Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 631950, 15 pages

http://dx.doi.org/10.1155/2013/631950

Research Article

Simultaneous Reconstruction of Coefficients and Source Parameters in Elliptic Systems Modelled with Many Boundary Value Problems

Nilson C. Roberty

Programa de Engenharia Nuclear-COPPE, Federal University of Rio de Janeiro, 21945-970 Rio de Janeiro, RJ, Brazil

Received 31 January 2013; Accepted 15 March 2013

Academic Editor: Ben T. Nohara

Copyright © 2013 Nilson C. Roberty. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this work the inverse problem for determination of unknown parameters related to both intensities and support of sources and materials coefficients in second-order elliptic equations models is posed with over specification of data on the boundary. A discrepancy function based on difference of two mixed problems formulated by splitting the Cauchy data is introduced. This function controls the measured difference between the two solutions for the same set of Cauchy data. Parameters can be determined by minimization of this function under guess values. The concept of Calderón projector gap is introduced as a tool for checking the consistency of Cauchy data. Numerical implementations based on quadratic finite elements are presented in a two dimensional square (−1, +1) × (−1, +1) model with unknown source, conductivity, and absorption supported by an also unknown characteristic square shape interior domain. Since this minimization involves the iterative solution of a huge number of direct boundary value problems, the adoption of a non-differentiable minimization algorithm is recommended and the Nelder-Mead simplex method is used to search for optimal parameters.

1. Introduction

The practical expression of linear elliptic partial differential equations found in most engineering applications is represented by the following system, in which the fields may be a vector and coefficients can be represented by matrices and vectors according to the following. To find such that

where is the outward unit normal vector on , the boundary of the domain is supposed to have a Lipschitz dissection in Dirichlet and Neumann boundaries. The subscript coefficients in these equations give physical information about the model and are

(i)the diffusive flux term, , which is also known as conductivity; (ii)the conservative convective flux term, ; (iii)the conservative flux source term, ; (iv)the convection term, ; (v)the absorption term, (also known as potential);

and the the source term . Boundary conditions data are

(i)the normal trace data, ; (ii)the function trace data, physically known as boundary source, ; (iii)the constraint that appears when Dirichlet and Neumann boundaries are active simultaneously, ; (iv)the boundary absorption term, .

In a large set of applications, this system fits in the class of strongly elliptic second-order partial differentials operators with representation

where coefficients are functions from into . They contain functions for the physical quantities presented in the previous paragraph. Thus, is a column vector with scalar fields, and is a vector-valued function. The principal part of , contains the diffusive part of the model and gives the strongly elliptic characteristic to the system. It can be written in divergence form as

Also, if is a Lipschitz domain and is the trace operator, then the conormal derivative

is well defined on .

Coefficients and source have specific meaning in the various technological areas and are given by functions containing parameters which in a specific application may not be a priori known. This work is addressed to investigate the class of problems in which we want to determine unknown parameters in the functions that characterize these coefficients and sources. To compensate this incomplete information that ill posed the problem, we suppose that both Neumann and Dirichlet data are prescribed for many boundary value problems. These problems are formulated for the same physical coefficients and source which depend on the same set of unknown parameters. In Section 2 we introduce the definition of Lipschitz domain and the concept of Lipschitz boundary dissection. Also some basic mathematical concepts necessary for understanding the well-posedness of strong elliptic systems are presented. In Section 3 we use the Lagrange-Green identity to derive the boundary integral and the reciprocity gap equations by alternatively using the singular and a regular part of the fundamental solution of this kind of operator. In Section 4 the integral Green’s function and the boundary integral method are discussed. The concept of Calderón projector gap as a tool to check Cauchy data consistency is discussed in Section 5. Section 6 demonstrates the main Theorem 18 that justifies the methodology proposed in Section 8. Section 7 discusses the many boundary value methodology treatment for the inverse problem parameter determination. In Section 8 we present the optimization problem based on minimization of a discrepancy function. This function is defined with solutions of complementary problems constructed with Lipschitz dissection of the many boundary value Cauchy data. Finally, in Section 9, a numerical experiment based on quadratic finite elements solutions of the direct problem and the Nelder-Mead simplex nondifferentiable algorithm are implemented with success according to part of the ideas here introduced.

2. Mathematical Preliminaries

2.1. Lipschitz Domains

Let be an open set with boundary . In order to use main results from analysis, we suppose that can be represented locally as the graph of a Lipschitz function, that is, a Holder continuous function. A domain is a Lipschitz hypograph when there is a Lipschitz function such that

Consider a disjoint union

where and are relatively open subsets of , and is their common boundary in .

When is a Lipschitz hypograph, we call this disjoint union a Lipschitz dissection of if there is a Lipschitz function such that

Definition 1. The open set is a Lipschitz domain when its boundary is compact and there exist finite open families and in such that

(i); (ii)each can be transformed to a Lipschitz hypograph by a rigid motion; (iii)for each , .

Definition 2. Consider a Lipschitz domain. We say that (6) is a Lipschitz domain dissection of if there are Lipschitz dissections such that

for all . Note that the subsets and need not be connected.

Let denote the conjugate transpose of a matrix or a vector. When the leading coefficients are Lipschitz functions and the lower order are functions, the operator (2), is a bounded linear operator.

2.2. The Lagrange-Green Identity

By defining , we state the divergence theorem as

where is a Lipschitz domain and . This result can be generalized to by approximating from the interior. The integration by parts identity in this situation is

Let be the linear differential operator (2). Repeated application of integration by parts gives a sesquilinear form that is known as Lagrange-Green identity:

which is bounded in :

The formal adjoint of will be denoted by :

Also, a dual of the conormal derivative (4) can be defined in :

Lemma 3. Let be a Lipschitz domain, and ; let the coefficients , , and be functions, and be, respectively, the duality pairs in and in , and . If:

(i) are Lipschitz and , then

(ii) and are Lipschitz and , then

(iii), then the first Green identity (15) is verified for and ; (iv), then the first adjoint Green identity (16) is verified for and ;(v) both and , then the second Green identity

is verified for and ; (vi) in and , then there exist such that

Furthermore, is uniquely determined by both and and not only by , and one has the estimate

Proof. See Lemmas , , , and in [1].

Definition 4. Let be the closed subspace dense in . We say that and are coercive on if

Lemma 5. (i) is coercive if and only if its principal part is coercive on .

(ii) Assume that the coefficients are bounded and uniformly continuous on , then is strongly elliptic if and only if it is coercive on .

Proof. See Lemmas and in [1].

Definition 6. We say that a differential operator is strongly elliptic on if

The Trace Operator

Definition 7. The trace operator is defined by the restriction of the function to .

Sometimes, the function is the restriction of a function and we must distinguish the trace from the interior of and the trace from the exterior . In this case we use the standard notation .

Lemma 8. If is domain, and if , then has a unique extension to a bounded linear operator with continuous right inverse

2.3. Direct Problem with Strongly Elliptic Operators

Let be a domain with Lipschitz dissection boundary . The mixed boundary value problem for the physical model given by (1) is given by the well-posed problem : to find such that

we can show that (24) has the following weak formulation :

3. Fundamental Solution, Boundary Integral, and Reciprocity Gap Equations

In order to understand the methodology used to solve the many boundary data inverse problem that is proposed in this work, let us introduce some concepts related to the integral representation of strongly elliptic model (24). Basic information about partial differential equations can be found in [2], in linear integral equations in [3], strongly elliptic systems in [1], partial differentials equations with many boundary measurements in [4], and boundary integral in [5]. Applications to reconstruction of sources are investigated in [6–8]. Also the works [9, 10] give an important contribution to integral methods. First of all, the function in the model must be locally integrable, that is, absolutely integrable in every compact subset of . We represent this set of functions as . Also, coefficients must be supposed to be functions in order to treat the problem in the context of Schwartz distributions. The second Green identity in Lemma 3(v) can be restated in a distributional framework with , then the second Green identity

is verified for . Note that here , , are one side trace and conormal derivative at the boundary. The complete theory must be done in the context of the transmission problem at the interface , which is beyond the scope of this work.

3.1. Fundamental Solution

Definition 9. A Fundamental solution to the operator is a distributional solution of the equation

where is the delta Dirac operator defined as , for and .

Note that the fundamental solution can be decomposed into two parts. One part is a regular function and may not have dependence on the point , and the other is the so-called fundamental solution with pole for the operator :

where

and is the particular solution corresponding to the singular source at . The second Green identity in Lemma 3(v) can be restated in a distributional framework with and gives for

As a trivial consequence of the definition of adjoint of operator , the operator

is a smoothing integral operator with kernel . It is the two sides inverse for operator , that is,

The operator is also known as the volume potential associated with .

3.2. The Third Green Identity and Jumps Relations

The bounded Lipschitz domain has a complementary and unbounded domain . The Lagrange-Green identity (33) can accordingly be extended in all as

The field can be viewed as restriction from some defined in both interior and exterior domains and jumps in the traces

and one side conormal derivatives

The normal points to the exterior of . The jumps of these quantities are

They can be used to define a new two sides trace operator and two sides conormal derivatives operators as distributions in and include possible jumps resulting from transmission of discontinuities in the surface as distributions sources for discontinuities

Definition 10. The single layer and the double layer potential are defined, respectively, by

By introducing the singular solution of (27), we can write the single layer potential as

and the double layer potential as

Since , they are locally bounded operators and satisfy the following mapping properties and jump relations.

Lemma 11. Let . Then

(i);(ii);(iii)if , then and ;(iv)if , then and .

Proof. See Costabel in [9].

Definition 12. When , with , has compact support in and , we can enunciate the third Green identity

3.3. Boundary Integral and Reciprocity Gap Integral Equations

The Lagrange-Green identity (26) can be used to search for locally integrable solutions to the operator equation . Depending on which part of the fundamental solution for the adjoint operator we use as test function in (26), that is, the regular or the singular part, we obtain the boundary integral equation or the reciprocity integral equation, respectively. These two equations are both integrals equations associated with the same operator and can be used alternatively.

Boundary Integral Equation. By using the third Green identity (41) with extension by zero in the exterior domain, we obtain integral equation

The interior trace acting on (42) produces the boundary integral equation, that is, for

Substituting the double and the single layer potentials by explicit integral representation and omitting the interior trace operator, we have

which is valid for .

Reciprocity Gap Integral Equation. We alternatively may use only the regular part of (28), which is given by functions in the subspace defined by the null set (29) of , and obtain the reciprocity gap equation

which can be written explicitly as

for all .

Remark 13. When we do numerical implementation of integral (42) for , we must take care in the evaluation of the double layer potential. This must be done in the sense of the Cauchy principal value. In this case the interior and exterior traces must be distinguished.

Remark 14. Note some difference between these two solutions. Equation (42) is derived by introducing a particular solution, that is, one unique function in the Lagrange-Green identity (26), while by choosing test functions in the set (29) to introduce in (26), we obtain the weak variational equation (45). Another important property related to the reciprocity gap equation is that when is sufficiently regular there exists a homeomorphism between the space of test functions and some spaces of functions in the boundary, such as , [8]. This means that the set of traces of functions in is dense in . We can use this result to constructs Green’s functions for direct problems by using both the singular and the regular part (28) of the fundamental solution.

4. Integral Equations-Based Methodologies for Direct Problem Solution

4.1. Green’s Function Methodology

Direct problems are well posed by satisfying only one part of the Cauchy data at the boundary; that is, when we have information about the Dirichlet part, we do not know the Neumann part, and vice versa. The same situation also occurs when linear combinations of Dirichlet and Neumann data are known in the Robin boundary condition. Let us consider the already defined mixed boundary value problem (24), : to find such that

Let us consider its solution given by the third Green identity supposing that is extended by zero outside :

We can solve this problem in the integral equation framework with two methodologies: the Green’s function one and the boundary integral equation formulation. In the first one, some particular regular solution of the fundamental solution that depends on the domain boundary needs to be chosen in order to null the unknown part of the Cauchy data in that boundary. Let us consider the layer potentials in the context of extension by zero outside . By perturbing the singular solution, , of (27) with some regular solution and since, in this problem, the boundary presents the following Lipschitz dissection:

then a modified single layer potential

and a modified double layer potential

make explicit the contributions of each part of the boundary to the potentials. Since in the Dirichlet boundary we know

and in the Neumann boundary we know

we must choose as solution of the auxiliary problem

in order to cancel, in the single layer, the contribution of the unknown part and, in the double layer, the contribution of the unknown part . We have defined the Green’s function associated with the mixed problem (24), , where is a solution of (54). The support of its trace is , while the support of the trace of its conormal derivative is .

The Green’s function single layer potential is

and the Green’s function double layer potential is

Combining them we obtain the Green’s function solution for the mixed problem (24)

Remark 15. The difficulties in determining the Green’s function associated with a given pair, operator, and boundary are mainly concentrated in the necessity of approximating the singular part of the fundamental solution by a basis for the null space that characterizes the regular part. Since the space is homeomorphic to for smooth boundaries , [8], despite numerical questions, this can be done when the boundary is Lipschitz.

4.2. Boundary Integral Methodology

In the Boundary integral equation methodology for the mixed boundary value problem (24), , we again consider extension by zero outside and use the system made with the Lipschitz dissection of the third Green’s identity (41) at the boundary

and its conormal derivative

For clearness we separate these restrictions. The respective restrictions are

(i) trace on :

(ii)trace on :

(iii)trace of conormal derivative on :

(iv)trace of conormal derivative on :

The compositions involving traces of the layers potentials can be expressed in terms of the Fredholm integrals index zero operators

(i);(ii);(iii);(iv).

Substituting these operators in (60), (61), (62), and (63) and observing the appropriated restrictions and that , we have

(i) trace on :

(ii) trace on :

(iii) trace of conormal derivative on :

(iv) trace of conormal derivative on :

In the mixed boundary value problem (24), ,

So we can select (64) and (67) to substitute the trace and the conormal derivative by the Cauchy data dissection

and we use it then to form the following system:

This system can be solved to determinate the unknown quantities in terms of the known. By putting

the problem reduces to the solution of the formal equation

The extension of the partially known Cauchy data (68) to all can be done by solving (73). This problem has been investigated in [1], which showed that when is formally self-adjoint and coercive on , then is a sum of a positive and bounded below operator with a compact operator. So, it is a Fredholm operator with index zero, and consequently the Fredholm theory can be applied.

5. The Calderon Projector Gap

The adoption of the integral equation (43) at the boundary or its regular reciprocity gap counterpart (45) for modelling inverse problems is facilitated by the fact that in most inverse problems the Cauchy data is known. Meanwhile, (42) can also be operated by the interior conormal derivative and

Let us now consider the mixed boundary value problem (24) with data . The Cauchy data in the unknown part of the boundary can be determined by the boundary integral methodology and is supposedly known. The Calderón operator is the linear operator defined by

and if , for some , then the function

satisfies the mixed boundary value problem (24) with data . Since for these data

then is a projection operator. The fact that the second equation (74) in the Calderón system is the conormal derivative of the same equation whose trace give the first equation, by taking its trace, is responsible for the noted behaviour of projector when the source is zero. In terms of the Fredholm integrals operators with index zero, the Calderón projector can be written as

When the mixed boundary value problem (24) is posed with a nonnull source , that is, , (77) is no longer valid and modifies to

which is the Calderón projector gap for the source . Considering the Lipschtiz dissection, this matrix equation can be written as the matrix form of the systems (64), (65), (66), and (67) in the following matrix equation:

and also writen as the integral equation system

Remark 16. For a given association of a Lipschitz domain with a source distribution, the Calderón projector gap can be understood as a restriction which the Cauchy data must satisfy in order to be consistent data with boundary value problems.

6. Complementary Problems on Lipschitz Domains

Definition 17. Let us consider two mixed boundary value problems and defined on the same Lipschitz domain . We say that these problems are complementary if , , and there exist Cauchy data such that

Theorem 18. Suppose that two mixed boundary value problems and have solutions and , respectively. If they are complementary, then

Proof. Note that is compact and

With the characteristic function for , in , we can introduce the extended Dirichlet data function which is the extension of to , with , and consequently, . Similar definitions can be done to treat the Neumann part of Cauchy data , . So,

Denoting , the solution will be, via boundary integral equation method,

By taking the trace and the conormal trace of (86), we see that they satisfy the Calderón gap projection dissection equation (80). So, Cauchy data obtained by the extension formulate a unique problem with integral representation (86).

7. Inverse Problem for Parameter Determination

Materials properties and sources in problems modelled with the strongly elliptic operator (2) can depend on some unknown parameters related to the support of inclusions inside where the coefficient has some different functional description, or even with the functional description itself. We consider these parameters collected in a parameter vector and that the coefficients and source are represented as

which for are functions from into . The parameters can be supposed to have values in some known interval . In this way, the strongly elliptic operator model with parameter dependence is formally written as

Modifications in the notation for (24) and (25) are straightforward.

7.1. The Many Boundary Value Problems Treatment for the Inverse Problem of Parameter Determination

For the same operator equation we suppose that some parameters are unknown and can be compensated with the overprescription of boundary conditions. In this way, Cauchy data are known in many boundary value system of problems indexed by ,

In this kind of problem we are supposing the knowledge of the Steklov-Poincaré operator, which is an extended definition of the Dirichlet to Neumann map for this kind of system, at some points in the trace space

This set of Cauchy data fully prescribed at the boundary can be used to formulate a nonunique set with well-posed direct problems in the following way.

Choose some Lipschitz dissection of associated with problem and given by

where and are disjoint, eventually empty or nonconnected, relatively open subsets of , having as their common boundary. Consider also the restriction for Cauchy data for problem associated with this partition:

For each one of these Cauchy data of the Lipschitz dissection, we can formulate two complementary well-posed sets of mixed boundary value problems, respectively, and , in such way that given a guess set of parameter , for problems , to find complementary solutions and such that

Note that since the partition done with the Lipschitz dissection is arbitrary, it can also be different for different problems in the many boundary value problems set, or, if necessary, we can do two or more dissections for the same problem. The correctness of this procedure will depend on the information about the parameters that it produces. One basic rule of thumb is that partitions must be chosen in a way to avoid the guess direct problems to be non-well-posed. We introduce a lemma that is an application of Theorem 18 to the set of many boundary value problems and that will be used later to introduce a methodology for solution to this kind of inverse problem.

Lemma 19. Suppose that in the model given by operator and source , characterized by the parameter set , the associated Cauchy boundary data are given by (90). If for some and for some Lipschitz dissection one has and solutions of problem (93), then

8. A Methodology Based on Discrepancy Function

In Section 7.1 we introduced the idea of decomposing the set of many boundary value problems, for which we knew the Cauchy data, in two subsets of complementary well-posed mixed direct problems. The idea now is exploring the fact that these two sets of solutions indexed by and must be, under ideal conditions, equal for each problem , as has been stated in Theorem 18 and Lemma 19, and create some discrepancy function that measures observed differences for guess value of the parameters. Norms in the solution space for the direct problems can be adopted as measures; that is,

where can be the norm of continuous functions solutions , or square integrable solutions, or even with some first derivative control solutions.

We now posed the optimization problem: in the guess set of parameters , to find that minimizes the discrepancy (95) between Lipschitz dissected solutions.

The simplest way to do numerical implementation of this methodology is by using the well-established finite elements method, since the two direct problems associated with each Cauchy data in the set can be solved for guess a priori values of the parameters. Of course, the boundary integral methodology or the Green’s function methodologies can also be used, but this is not the more conventional procedure. From computational point of view, minimization of the discrepancy function can be easily implemented if the algorithm does not require the computations of gradients of the solution with respect to the parameters. The Nelder-Mead simplex method in low dimensions, [11], is an ideal algorithm to be used when the parameters space dimension is small but with the drawback that it is slow, and there are many examples where it does not converge.

8.1. The Nelder-Mead Simplex Algorithm

The discrepancy function (95) is a nonlinear function of the parameters . The minimization problem can be solved with different optimization algorithms, but the cost of determining derivatives with respect to parameters defining the source makes the utilization of gradient-based algorithms more difficult to implement than those algorithms based only on functional evaluation, such as the case of the Nelder-Mead method introduced in [11]. This algorithm attempts only to minimize the scalar-value nonlinear discrepancy function of characteristic sources parameters that can be obtained from solutions of sets of problems II and II. It falls in the general class of direct search methods. A nondegenerated simplex with the same dimension of the number of parameters to be determined is established at each step. Each iteration begins with this simplex, which is the hull of vertices in the dimensional parameters space. The Nelder-Mead simplex algorithm is based on four operations:

(1)reflection with algorithm parameter ; (2)expansion with algorithm parameter ;(3)contraction with algorithm parameter :

(i)outside;(ii)inside;

(4)shrink with algorithm parameter .

Steps are used to create a new simplex by attempting to replace the vertex with the highest functional values with a smaller one. If this attempt is unsuccessful, then the current simplex is reduced in size using Step , and the entire procedure is repeated. We adopted here the universal typical algorithm parameters values: , , , and .

9. Some Numerical Results

The numerical experiment that illustrates this work is a model in which the square has in its interior a small rectangle that has unknown center and unknown edges and , which support unknown parameters related with the conductivity, , the potential, , and the source intensity, . Cauchy data are synthetically generated with a problem in which parameters values are known to be equal to in the exterior of the small rectangle, and all equal in the interior. Also the unknown information about the rectangle used are center at the origin and side . The parameters in operator are

and the set of parameters are .

The rectangle has boundaries

and is counterclockwise oriented.

Cauchy data are synthetically produced by solving a set of Dirichlet direct problems with parameters with quadratic Lagrange finite elements method:

; ; ; ; ,

generating Neumann data ; ; ; ; .

These synthesized Cauchy data are shown in Figures 1 and 2.

Figure 1: Cauchy data Dirichlet problems (0).

Figure 2: Cauchy data Dirichlet problems , , , and .

They have been interpolated with piecewise cubic splines and used in the inverse algorithm. The boundary of the square has been dissected in two nonconnected parts composed by

Ten problems formulated with the dissection of these Cauchy data can now be used to evaluate the discrepancy function based on the following norm:

The search starts with random generated initial data in the intervals for the unknown parameters, and the convergence results are presented in Figure 3.

Figure 3: Iterative simultaneous reconstruction of rectangle shape, conductivity, absorption, and source.

10. Conclusions

We proposed a methodology for reconstruction of unknown parameters associated with coefficients and source in strongly elliptic system. To make it clear, we also introduce the most important mathematical concepts involved in the solution of the strongly elliptic problem with integral equations at the boundary of a Lipschitz domain. In the inverse problem, the existence of the unknown parameters is compensated with the prescription of many Cauchy data related experimentally with the same set of parameters. We demonstrate that a discrepancy function depending on the parameters must be minimized in order to be consistent with the given Cauchy data. The main ideas used to develop this formulation are Lipschitz dissection and Calderón projector gap. In this first work, the optimization methodology is numerically investigated with nondifferentiable Nelder-Mead search algorithm. Numerical results are presented to illustrate the ideas. Further research involving differentiability and the use of differentiable algorithms is currently being investigated.

Acknowledgments

This work is supported by Brazilian agencies CNPq, CAPES, and COPPETEC Foundation.

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