion D inside a non steady state heat conductor Ω in 1-dimensional case is formulated as follows. Let Ω be an open connected interval in R and T>0 be a fixed given time. For our forward problem, we consider the heat conduction problemtu-·(γ(x)u)=0,(x,t)ΩT:=Ω×(0,T),νu=f(x,t),(x,t)ΩT:=Ω×(0,T),u(x,0)=0,xΩ,where Ω denotes the boundary of Ω. We assume that the heat conductivity γ(x) has the following formγ(x)=1+(k-1)χD(x)for some constant k satisfying 0<k1, where DΩ is some unknown open, connected interval and χD is the characteristic function of D. In other words, the function γ(x) has jump discontinuity across D, the boundary of D. For given fL2((0,T);(H1/2(Ω))), there exists a unique weak solutionuW(ΩT):={uL2((0.T);H1(Ω));tuL2((0,T);(H1(Ω)))}to the forward problem (1.1) in the senseΩT(-utϕ+γu·ϕ)dxdt=ΩTfϕds(x)dtfor all ϕ(x,t)W(ΩT) with ϕ(x,T)=0, see [14]. Here, (H1/2(Ω)) and (H1(Ω)) are the dual spaces of H1/2(Ω) and H1(Ω), respectively. We also denote the solution u of (1.1) by u(f). Based on this weak solution, we define the Neumann to Dirichlet mapΛDT:f(x,t)u(f)|ΩT,which maps from L2((0,T);(H1/2(Ω))) to L2((0,T);H1/2(Ω)). The inverse problem considered in this paper is to identify DT from ΛDT numerically.

This paper is organized as follows. The indicator function (i.e., a mathematical testing machine to identify the unknown inclusion) and pre-indicator function are defined in Section 2. We also give the behavior of the pre-indicator function in the 1-dimensional case using the behavior of the so called reflected solution. The key to define the pre-indicator function is the Runge approximation. In Sections 3 Realization of Runge approximation by regularization scheme, 4 Denseness of, the Runge approximation is given in a constructive way by using a regularizing method. That is in Section 3, we propose a regularizing scheme to determine the density function of the integral operator A, and in Section 4, we prove the denseness of the range of operator A which guarantees the existence of this density function. We also give in Section 4 the discrete argument for the numerical realization of the dynamical probe method. Finally in Section 5, some examples are presented to show the validity of the dynamical probe method.

Section snippets

Pre-indicator function and indicator function of dynamical probe method

Let (y,s),(y,s)ΩT with s-s>0 and DT¯ be the closure of open domain DT=D×(0,T). Also, let V(y,s;y,s)ΩTδ:=Ω×(-δ,T+δ) with a small δ>0 be a domain

  • (H1)

    (y,s),(y,s)ΩTδV(y,s;y,s)¯,

  • (H2)

    V(y,s:y,s) is Lipschitz,

  • (H3)

    (ΩTδV(y,s;y,s)¯){t=τ} is an empty set, or connected to Ω×{t=τ} for each τ(-δ,T+δ), which means that each point in (ΩTδV(y,s;y,s)¯){t=τ} can be connected to a point in Ω×{t=τ} by a continuous curve in (ΩTδV(y,s;y,s)¯){t=τ}.

In the rest of this paper, we call such a V(y,s;y,s

Realization of Runge approximation by regularization scheme

Let Ω,D be intervals Ω:=[A0,A1],D:=[D0,D1], with the two sets of ends {A0,A1} and {D0,D1}. We assume D¯Ω. In this case, for any given y(D1,A1) or y(A0,D0) and s,s(0,T) with s>s, we can construct V(y,s;y,s) in a very simple way, that is, the configuration of V(y,s;y,s) is independent of y, which will be denoted by V(y,s,s) in the sequel. Moreover, when we approximate D1 by y(V1y,A1) with V1y being the right-hand side boundary of V(y,s,s), the left-hand side V0 keeps unchanged, see

Denseness of Range(A) and discrete scheme

In this section, we prove the denseness of the range of map A and the discrete scheme, that is the detailed discretization for computing I(y,s;y,s) and the simulation process for generating the boundary measurement data. As stated previously, we begin with ((3.7), (3.8), (3.9), (3.10), (3.11), (3.12), (3.13)). It is easy to see thatA:L2(s,T)×L2(s,T)L2(s,T)×L2(s,T)defined by (3.6) has the adjoint operator(Aψ)(t)=tTAT(τ-t)ψ(τ)dτ.Therefore, the regularizing equation in (3.8) for given α>0

Numerical results

From the simulation procedure for ΛD and the construction of v(y,s)j,ϕ(y,s)j given in Sections 3 Realization of Runge approximation by regularization scheme, 4 Denseness of, we can finally compute the indicator function and check the numerical performance of dynamical probe method in 1-dimensional case.

Based on the above scheme, we use the configurations[A0,A1]=[-1,1],k=3,T=1.0,s=0.3,s=0.34,y=y+0.2in the following Example 1, Example 2.

Example 1

For a given inclusion [D0,D1]=[-0.5,0.5][A0,A1], we

Acknowledgements

The authors would like to thank one of the referee for pointing out the mistakes in the original proof of Theorem 2.1 and providing the references [4], [5], which leads to an improved version of this paper. The third author is supported by Natural Science Foundation of China, No.10771033. The other authors are supported by Grant-in-Aid for Scientific Research (B) (No. 19340028) of Japan Society for Promotion of Science. The second and third authors thank Faculty of Science of Hokkaido

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