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تعداد صفحات این فایل: ۴۶ صفحه

بخشی از مقاله انگلیسیعنوان انگلیسی:Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient~~en~~

Abstract

An inverse problem for the identification of an unknown coefficient in a quasilinear parabolic partial differential equation is considered. We present an approach based on utilizing adjoint versions of the direct problem in order to derive equations explicitly relating changes in inputs (coefficients) to changes in outputs (measured data). Using these equations it is possible to show that the coefficient to data mappings are continuous, strictly monotone and injective. The equations are further exploited to construct an approximate solution to the inverse problem and to analyse the error in the approximation. Finally, the results of some numerical experiments are displayed.

۱ Introduction

Using partial differential equations to model physical systems is one of the oldest activities in applied mathematics. A complete model requires certain state inputs in the form of initial and/or boundary data together with what might be called structure inputs such as coefficients or source terms which are related to the physical properties of the system. Obtaining a unique solution for the associated well-posed problem constitutes what we will call solving the direct problem. Solving the direct problem permits the computation of various system outputs of physical interest. On the other hand, when some of the required inputs are not available we may instead be able to determine the missing inputs from outputs that are measured rather than computed by formulating and solving an appropriate inverse problem. In particular, when the missing inputs are one or more unknown coefficients in the partial differential equation, the problem is called a coefficient identification problem. The identification of a diffusion coefficient in a quasilinear diffusion equation is chosen here as a prototype coefficient identification problem that has been approached by various methods.

The most common technique for identifying an unknown coefficient from some measured output is the method of output least squares (OLS) [1, 4, 8–۱۰]. Here the unknown coefficient, C, is chosen from an appropriate space K and the output,
[C], is computed by solving the direct problem. One defines an error functional, J [C] =

[C] f
۲ F , comparing the computed output to the measured value, f , in the norm of the output space, F, and seeks to minimize J over K. OLS methods are very general and can be efficiently programmed for computer implementation. Typically there are problems with lack of uniqueness, convergence to false minima and instability under parameter mesh refinement, although a skilful user may be able to incorporate a priori information about the solution into the parametric description of the unknown coefficient in order to lessen some of these difficulties [1, 9]. Since the connection between the inputs and outputs is expressed only indirectly through the solver, general information about an input-to-output mapping is not readily available by OLS methods.

An alternative to coefficient identification by output least squares is the so-called equation error method [3, 6, 7, 11, 12]. Here the measured overspecification is used as input to the differential equation in the direct problem which is then viewed as an equation for the unknown coefficient. This equation expresses a direct relationship between the unknown coefficient values and the measured data values. Since the relationship is frequently quite complicated, it is not easy to discern from it properties of an input-to-output mapping. Equation error methods are quite problem dependent and produce varying degrees of success.

The method described in this paper is based on an integral equation relating changes in the unknown coefficient to corresponding changes in the measured output. The integral equation is derived by exploiting a problem which is adjoint to the direct problem, an idea close to the techniques often used to estimate sensitivity in the OLS approach [8, 9]. However, this integral equation provides information about the input/output mapping itself rather than the error functional. It is then possible to prove that the input-to-output map is continuous, monotone and injective. Moreover, it is shown that when the input/output map is restricted to a (finite-dimensional) space of polygonal coefficients, it is explicitly invertible. This observation provides the basis for a method for numerically approximating the unknown coefficient. It is shown that a unique polygonal approximation to the unknown coefficient is obtained by solving a triangular system of linear algebraic equations. Error estimates show that the accuracy of the approximation is limited by the precision of the data measurements so that there is an optimal attainable accuracy but exact determination of the coefficient is never possible.

The results of a few numerical experiments are provided here to illustrate the working of the method. A more extensive presentation of numerical experiments will be included in a later publication.

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