A computational procedure for estimation of an unknown coefficient in an inverse boundary value problem

doi:10.1016/j.amc.2006.09.015

Applied Mathematics and Computation

Volume 187, Issue 2, 15 April 2007, Pages 1120-1125

https://doi.org/10.1016/j.amc.2006.09.015 Get rights and content

Abstract

This study deals with the use of conjugate gradient method in conjunction with an adjoint problem formulation for the estimation of an unknown coefficient in a boundary value problem (BVP) of second-order. in the present approach, no a priori assumption is required regarding the functional form of the unknown. The accuracy of the inverse analysis is examined by simulated exact and inexact data. The numerical results show that good estimation on unknown coefficient can be obtained for all the test cases considered in this study.

Introduction

Inverse problems play nowadays an important role in the solution of a number of practical problems. The use of inverse methods represent a new research direction, where the results obtained from numerical simulation and from experiments are not simply compared a posteriori, but a closed synergism exist between experimental and theoretical researches during a course of study, in order to obtain the maximum of information regarding the physical problem under consideration. Most of the methods for the solutions of inverse problems, which are currently in common use, were formalized in the last four decades in terms of their capabilities to treat ill-posed unstable problems. The basis of such formal methods resides on the idea of reformulating an inverse problem in terms of an approximate well-posed problem, by utilizing some kind of regularization techniques [1], [2], [3], [4], [5], [6], [7], [8], [9].

In [1], the in-cylinder pressure in an internal combustion engine was modeled as a function of crank angle. This model was stated as a boundary value problem of second-order and it was studied as an inverse problem. In this work, we investigate the use of conjugate gradient method together with an adjoint problem formulation for the estimation of an unknown function that appeared in the coefficients of a boundary value problem of second-order. The inverse problem considered in this work is solved by using a function estimation approach [4], [5], [6], where no information is a priori available regarding the forms of the unknown function, except for the functional space that they belong to. It is assumed that the unknown function belong to Hilbert space of square integrable function in the spatial domain of interest (L₂[D]).

This article is organized as follows: (i) problem definition, (ii) variational formulation, (iii) iterative regularization procedure, (iv) stopping criterion, and (v) numerical results.

Section snippets

Problem definition

In this study we consider the following boundary value problem: $\{\begin{matrix} \frac{d^{2} y}{d x^{2}} + p (x) \frac{d y}{d x} + q (x) y = g (x), & a < x < b, \\ y (a) = α, & y (b) = β, \end{matrix}$ where p(x) ∈ C¹[a, b] and q(x),g(x) ∈ L₂[a,b].

Notice that in the direct problem, all coefficients p(x), q(x), and g(x) are regarded as known quantities, so that a direct problem is concerned with the computation of y(x).

For the inverse problem of interest here, the function q(x) is regarded as unknown. Such function will be estimated by using measurements of y(x) taken at the locations x_i $y (x_{i}$

Variational formulation

For estimation of the unknown function q(x), we use the variational formulation of the inverse problem of interest. For this purpose we make use of minimization procedure involving the following objective functional $J (q (x)) = \frac{1}{2} \sum_{i = 1}^{N} [y (x_{i}; q) - Y_{i}]^{2} .$ The estimated dependent variable y(x_i; q) is obtained from the solution of the direct problem at the measurement positions x_i, i = 1, 2, …, N with the estimates for q(x).

The conjugate gradient method with an adjoint problem is used for the minimization of the

Iterative regularization procedure

For the estimation of q(x), the iterative procedure of the conjugate gradient method is written as follows [4], [5], [6]: $q^{n + 1} (x) = q^{n} (x) - β^{n} d^{n} (x), n = 0, 1, \dots,$ where dⁿ(x) is the direction of descent, β is the search step size, and n is the number of iterations.

For the iterative procedure, the direction of descent is obtained as a linear combination of the gradient direction with directions of descent of the previous iterations. The direction of descent for the conjugate gradient method can be written as

Stopping criterion

In Section 3, it was pointed out that the conjugate gradient method of function estimation belongs to the class of iterative regularization methods. In such class of methods, the stopping criterion for the computational procedure is used as a regularization parameter, so that sufficiently accurate and smooth solution is obtained for the unknown functions. For the stopping criterion, we illustrate in this work the use of the discrepancy principle [4], [5], [6], [8], [9].

With the use of the

Numerical results

The accuracy of the present solution approach is examined by using simulated measurements of y(x) containing random errors in the inverse analysis in the following examples.

For the test cases examined, the direct, sensitivity and adjoint problems were numerically solved by using MATLAB software. The simulated measurements of y(x) in the following examples contained random errors with standard deviation σ = 0.01Y_max, where Y_max is the maximum absolute value of the measured variable.

Example 1

For the first

Conclusion

In this article, a function estimation approach based on the conjugate gradient method is applied for the estimation of the function that appeared in the coefficient of a boundary value problem of second kind. The accuracy of the proposed solution method is addressed by using simulated measurements containing random errors.

This method can be applied for a lot of industrial and engineering problems that deals with boundary value problems with unknown boundaries, such as the inverse problem that

References (9)

A. Shidfar et al.
Numerical study of in-cylinder pressure in an internal combustion engine
Appl. Math. Comput.
(2005)
A. Shidfar et al.
Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials
Appl. Math. Comput.
(2006)
A. Shidfar et al.
Numerical solution of inverse heat conduction problem with nonstationary measurements
Appl. Math. Comput.
(2005)
O.M. Alifanov et al.
Extreme Method for Solving Ill-posed Problems with Application to Inverse Heat Transfer Problems
(1995)

There are more references available in the full text version of this article.

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A computational procedure for estimation of an unknown coefficient in an inverse boundary value problem

Abstract

Introduction

Section snippets

Problem definition

Variational formulation

Iterative regularization procedure

Stopping criterion

Numerical results

Conclusion

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Extreme Method for Solving Ill-posed Problems with Application to Inverse Heat Transfer Problems