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

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 𝑢(𝑥) such that∇⋅−𝑐∇𝑢−𝛼𝑢+𝛾+𝛽⋅∇𝑢+𝑎𝑢=𝑓if𝑥∈Ω,ℎ𝑢=𝑔if𝑥∈𝜕Ω_𝐷,𝜈⋅𝑐∇𝑢+𝛼𝑢−𝛾+𝑞𝑢=𝑔_𝜈−ℎ^∗𝜇if𝑥∈𝜕Ω_𝑁, (1) where 𝜈 is the outward unit normal vector on 𝜕Ω∶=𝜕Ω_𝐷∪Π∪𝜕Ω_𝑁, the boundary of the domain Ω 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, 𝑐, which is also known as conductivity; (ii)the conservative convective flux term, 𝛼; (iii)the conservative flux source term, 𝛾; (iv)the convection term, 𝛽; (v)the absorption term, 𝑎 (also known as potential); and the the source term 𝑓. Boundary conditions data are(i)the normal trace data, 𝑔_𝜈; (ii)the function trace data, physically known as boundary source, 𝑔; (iii)the constraint that appears when Dirichlet and Neumann boundaries are active simultaneously, 𝜇,ℎ; (iv)the boundary absorption term, 𝑞.

In a large set of applications, this system fits in the class of strongly elliptic second-order partial differentials operators with representation ℒ𝑢=−^𝑑_𝑗=1^𝑑_𝑘=1𝜕_𝑗𝐴_𝑗𝑘𝜕_𝑘𝑢+𝐴_𝑗𝜕_𝑗𝑢+𝐴𝑢, (2) where coefficients (𝐴_𝑗𝑘,𝐴_𝑗,𝐴) are functions from Ω into ℝ^𝑚×𝑚. They contain functions for the physical quantities presented in the previous paragraph. Thus, 𝑢 is a column vector with 𝑚 scalar fields, and ℒ𝑢∶Ω→ℝ^𝑚 is a vector-valued function. The principal part of ℒ, ℒ₀ contains the diffusive part of the model and gives the strongly elliptic characteristic to the system. It can be written in divergence form as ℒ₀𝑢=−^𝑑_𝑗=1𝜕_𝑗ℬ_𝑗𝑢,whereℬ_𝑗=^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘. (3) Also, if Ω is a Lipschitz domain and 𝛾 is the trace operator, then the conormal derivative ℬ_𝜈𝑢=^𝑑_𝑗=1𝜈_𝑗𝛾ℬ_𝑗𝑢 (4) is well defined on 𝜕Ω.

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 Ω⊂ℝ^𝑑 be an open set with boundary Γ=𝜕Ω=Ω∩ℝ^𝑑⧵Ω. In order to use main results from analysis, we suppose that Γ can be represented locally as the graph of a Lipschitz function, that is, a Holder continuous 𝐶^0,1 function. A domain Ω is a Lipschitz hypograph when there is a Lipschitz function 𝜉∶ℝ^𝑑−1→ℝ such that Ω=𝑥=𝑥^′,𝑥_𝑑∈ℝ^𝑑∶𝑥_𝑑<𝜉𝑥^′,𝑥^′=𝑥₁,…,𝑥_𝑑−1∈ℝ^𝑑−1. (5) Consider a disjoint union Γ=Γ_𝐼∪Π∪Γ_𝐼𝐼, (6) where Γ_𝐼 and Γ_𝐼𝐼 are relatively open subsets of Γ, and Π=Γ_𝐼∩Γ_𝐼𝐼 is their common boundary in Γ.

When Ω is a Lipschitz hypograph, we call this disjoint union a Lipschitz dissection of Γ if there is a Lipschitz function 𝜌∶ℝ^𝑑−2→ℝ such that Γ_𝐼=𝑥=𝑥^′′,𝑥_𝑑−1,𝑥_𝑑∈Γ∶𝑥_𝑑−1<𝜌𝑥^′′,Π=𝑥=𝑥^′′,𝑥_𝑑−1,𝑥_𝑑∈Γ∶𝑥_𝑑−1=𝜌𝑥^′′,Γ_𝐼𝐼=𝑥=𝑥^′′,𝑥_𝑑−1,𝑥_𝑑∈Γ∶𝑥_𝑑−1>𝜌𝑥^′′. (7)

Definition 1. The open set Ω is a Lipschitz domain when its boundary Γ is compact and there exist finite open families {𝑊_𝑗} and {Ω_𝑗} in ℝ^𝑑 such that(i)Γ⊆⋃_𝑗𝑊_𝑗; (ii)each Ω_𝑗 can be transformed to a Lipschitz hypograph by a rigid motion; (iii)for each 𝑗, 𝑊_𝑗⋂Ω=𝑊_𝑗⋂Ω_𝑗.

Definition 2. Consider Ω a Lipschitz domain. We say that (6) is a Lipschitz domain dissection of Γ if there are Lipschitz dissections 𝜕Ω_𝑗=Γ_𝐼𝑗∪Π_𝑗∪Γ_𝐼𝐼𝑗 such that 𝑊_𝑗∩Γ_𝐼=𝑊_𝑗∩Γ_𝐼𝑗,𝑊_𝑗∩Π=𝑊_𝑗∩Π_𝑗,𝑊_𝑗∩Γ_𝐼𝐼=𝑊_𝑗∩Γ_𝐼𝐼𝑗, (8) for all 𝑗. Note that the subsets Γ_𝐼 and Γ_𝐼𝐼 need not be connected.

Let (⋅)^∗ denote the conjugate transpose of a matrix or a vector. When the leading coefficients 𝐴_𝑗𝑘 are Lipschitz functions and the lower order are 𝐿^∞(Ω)^𝑚×𝑚 functions, the operator (2), ℒ∶𝐻²(Ω)^𝑚→𝐿²(Ω)^𝑚 is a bounded linear operator.

2.2. The Lagrange-Green Identity

By defining 𝜕_𝑘=𝜕/𝜕𝑥_𝑘,𝑘=1,…,𝑑, we state the divergence theorem as _Ω𝜕_𝑘𝑢(𝑥)𝑑𝑥=_𝜕Ω𝑢(𝑥)𝜕𝑥_𝑘𝜕𝜈𝑑𝑆_𝑥=_𝜕Ω𝑢(𝑥)𝜈_𝑘𝑑𝜎_𝑥, (9) where Ω is a Lipschitz domain and 𝑢∈𝐶¹(Ω). This result can be generalized to 𝑢∈𝐶¹(Ω)∩𝐶⁰(Ω) by approximating Ω from the interior. The integration by parts identity in this situation is _Ω𝑣^∗𝜕_𝑘𝑢(𝑥)𝑑𝑥=_𝜕Ω𝑣^∗(𝑥)𝑢(𝑥)𝜈_𝑘𝑑𝜎_𝑥−_Ω𝜕_𝑘𝑣^∗𝑢(𝑥)𝑑𝑥,𝑘=1,…,𝑑. (10) Let ℒ be the linear differential operator (2). Repeated application of integration by parts gives a sesquilinear form that is known as Lagrange-Green identity: Φ(𝑢,𝑣)=_Ω^𝑑𝑑_𝑗=1𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑢^∗𝜕_𝑗𝑣+^𝑑_𝑗=1𝐴_𝑗𝜕_𝑗𝑢^∗𝑣+(𝐴𝑢)^∗𝑣𝑑𝑥 (11) which is bounded in 𝐻¹(Ω)^𝑚: |Φ(𝑢,𝑣)|≤𝐶‖𝑢‖_𝐻¹_(Ω)^𝑚‖𝑣‖_𝐻¹_(Ω)^𝑚. (12)

The formal adjoint of ℒ will be denoted by ℒ^∗: ℒ^∗𝑢=−^𝑑𝑑_𝑗=1𝑘=1𝜕_𝑗𝐴^∗_𝑘𝑗𝜕_𝑘𝑢−^𝑑_𝑗=1𝜕_𝑗𝐴^∗_𝑗𝑢+𝐴^∗𝑢. (13)

Also, a dual of the conormal derivative (4) can be defined in 𝜕Ω: ℬ^∼_𝜈𝑢=^𝑑_𝑘=1𝜈_𝑗𝛾ℬ^∼_𝑗𝑢,whereℬ^∼_𝑗𝑢=^𝑑_𝑘=1𝐴^∗_𝑘𝑗𝜕_𝑘𝑢+𝐴^∗𝑢. (14)

Lemma 3. Let Ω be a Lipschitz domain, 𝑢 and 𝑣∈𝐻¹(Ω)^𝑚; let the coefficients 𝐴_𝑗𝑘, 𝐴_𝑗, and 𝐴 be 𝐿^∞(Ω)^𝑚×𝑚 functions, (⋅,⋅)_Ω and (⋅,⋅)_𝜕Ω be, respectively, the duality pairs in Ω and in 𝜕Ω, and ℒ𝑢=𝑓. If:(i)A_𝑗𝑘 are Lipschitz and 𝑢∈𝐻²(Ω)^𝑚, then Φ(𝑢,𝑣)=(ℒ𝑢,𝑣)_Ω+ℬ_𝜈𝑢,𝛾[𝑣]_𝜕Ω; (15)(ii)𝐴_𝑗𝑘 and 𝐴_𝑗 are Lipschitz and 𝑣∈𝐻²(Ω)^𝑚, then Φ(𝑢,𝑣)=𝑢,ℒ^∗𝑣_Ω+𝛾[𝑢],ℬ^∼_𝜈𝑣_𝜕Ω; (16)(iii)ℒ𝑢∈𝐿₂(Ω)^𝑚, then the first Green identity (15) is verified for 𝑢 and 𝑣∈𝐻¹(Ω)^𝑚; (iv)ℒ^∗𝑢∈𝐿₂(Ω)^𝑚, then the first adjoint Green identity (16) is verified for 𝑢 and 𝑣∈𝐻¹(Ω)^𝑚;(v) both ℒ𝑢 and ℒ^∗𝑢∈𝐿₂(Ω)^𝑚, then the second Green identity (ℒ𝑢,𝑣)_Ω−𝑢,ℒ^∗𝑣_Ω=𝛾[𝑢],ℬ^∼_𝜈𝑣_𝜕Ω−ℬ_𝜈𝑢,𝛾[𝑣]_𝜕Ω (17) is verified for 𝑢 and 𝑣∈𝐻¹(Ω)^𝑚; (vi)ℒ𝑢=𝑓 in Ω and 𝑓∈𝐻^∼−1(Ω)^𝑚, then there exist 𝑔∈𝐻^−1/2(Ω)^𝑚 such that Φ(𝑢,𝑣)=𝑓,𝑣_Ω+𝑔,𝑣_𝜕Ω𝑓𝑜𝑟𝑣∈𝐻¹(Ω)^𝑚. (18) Furthermore, 𝑔 is uniquely determined by both 𝑢 and 𝑓 and not only by 𝑢, and one has the estimate 𝑔_{𝐻−1/2(𝜕Ω)}≤𝐶‖𝑢‖_𝐻¹_(Ω)^𝑚+𝐶𝑓_∼𝐻⁻¹_(𝜕Ω)𝑚. (19)

Proof. See Lemmas (4.1), (4.2), (4.3), and (4.4) in [1].

Definition 4. Let 𝐻¹₀(Ω)^𝑚={𝑢∈𝐻¹(Ω)^𝑚;𝛾[𝑢]=0} be the closed subspace dense in 𝐿²(Ω)^𝑚. We say that ℒ and Φ are coercive on 𝐻¹₀(Ω)^𝑚 if ReΦ(𝑢,𝑢)≥𝑐‖𝑢‖²_𝐻1(Ω)^𝑚−𝐶‖𝑢‖²_𝐻2(Ω)^𝑚. (20)

Lemma 5. (i)ℒ is coercive if and only if its principal part ℒ₀ is coercive on 𝐻¹₀(Ω)^𝑚.
(ii) Assume that the coefficients 𝐴_𝑗𝑘 are bounded and uniformly continuous on Ω, then ℒ is strongly elliptic if and only if it is coercive on 𝐻¹₀(Ω)^𝑚.

Proof. See Lemmas (4.5) and (4.6) in [1].

Definition 6. We say that a differential operator ℒ is strongly elliptic on Ω if Re^𝑑𝑑_𝑗=1𝑘=1𝐴_𝑗𝑘(𝑥)𝜉_𝑘𝜂^∗𝜉_𝑗𝜂≥𝑐𝜉²𝜂²∀𝑥∈Ω,𝜉∈ℝ^𝑑,𝜂∈ℂ^𝑚. (21)The Trace Operator

Definition 7. The trace operator 𝛾∶𝒟(Ω)→𝒟(𝜕Ω) is defined by the restriction of the function 𝑢∈𝒟(Ω) to 𝜕Ω. 𝛾[𝑢]=𝑢|_𝜕Ω (22) Sometimes, the function 𝑢=𝑈|_Ω is the restriction of a function 𝑈∈𝒟(ℝ^𝑑) and we must distinguish the trace from the interior of Ω and the trace from the exterior ℝ^𝑑⧵Ω. In this case we use the standard notation 𝛾^±.

Lemma 8. If Ω is 𝐶^𝑘−1,1 domain, and if 1/2<𝑠≤𝑘, then 𝛾 has a unique extension to a bounded linear operator with continuous right inverse 𝛾∶𝐻^𝑠(Ω)⟶𝐻^𝑠−1/2(Ω). (23)

2.3. Direct Problem with Strongly Elliptic Operators

Let Ω be a domain with Lipschitz dissection boundary 𝜕Ω=𝜕Ω_𝑁∪Π∪𝜕Ω_𝑁. The mixed boundary value problem for the physical model given by (1) is given by the well-posed problem 𝑃_{𝑓,𝑔𝐷,𝑔𝑁}: to find 𝑢∈𝐻¹(Ω)^𝑚 such that 𝑃_{𝑓,𝑔𝐷,𝑔𝑁}𝛾[𝑢]ℒ𝑢=𝑓if𝑥∈Ω,=𝑔_𝐷if𝑥∈𝜕Ω_𝐷,ℬ_𝜈𝑢=𝑔_𝑁if𝑥∈𝜕Ω_𝑁; (24) we can show that (24) has the following weak formulation 𝑊_{𝑓,𝑔𝐷,𝑔𝑁}: 𝑊_{𝑓,𝑔𝐷,𝑔𝑁}(ℒ𝑢,𝑣)_Ω+ℬ_𝜈𝑢,𝛾[𝑣]_𝜕Ω=Φ(𝑢,𝑣)=𝑓,𝑣_Ω+𝑔_𝑁,𝛾[𝑣]_𝜕Ω𝑁if𝑣∈𝐻¹_𝐷(Ω)^𝑚,𝛾[𝑢]=𝑔_𝐷if𝑥∈𝜕Ω_𝐷. (25)

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 𝑢 in the model must be locally integrable, that is, absolutely integrable in every compact subset of Ω. We represent this set of functions as 𝐿¹_loc(Ω). Also, coefficients {𝐴_𝑗𝑘,𝐴_𝑗,𝐴} must be supposed to be 𝐶^∞(Ω) 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 ℒ^∗𝑣=𝛿_𝜉∈((𝐶⁰(Ω))^∗)^𝑚, then the second Green identity (ℒ𝑢,𝑣)_Ω−𝑢,ℒ^∗𝑣_Ω=𝛾[𝑢],ℬ^∼_𝜈𝑣_𝜕Ω−ℬ_𝜈𝑢,𝛾[𝑣]_𝜕Ω (26) is verified for 𝑢∈𝐻²(Ω). Note that here 𝛾, ℬ_𝜈, ℬ^∼_𝜈 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 𝜕Ω, which is beyond the scope of this work.

3.1. Fundamental Solution

Definition 9. A Fundamental solution to the operator ℒ is a distributional solution of the equation ℒ^∗𝑢_𝜉=𝛿_𝜉, (27) where 𝛿_𝜉 is the delta Dirac operator defined as 𝛿_𝜉[𝜙]=𝜙(𝜉), for 𝜉∈Ω and 𝜙∈(𝐶^∞_compact(Ω))^𝑚=(𝒟(Ω))^𝑚.

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 𝜉, and the other is the so-called fundamental solution with pole 𝜉 for the operator ℒ: 𝑢_𝜉(𝑥)=𝑣(𝑥)+𝐺_𝜉(𝑥),𝑥∈Ω, (28) where 𝑣(𝑥)∈𝐻_ℒ^∗(Ω)=𝑣suchthatℒ^∗𝑣=0 (29) and 𝐺_𝜉(𝑥) is the particular solution corresponding to the singular source 𝛿_𝜉 at 𝑥=𝜉. The second Green identity in Lemma 3(v) can be restated in a distributional framework with ℒ^∗𝐺_𝜉=𝛿_𝜉∈ℰ^∗(𝐑^𝑑)^𝑚∶=((𝐶^∞(Ω))^∗)^𝑚 and gives for ℒ𝑢∈ℰ^𝑚=(𝐶^∞(Ω))^𝑚ℒ𝑢,𝐺_𝜉Ω−𝑢,ℒ^∗𝐺_𝜉Ω=ℒ𝑢,𝐺_𝜉Ω−𝑢(𝜉)=0. (30) As a trivial consequence of the definition of adjoint of operator ℒ, the operator 𝒢𝑢(𝜉)=_Ω𝐺(𝜉,𝑥)𝑢(𝑥)𝑑𝑥for𝜉∈𝐑^𝑑 (31) is a smoothing integral operator with kernel 𝐺(𝜉,𝑥)=𝐺_𝜉(𝑥). It is the two sides inverse for operator ℒ, that is, ℒ𝒢𝑢=𝒢ℒ𝑢=𝑢for𝑢∈ℰ^∗𝐑^𝑑𝑚. (32) The operator 𝒢 is also known as the volume potential associated with ℒ.

3.2. The Third Green Identity and Jumps Relations

The bounded Lipschitz domain Ω=Ω⁻⊂ℝ^𝑑 has a complementary and unbounded domain Ω⁺=ℝ^𝑑⧵Ω. The Lagrange-Green identity (33) can accordingly be extended in all ℝ^𝑑 asΦ^±(𝑢,𝑣)=_Ω±^𝑑𝑑_𝑗=1𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑢^∗𝜕_𝑗𝑣+^𝑑_𝑗=1𝐴_𝑗𝜕_𝑗𝑢^∗𝑣+(𝐴𝑢)^∗𝑣𝑑𝑥. (33) The field 𝑢 can be viewed as restriction from some 𝑈∈𝒟(ℝ^𝑑) defined in both interior and exterior domains and jumps in the traces 𝛾^±[𝑢]=𝑈^±_𝛾forsome𝑈^±∈𝒟ℝ^𝑑 (34) and one side conormal derivatives ℬ^±_𝜈[𝑢]=^𝑑_𝑗=1𝜈_𝑗𝛾^±𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑢,_∼ℬ^±_𝜈[𝑢]=^𝑑_𝑗=1𝜈_𝑗𝛾^±𝑑_𝑘=1𝐴^∗_𝑘𝑗𝜕_𝑘𝑢+𝐴^∗_𝑗𝑢. (35) The normal 𝜈 points to the exterior of Ω. The jumps of these quantities are [𝑢]_Γ=𝛾⁺𝑢−Γ⁻𝑢,ℬ_𝜈𝑢_Γ=ℬ⁺_𝜈𝑢−ℬ⁻𝑢,_∼ℬ_𝜈𝑢_Γ=ℬ^∼+_𝜈𝑢−ℬ^∼−𝑢. (36) They can be used to define a new two sides trace operator 𝛾^∗ and two sides conormal derivatives ℬ^∼∗_𝜈 operators as distributions in (𝐶^∞(ℝ^𝑑)^𝑚)^∗ and include possible jumps resulting from transmission of discontinuities in the surface Γ as distributions sources for discontinuitiesℒ𝑢=𝑓+ℬ^∼∗_𝜈[𝑢]_Γ−𝛾^∗ℬ_𝜈𝑢_Γonℝ^𝑑. (37)

Definition 10. The single layer and the double layer potential are defined, respectively, by SL=𝒢𝛾^∗,DL=𝒢ℬ^∼∗_𝜈. (38)

By introducing the singular solution 𝐺_𝜉 of (27), we can write the single layer potential as SLℬ_𝜈𝜓(𝜉)=_Γ𝐺_𝜉(𝑥)ℬ_𝜈𝜓(𝑥)𝑑𝜎_𝑥=_Γ𝐺_𝜉(𝑥)^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝜓(𝑥)𝑑𝜎_𝑥,𝜉∈Ω (39) and the double layer potential as DL𝜓(𝜉)=ℬ_Γ^∼∗_𝜈𝜉𝐺_𝜉(𝑥)^∗𝜓(𝑥)𝑑𝜎_𝑥=_Γ^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝐺_𝜉(𝑥)+𝐴_𝑗𝐺_𝜉(𝑥)×𝜓(𝑥)𝑑𝜎_𝑥,𝜉∈Ω. (40)

Since 𝐺_𝜉(𝑥)∈(𝐶^∞(ℝ^𝑑)⧵{0})^𝑚, they are locally bounded operators and satisfy the following mapping properties and jump relations.

Lemma 11. Let 𝜒∈𝐶^∞_compact(ℝ^𝑑). Then(i)(𝜒SL,𝛾SL,ℬ^±_𝜈SL)∶H^−1/2(Γ)^m×H^−1/2(Γ)^m×H^−1/2(Γ)^m→H¹(ℝ^d)×H^1/2(Γ)^m×H^−1/2(Γ)^m;(ii)(𝜒DL,𝛾^±DL,ℬ_𝜈DL)∶H^1/2(Γ)^m×H^1/2(Γ)^m×H^1/2(Γ)^m→H¹(Ω^±)×H^1/2(Γ)^m×H^−1/2(Γ)^m;(iii)if 𝜓∈𝐻^−1/2(Γ)^𝑚, then [SL𝜓]_Γ=0 and [ℬ_𝜈SL𝜓]_Γ=−𝜓;(iv)if 𝜓∈𝐻^1/2(Γ)^𝑚, then [DL𝜓]_Γ=𝜓 and [ℬ_𝜈DL𝜓]_Γ=0.

Proof. See Costabel in [9].

Definition 12. When 𝑢=𝑢⁺+𝑢⁻∈𝐿²(ℝ^𝑑)^𝑚, with 𝑢^±∈𝐻¹(Ω^±)^𝑚, has compact support in ℝ^𝑑 and 𝑓=𝑓⁺+𝑓⁻∈𝐻⁻¹(ℝ^𝑑)^𝑚, we can enunciate the third Green identity 𝑢=𝒢𝑓+DL[𝑢]_Γ−SLℬ_𝜈𝑢_Γonℝ^𝑑. (41)

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 ℒ𝑢=𝑓. Depending on which part of the fundamental solution for the adjoint operator ℒ^∗𝑣=𝛿_𝜉 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 ℒ 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 equation𝑢(𝜉)=_Ω𝐺_𝜉(𝑥)𝑓𝑑𝑥+SLℬ_𝜈[𝑢](𝜉)−DL[𝑢](𝜉),𝜉∈Ω. (42)

The interior trace acting on (42) produces the boundary integral equation, that is, for 𝜉∈Γ𝛾⁻_𝜉𝑢(𝜉)=𝛾⁻_𝜉Ω𝐺_𝜉(𝑥)𝑓𝑑𝑥+𝛾⁻_𝜉SLℬ_𝜈[𝑢](𝜉)−𝛾⁻_𝜉DL[𝑢](𝜉). (43) Substituting the double and the single layer potentials by explicit integral representation and omitting the interior trace operator, we have 𝑢(𝜉)=_Ω𝐺_𝜉(𝑥)𝑓𝑑𝑥+_Γ𝐺_𝜉(𝑥)^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑢(𝑥)𝑑𝜎_𝑥−_Γ^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝐺_𝜉(𝑥)+𝐴_𝑗𝐺_𝜉(𝑥)𝑢(𝑥)𝑑𝜎_𝑥, (44) which is valid for 𝜉∈Γ.

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 ℒ^∗, and obtain the reciprocity gap equation _Ω𝑣(𝑥)𝑓(𝑥)𝑑𝑥=−ℬ_Γ^∼_𝜈𝑥[𝑣(𝑥)]^∗𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝑣(𝑥)ℬ_𝜈𝑥[𝑢(𝑥)]𝑑𝜎_𝑥, (45) which can be written explicitly as _Ω𝑣(𝑥)𝑓(𝑥)𝑑𝑥=−_Γ^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑣(𝑥)+𝐴_𝑗𝑣(𝑥)𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝑣(𝑥)^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝑢(𝑥)𝑑𝜎_𝑥 (46) for all 𝑣∈(𝐻_ℒ^∗(Ω)^𝑚)^∗.

Remark 13. When we do numerical implementation of integral (42) for 𝜉∈Γ, 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 Ω is sufficiently regular there exists a homeomorphism between the space of test functions 𝐻_ℒ^∗(Ω) and some spaces of functions in the boundary, such as 𝐻^1/2(𝜕Ω), [8]. This means that the set of traces of functions in 𝐻_ℒ^∗(Ω) is dense in 𝐻^1/2(𝜕Ω). 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), 𝑃_{𝑓,𝑔𝐷,𝑔𝑁𝜈}: to find 𝑢∈𝐻¹(Ω)^𝑚 such that 𝑃_{𝑓,𝑔𝐷,𝑔𝑁}𝛾[𝑢]ℒ𝑢=𝑓if𝑥∈Ω,=𝑔^𝐷if𝑥∈Γ_𝐷,ℬ_𝜈𝑢=𝑔^𝑁_𝜈if𝑥∈Γ_𝑁. (47) Let us consider its solution given by the third Green identity supposing that 𝑢 is extended by zero outside Ω: 𝑢=𝒢𝑓−DL[𝑢]_Γ+SLℬ_𝜈𝑢_Γonℝ^𝑑. (48) 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 Γ 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 Ω. By perturbing the singular solution, 𝐺_𝜉, of (27) with some regular solution 𝑣(𝑥) and since, in this problem, the boundary presents the following Lipschitz dissection: Γ=Γ_𝐷∪Π∪Γ_𝑁, (49) then a modified single layer potentialSL_𝑣ℬ_𝜈𝑢𝜉)=_Γ𝐷(+_Γ𝑁𝐺_𝜉(𝑥)+𝑣(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥,𝜉∈Ω (50) and a modified double layer potentialDL_𝑣[𝑢𝜉)=](_Γ𝐷+ℬ_Γ𝑁^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)+𝑣(𝑥)^∗𝑢(𝑥)𝑑𝜎_𝑥,𝜉∈Ω (51) make explicit the contributions of each part of the boundary to the potentials. Since in the Dirichlet boundary we know 𝑢|_Γ𝐷=𝑔_Γ𝐷anddonotknowℬ_𝜈[𝑢](𝑥)_Γ𝐷, (52) and in the Neumann boundary we know ℬ_𝜈[𝑢]_Γ𝑁=𝑔_𝜈Γ𝑁anddonotknow𝑢|_Γ𝑁, (53) we must choose 𝑣 as solution of the auxiliary problem𝑃^∗_{0,−𝐺∗𝜉(𝑥)|Γ𝐷,−}^∼_{ℬ𝜈𝐺∗𝜉𝑥()Γ𝑁=}ℒ^∗𝛾𝑣=0if𝑥∈Ω,₋[𝑣]=−𝐺^∗_𝜉(𝑥)_Γ𝑁if𝑥∈Γ_𝑁,_∼ℬ_𝜈𝑣=−ℬ^∼_𝜈𝐺^∗_𝜉(𝑥)_Γ𝐷if𝑥∈Γ_𝐷, (54) in order to cancel, in the single layer, the contribution of the unknown part ℬ_𝜈𝑢|_Γ𝐷 and, in the double layer, the contribution of the unknown part 𝛾𝑢|_Γ𝑁. We have defined the Green’s function associated with the mixed problem (24), 𝐺^𝑣_𝜉(𝑥)=𝐺_𝜉(𝑥)+𝑣(𝑥), where 𝑣(𝑥) is a solution of (54). The support of its trace is Γ_𝐷∪Π, while the support of the trace of its conormal derivative is Γ_𝑁∪Π.

The Green’s function single layer potential is SL_𝑣[𝑢](𝜉)=_Γ𝐷𝐺^𝑣_𝜉(𝑥)𝑢(𝑥)=𝑑𝑥_Γ𝑁𝐺_𝜉(𝑥)+𝑣(𝑥)^𝑑_𝑗=1𝜈_𝑗^𝑑_𝑘=1A_𝑗𝑘𝜕_𝑘𝑢(𝑥)𝑑𝜎_𝑥,𝜉∈Ω, (55) and the Green’s function double layer potential is DL_𝑣[𝑢𝜉)=](_Γ𝐷∼ℬ𝐺_𝜉^𝑣∗(𝑥)^∗𝑢(𝑥)=𝑑𝑥_{Γ𝐷𝑑𝑗=1}𝜈_𝑗^𝑑_𝑘=1𝐴_𝑗𝑘𝜕_𝑘𝐺_𝜉(𝑥)+𝑣(𝑥)+𝐴_𝑗𝐺_𝜉(𝑥)+𝑣(𝑥)𝑢(𝑥)𝑑𝜎_𝑥,𝜉∈Ω. (56) Combining them we obtain the Green’s function solution for the mixed problem (24) 𝑢(𝜉)=_Ω𝐺^𝑣_𝜉(𝑥)𝑓𝑑𝑥−DL_𝑣𝑔^𝐷(𝜉)+SL_𝑣𝑔^𝑁_𝜈𝑢(𝜉),𝜉∈Ω. (57)

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 𝐻_ℒ^∗(Ω) is homeomorphic to 𝐻^1/2(𝜕Ω) for smooth boundaries 𝜕Ω, [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), 𝑃_{𝑓,𝑔𝐷,𝑔𝑁𝜈}, we again consider extension by zero outside Ω and use the system made with the Lipschitz dissection of the third Green’s identity (41) at the boundary 𝛾_𝜉𝑢(𝜉)|_Γ𝐷+|_Γ𝑁=_Ω𝛾_𝜉𝐺_𝜉(𝑥)𝑓(𝑥)−𝑑𝑥_Γ𝐷+_Γ𝑁𝛾ℬ_𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗𝑢(𝑥)𝑑𝜎_𝑥×|_Γ𝐷+|_Γ𝑁+_Γ𝐷+_Γ𝑁𝛾_𝜉𝐺_𝜉(𝑥)ℬ_𝜈[𝑢]𝑥𝑑𝜎_𝑥×|_Γ𝐷+|_Γ𝑁 (58) and its conormal derivative ℬ_𝜈𝜉𝑢(𝜉)|_Γ𝐷+|_Γ𝑁=ℬ_𝜈𝜉Ω𝐺_𝜉(𝑥)𝑓(𝑥)𝑑𝑥−ℬ_{𝜈𝜉Γ𝐷}+ℬ_Γ𝑁^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗𝑢(𝑥)𝑑𝜎_𝑥|_Γ𝐷+|_Γ𝑁+ℬ_{𝜈𝜉Γ𝐷}+_Γ𝑁𝐺_𝜉(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥|_Γ𝐷+|_Γ𝑁. (59)

For clearness we separate these restrictions. The respective restrictions are(i) trace on Γ_𝐷: 𝛾_𝜉𝑢(𝜉)_Γ𝐷=_Ω𝛾_𝜉𝐺_𝜉Γ𝐷(𝑥)𝑓(𝑥)−𝑑𝑥_Γ𝐷𝛾ℬ_𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝐷𝑢(𝑥)𝑑𝜎_𝑥−_Γ𝑁𝛾ℬ_𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝐷𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝐷𝛾_𝜉𝐺_𝜉Γ𝐷(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥+_Γ𝑁𝛾_𝜉𝐺_𝜉Γ𝐷(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥; (60)(ii)trace on Γ_𝑁: 𝛾_𝜉𝑢(𝜉)_Γ𝑁=_Ω𝛾_𝜉𝐺_𝜉Γ𝑁(𝑥)𝑓(𝑥)−𝑑𝑥_Γ𝐷𝛾ℬ_𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝑁𝑢(𝑥)𝑑𝜎_𝑥−_Γ𝑁𝛾ℬ_𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝑁𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝐷𝛾_𝜉𝐺_𝜉Γ𝑁(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥+_Γ𝑁𝛾_𝜉𝐺_𝜉Γ𝑁(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥; (61)(iii)trace of conormal derivative on Γ_𝐷: ℬ_𝜈𝜉𝑢(𝜉)_Γ𝐷=_Ωℬ_𝜈𝜉𝐺_𝜉Γ𝐷(𝑥)𝑓(𝑥)−𝑑𝑥_Γ𝐷ℬℬ_𝜈𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝐷𝑢(𝑥)𝑑𝜎_𝑥−_Γ𝑁ℬℬ_𝜈𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝐷𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝐷ℬ_𝜈𝜉𝐺_𝜉Γ𝐷(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥+_Γ𝑁ℬ_𝜈𝜉𝐺_𝜉Γ𝐷(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥; (62)(iv)trace of conormal derivative on Γ_𝑁:ℬ_𝜈𝜉𝑢(𝜉)_Γ𝑁=_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝑁(𝑥)𝑓(𝑥)−𝑑𝑥_Γ𝐷ℬℬ_𝜈𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝑁𝑢(𝑥)𝑑𝜎_𝑥−_Γ𝑁ℬℬ_𝜈𝜉^∼_𝜈𝜉𝐺^∗_𝜉(𝑥)^∗_Γ𝑁𝑢(𝑥)𝑑𝜎_𝑥+_Γ𝐷ℬ_𝜈𝜉𝐺_𝜉|_Γ𝑁(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥+_Γ𝑁ℬ_𝜈𝜉𝐺_𝜉|_Γ𝑁(𝑥)ℬ_𝜈[𝑢](𝑥)𝑑𝜎_𝑥. (63)

The compositions involving traces of the layers potentials can be expressed in terms of the Fredholm integrals index zero operators(i)𝑆_𝑥→𝜉=𝛾_𝜉SL_𝑥→𝜉=∫_Γ𝛾_𝜉(ℬ^∼_𝜈𝜉𝐺^∗_𝜉)^∗[⋅]𝑑𝜎_𝑥∶𝐻^−1/2(Γ)→𝐻^1/2(Γ);(ii)𝑇^∼∗_𝑥→𝜉=−𝐼_𝑥→𝜉+2ℬ_𝜈𝜉SL_𝑥→𝜉=∫_Γℬ_𝜈𝜉(ℬ^∼_𝜈𝜉𝐺^∗_𝜉)^∗[⋅]𝑑𝜎_𝑥∶𝐻^1/2(Γ)→𝐻^−1/2(Γ);(iii)𝑇_𝑥→𝜉=𝐼_𝑥→𝜉+2𝛾_𝜉DL_𝑥→𝜉=∫_Γ𝛾_𝜉𝐺_𝜉(𝑥)[⋅]𝑑𝜎_𝑥∶𝐻^1/2(Γ)→𝐻^1/2(Γ);(iv)−𝑅_𝑥→𝜉=ℬ_𝜈𝜉DL_𝑥→𝜉=∫_Γℬ_𝜈𝜉(ℬ^∼_𝜈𝜉𝐺^∗_𝜉)^∗[⋅]𝑑𝜎_𝑥∶𝐻^1/2(Γ)→𝐻^−1/2(Γ).Substituting these operators in (60), (61), (62), and (63) and observing the appropriated restrictions and that 𝐼^𝑁𝐷_𝑥→𝜉=𝐼^𝐷𝑁_𝑥→𝜉=0, we have(i) trace on Γ_𝐷: 𝛾_𝜉𝑢(𝜉)_Γ𝐷=_Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥)+1𝑑𝑥2𝐼^𝐷𝐷_𝑥→𝜉−𝑇^𝐷𝐷_𝑥→𝜉𝑢(𝑥)_Γ𝐷−𝑇^𝑁𝐷_𝑥→𝜉𝑢(𝑥)_Γ𝑁+𝑆^𝐷𝐷_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝐷+𝑆^𝑁𝐷_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝑁; (64)(ii) trace on Γ_𝑁: 𝛾_𝜉𝑢(𝜉)_Γ𝑁=_Ω𝛾_𝜉𝐺_𝜉Γ𝑁(𝑥)𝑓(𝑥)𝑑𝑥−𝑇^𝐷𝑁_𝑥→𝜉𝑢(𝑥)_Γ𝑁+12𝐼^𝑁𝑁_𝑥→𝜉−𝑇^𝑁𝑁_𝑥→𝜉𝑢(𝑥)_Γ𝑁+𝑆^𝐷𝑁_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝐷+𝑆^𝑁𝑁_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝑁; (65)(iii) trace of conormal derivative on Γ_𝐷: ℬ_𝜈𝜉𝑢(𝜉)_Γ𝐷=_Ωℬ_𝜈𝜉𝐺_𝜉Γ𝐷(𝑥)𝑓(𝑥)𝑑𝑥+𝑅^𝐷𝐷_𝑥→𝜉𝑢(𝑥)_Γ𝐷+𝑅^𝑁𝐷_𝑥→𝜉𝑢(𝑥)_Γ𝑁−12𝐼^𝐷𝐷_𝑥→𝜉+𝑇^{∼∗𝐷𝐷}_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝐷−𝑇^{∼∗𝑁𝐷}_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝑁; (66)(iv) trace of conormal derivative on Γ_𝑁: ℬ_𝜈𝜉𝑢(𝜉)_Γ𝑁=_Ωℬ_𝜈𝜉𝐺_𝜉Γ𝑁(𝑥)𝑓(𝑥)𝑑𝑥+𝑅^𝐷𝑁_𝑥→𝜉𝑢(𝑥)_Γ𝐷+𝑅^𝑁𝑁_𝑥→𝜉𝑢(𝑥)_Γ𝑁−𝑇^{∼∗𝐷𝑁}_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝐷−12𝐼^𝑁𝑁_𝑥→𝜉+𝑇^{∼∗𝑁𝑁}_𝑥→𝜉ℬ_𝜈𝑢(𝑥)_Γ𝑁. (67)

In the mixed boundary value problem (24), 𝑃_{𝑓,𝑔𝐷,𝑔𝑁𝜈}, weknow𝛾𝑢|_Γ𝐷=𝑔^𝐷,ℬ_𝜈𝑢|_Γ𝑁=𝑔^𝑁_𝜈,butwedonotknow𝛾𝑢|_Γ𝑁=𝑔^𝑁,ℬ_𝜈𝑢|_Γ𝐷=𝑔^𝐷_𝜈. (68)(69) So we can select (64) and (67) to substitute the trace and the conormal derivative by the Cauchy data dissection 𝑔^𝐷=_Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥)1𝑑𝑥+2𝐼^𝐷𝐷_𝑥→𝜉−𝑇^𝐷𝐷_𝑥→𝜉𝑔^𝐷−𝑇^𝑁𝐷_𝑥→𝜉𝑔^𝑁+𝑆^𝐷𝐷_𝑥→𝜉𝑔^𝐷_𝜈+𝑆^𝑁𝐷_𝑥→𝜉𝑔^𝑁_𝜈,𝑔^𝑁_𝜈=_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝑁(𝑥)𝑓(𝑥)𝑑𝑥+𝑅^𝐷𝑁_𝑥→𝜉𝑔^𝐷+𝑅^𝑁𝑁_𝑥→𝜉𝑔^𝑁−𝑇^{∼∗𝐷𝑁}_𝑥→𝜉𝑔^𝐷_𝜈−12𝐼^𝑁𝑁_𝑥→𝜉+𝑇^{∼∗𝑁𝑁}_𝑥→𝜉𝑔^𝑁_𝜈, (70) and we use it then to form the following system: 𝑆^𝐷𝐷_𝜉→𝑥−12𝑇^𝑁𝐷_𝜉→𝑥12^∼𝑇^∗𝐷𝑁_𝜉→𝑥𝑅^𝑁𝑁_𝜉→𝑥𝑔^𝐷_𝜈𝑔^𝑁=−𝑆^𝑁𝐷_𝜉→𝑥12𝐼^𝐷𝐷_𝜉→𝑥+𝑇^𝐷𝐷_𝜉→𝑥12𝐼^𝑁𝑁_𝜉→𝑥+𝑇^{∼∗𝑁𝑁}_𝜉→𝑥−𝑅^𝑁𝑁_𝜉→𝑥𝑔^𝐷_𝜈𝑔^𝑁−∫_Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥)∫𝑑𝑥_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥).𝑑𝑥 (71) This system can be solved to determinate the unknown quantities in terms of the known. By putting 𝑆𝐴=^{𝐷𝐷𝜉→𝑥}−𝑇_∼^𝑁𝐷_𝜉→𝑥𝑇^∗𝐷𝑁_𝜉→𝑥𝑅^𝑁𝑁_𝜉→𝑥𝑔,𝜓=^𝐷𝜈𝑔^𝑁,ℎ=−𝑆^𝑁𝐷_𝜉→𝑥12+𝑇^𝐷𝐷_𝜉→𝑥12𝐼^𝑁𝑁_𝜉→𝑥+𝑇^{∼∗𝑁𝑁}_𝜉→𝑥−𝑅^𝑁𝑁_𝜉→𝑥𝑔^𝐷_𝜈𝑔^𝑁−∫_Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥)∫𝑑𝑥_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓(𝑥),𝑑𝑥 (72) the problem reduces to the solution of the formal equation 𝐴𝜓=ℎ. (73) The extension of the partially known Cauchy data (68) to all Γ can be done by solving (73). This problem has been investigated in [1], which showed that when ℒ is formally self-adjoint and coercive on 𝐻¹, then 𝐴∶𝐻^∼−1/2(Ω)^𝑚×𝐻^∼1/2(Ω)^𝑚→(𝐻^∼−1/2(Ω)^𝑚×𝐻^∼1/2(Ω)^𝑚)^∗ 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 ℬ⁻_𝜈𝑢(𝜉)=ℬ⁻_𝜈Ω𝐺_𝜉(𝑥)𝑓𝑑𝑥+ℬ⁻_𝜈SL[𝑢](𝜉)−ℬ⁻_𝜈DL[𝑢](𝜉),𝜉∈Ω. (74) Let us now consider the mixed boundary value problem (24) with data 𝑃_{0,𝑔𝐷,𝑔𝑁𝜈}. 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 2×2 linear operator 𝒞∶(𝐻^1/2(Ω))^𝑚×(𝐻^−1/2(Ω))^𝑚→(𝐻^1/2(Ω))^𝑚×(𝐻^−1/2(Ω))^𝑚 defined by 𝒞𝛾𝑢,ℬ_𝜈^𝑇=−𝛾DL𝛾𝑢𝛾SLℬ_𝜈𝑢−ℬ_𝜈DL𝛾𝑢ℬ_𝜈SLℬ_𝜈𝑢, (75) and if 𝜙=𝒞[𝜓], for some 𝜓=^𝑔_𝑔𝜈, then the function 𝑢=−DL𝑔+SL𝑔_𝜈 (76) satisfies the mixed boundary value problem (24) with data 𝑃_{0,𝑔𝐷,𝑔𝑁𝜈}. Since for these data 𝜙=𝒞𝑔,𝑔_𝜈=𝑔,𝑔_𝜈^𝑇=𝒞²𝑔,𝑔_𝜈^𝑇, (77) then 𝒞 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 1𝒞=2()𝑆𝑅1𝐼−𝑇2𝐼+𝑇^∼∗. (78)

When the mixed boundary value problem (24) is posed with a nonnull source 𝑓, that is, 𝑃_{𝑓,𝑔𝐷,𝑔𝑁𝜈}, (77) is no longer valid and modifies to (𝜉)ℬ𝛾𝑢_𝜈𝑢(𝜉)=∫_Ω𝛾_𝜉𝐺_𝜉𝑦𝑓𝑦∫𝑑𝑦_Ωℬ_𝜈𝜉𝐺_𝜉𝑦𝑓𝑦+1𝑑𝑦2𝐼_𝑥→𝜉−𝑇_𝑥→𝜉𝑆_𝑥→𝜉𝑅_𝑥→𝜉12𝐼_𝑥→𝜉+𝑇_𝑥→𝜉×(𝜉)ℬ𝛾𝑢_𝜈𝑢(𝜉),(𝑥,𝜉)∈Γ×Γ, (79) which is the Calderón projector gap for the source 𝑓. 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:𝛾𝑢(𝜉)_Γ𝐷𝛾𝑢(𝜉)_Γ𝑁ℬ_𝜈𝑢(𝜉)_Γ𝐷ℬ_𝜈𝑢(𝜉)_Γ𝑁=12𝐼^𝐷𝐷_𝑥→𝜉−𝑇^𝐷𝐷_𝑥→𝜉−𝑇^𝑁𝐷_𝑥→𝜉𝑆^𝐷𝐷_𝑥→𝜉𝑆^𝑁𝐷_𝑥→𝜉−𝑇^𝐷𝑁_𝑥→𝜉12𝐼^𝑁𝑁_𝑥→𝜉−𝑇^𝑁𝑁_𝑥→𝜉𝑆^𝐷𝑁_𝑥→𝜉𝑆^𝑁𝑁_𝑥→𝜉𝑅^𝐷𝐷_𝑥→𝜉𝑅^𝑁𝐷_𝑥→𝜉12𝐼^𝐷𝐷_𝑥→𝜉+𝑇^∼𝑇^∗𝐷𝐷_𝑥→𝜉^{∼∗𝑁𝐷}_𝑥→𝜉𝑅^𝐷𝑁_𝑥→𝜉𝑅^∼𝑁𝑁_𝑥→𝜉𝑇^∗𝐷𝑁_𝑥→𝜉12𝐼^𝑁𝑁_𝑥→𝜉+𝑇^{∼∗𝑁𝑁}_𝑥→𝜉×𝛾𝑢(𝑥)|_Γ𝐷𝛾𝑢(𝑥)|_Γ𝑁ℬ_𝜈𝑢(𝑥)|_Γ𝐷ℬ_𝜈𝑢(𝑥)|_Γ𝑁+∫_Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷𝑦𝑓𝑦∫𝑑𝑦_Ω𝛾_𝑥𝑖𝐺_𝜉|_Γ𝑁𝑦𝑓𝑦∫𝑑𝑦_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷𝑦𝑓𝑦∫𝑑𝑦_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷𝑦𝑓𝑦,()𝑑𝑦𝑥,𝜉∈Γ×Γ, (80)

and also writen as the integral equation system−12𝐼^𝐷𝐷_𝑥→𝜉+𝑇^𝐷𝐷_𝑥→𝜉−𝑇^𝑁𝐷_𝑥→𝜉𝑆^𝐷𝐷_𝑥→𝜉𝑆^𝑁𝐷_𝑥→𝜉−𝑇^𝐷𝑁_𝑥→𝜉−12𝐼^𝑁𝑁_𝑥→𝜉+𝑇^𝑁𝑁_𝑥→𝜉𝑆^𝐷𝑁_𝑥→𝜉𝑆^𝑁𝑁_𝑥→𝜉𝑅^𝐷𝐷_𝑥→𝜉𝑅^𝑁𝐷_𝑥→𝜉−12𝐼^𝐷𝐷_𝑥→𝜉−𝑇^∼𝑇^∗𝐷𝐷_𝑥→𝜉^{∼∗𝑁𝐷}_𝑥→𝜉𝑅^𝐷𝑁_𝑥→𝜉𝑅^∼𝑁𝑁_𝑥→𝜉𝑇^∗𝐷𝑁_𝑥→𝜉−12𝐼^𝑁𝑁_𝑥→𝜉−𝑇^{∼∗𝑁𝑁}_𝑥→𝜉𝛾𝑢(𝑥)|_Γ𝐷𝛾𝑢(𝑥)|_Γ𝑁ℬ_𝜈𝑢(𝑥)|_Γ𝐷ℬ_𝜈𝑢(𝑥)|_Γ𝑁∫=−^Ω𝛾_𝜉𝐺_𝜉|_Γ𝐷(𝑥)𝑓𝑦∫𝑑𝑦_Ω𝛾_𝜉𝐺_𝜉|_Γ𝑁(𝑥)𝑓𝑦∫𝑑𝑦_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷𝑦𝑓𝑦∫𝑑𝑦𝑥_Ωℬ_𝜈𝜉𝐺_𝜉|_Γ𝐷𝑦𝑓𝑦,()𝑑𝑦𝑥,𝜉∈Γ×Γ. (81)

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 𝑃_{𝑓𝐼,𝑔𝐼,𝑔𝐼𝜈} and 𝑃_{𝑓𝐼𝐼,𝑔𝐼𝐼,𝑔𝜈𝐼𝐼} defined on the same Lipschitz domain Ω. We say that these problems are complementary if 𝑓_𝐼=𝑓_𝐼𝐼, Γ^𝐼_𝐷=Γ_𝑁^𝐼𝐼, Γ_𝐷^𝐼𝐼=Γ^𝐼_𝑁 and there exist Cauchy data (𝑔,𝑔_𝜈) such that 𝑔^𝐼=𝑔𝜒_Γ𝐼𝐷,𝑔^𝐼𝐼=𝑔𝜒_{Γ𝐷𝐼𝐼},𝑔^𝐼_𝜈=𝑔_𝜈𝜒_Γ𝐼𝐷,𝑔_𝜈^𝐼𝐼=𝑔_𝜈𝜒_{Γ𝐷𝐼𝐼}. (82)

Theorem 18. Suppose that two mixed boundary value problems 𝑃_{𝑓𝐼,𝑔𝐼,𝑔𝐼𝜈} and 𝑃_{𝑓𝐼𝐼,𝑔𝐼𝐼,𝑔𝜈𝐼𝐼} have solutions 𝑢_𝐼 and 𝑢_𝐼𝐼, respectively. If they are complementary, then 𝑢_𝐼=𝑢_𝐼𝐼. (83)

Proof. Note that Γ is compact and Γ=Γ^𝐼_𝐷∪Π∪Γ^𝐼_𝑁=Γ_𝐷^𝐼𝐼∪Π∪Γ_𝑁^𝐼𝐼. (84) With the characteristic function for Γ^𝐼_𝐷, 𝜒_Γ𝐼𝐷 in Γ, we can introduce the extended Dirichlet data function 𝑔∈𝐻^1/2(Γ) which is the extension of 𝑔^𝐼∈𝐻^1/2(Γ^𝐼_𝐷) to 𝐻^1/2(Γ), with 𝑔^𝐼=𝑔𝜒_Γ𝐼𝐷, and consequently, 𝑔^𝐼𝐼=𝑔𝜒_{Γ𝑁𝐼𝐼}. Similar definitions can be done to treat the Neumann part of Cauchy data 𝑔_𝜈∈𝐻^−1/2(Γ), 𝑔^𝐼_𝜈=𝑔_𝜈𝜒_Γ𝐼𝐷. So, 𝑔(𝑥)=𝑔^𝐼(𝑥)𝜒_Γ𝐼𝐷(𝑥)+𝑔^𝐼𝐼(𝑥)𝜒_Γ𝐼𝑁(𝑥)=𝑔^𝐼(𝑥)𝜒_{Γ𝑁𝐼𝐼}(𝑥)+𝑔^𝐼𝐼(𝑥)𝜒_{Γ𝐷𝐼𝐼}(𝑥),𝑔_𝜈(𝑥)=𝑔^𝐼_𝜈(𝑥)𝜒_Γ𝐼𝐷(𝑥)+𝑔_𝜈^𝐼𝐼(𝑥)𝜒_Γ𝐼𝑁(𝑥)=𝑔^𝐼_𝜈(𝑥)𝜒_{Γ𝑁𝐼𝐼}(𝑥)+𝑔_𝜈^𝐼𝐼(𝑥)𝜒_{Γ𝐷𝐼𝐼}(𝑥). (85) Denoting 𝑓=𝑓_𝐼=𝑓_𝐼𝐼, the solution will be, via boundary integral equation method,𝑢(𝑥)=_Ω𝐺_𝜉(𝑥)𝑓(𝑥)𝑑𝑥−DL𝑔(𝑥)+SL𝑔_𝜈(𝑥). (86) 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 Ω where the coefficient has some different functional description, or even with the functional description itself. We consider these parameters collected in a parameter vector 𝛼=[𝛼₁,𝛼₂,…,𝛼_𝑁𝐴]^∗ and that the coefficients and source are represented as 𝐴_𝑗𝑘∶=𝑎^𝑗𝑘_𝑝𝑞(𝛼,𝑥),𝐴_𝑗∶=𝑎^𝑗_𝑝𝑞(),[𝑎((,𝛼,𝑥𝐴∶=𝛼,𝑥)],𝑓𝛼,𝑥) (87) which for 𝑝,𝑞=1,…,𝑚 are functions from Ω×[𝛽₁,𝛽₂] into ℝ^𝑚×𝑚. The parameters can be supposed to have values in some known interval [𝛽₁,𝛽₂]∈ℝ^𝑁𝐴. In this way, the strongly elliptic operator model with parameter dependence is formally written as ℒ_𝛼𝑢=𝑓_𝛼,𝑥∈Ω,𝛼∈𝛼₁,𝛼₂. (88)

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 𝑝=1,…,𝑁𝑃, 𝛾𝑢^(𝑝)=𝑔^(𝑝),ℬ_𝜈𝑢^(𝑝)=𝑔_𝜈^(𝑝). (89) 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 𝛾𝑢^(𝑝),ℬ_𝜈𝑢^(𝑝)=𝑔^(𝑝),𝑔_𝜈^(𝑝). (90) This set of (𝑁𝑃) Cauchy data fully prescribed at the boundary can be used to formulate a nonunique set with 2(𝑁𝑃) well-posed direct problems in the following way.

Choose some Lipschitz dissection of Γ associated with problem (𝑝) and given by Γ=Γ_𝐼^(𝑝)∪Π^(𝑝)∪Γ^(𝑝)_𝐼𝐼, (91) where Γ₁^(𝑝) and Γ^(𝑝)_𝐼𝐼 are disjoint, eventually empty or nonconnected, relatively open subsets of Γ, having Π^(𝑝) as their common boundary. Consider also the restriction for Cauchy data for problem (𝑝) associated with this partition: 𝑔^𝐼(𝑝),𝑔_𝜈^𝐼(𝑝)=𝛾𝑢^(𝑝),ℬ_𝜈𝑢^(𝑝)_Γ𝐼=𝑔^(𝑝)_Γ𝐼,𝑔_𝜈^(𝑝)_Γ𝐼,𝑔^{𝐼𝐼(𝑝)},𝑔_𝜈^{𝐼𝐼(𝑝)}=𝛾𝑢^(𝑝),ℬ_𝜈𝑢^(𝑝)_Γ𝐼𝐼=𝑔^(𝑝)_Γ𝐼𝐼,𝑔_𝜈^(𝑝)_Γ𝐼𝐼. (92)

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, 𝑃^𝛼_{𝑓𝛼,𝑔𝐼(𝑝),𝑔(𝑝)𝐼𝐼𝜈} and 𝑃^𝛼_{𝑓𝛼,𝑔(𝑝)𝐼𝐼,𝑔(𝑝)𝐼𝜈}, in such way that given a guess set of parameter 𝛼, for problems 𝑝=1,…,𝑁𝑃, to find complementary solutions 𝑢_𝐼^(𝑝) and 𝑢^(𝑝)_𝐼𝐼 such that 𝑢_𝐼^(𝑝)solutionof𝑃^𝛼_{𝑓𝛼,𝑔𝐼(𝑝),𝑔(𝑝)𝐼𝐼𝜈}𝑢^(𝑝)_𝐼𝐼solutionof𝑃^𝛼_{𝑓𝛼,𝑔(𝑝)𝐼𝐼,𝑔(𝑝)𝐼𝜈}. (93)

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 ℒ_𝛼 and source 𝑓_𝛼, characterized by the parameter set 𝛼, the associated Cauchy boundary data are given by (90). If for some 𝑝 and for some Lipschitz dissection one has 𝑢_𝐼^(𝑝) and 𝑢^(𝑝)_𝐼𝐼 solutions of problem (93), then 𝑢_𝐼^(𝑝)=𝑢^(𝑝)_𝐼𝐼. (94)

8. A Methodology Based on Discrepancy Function

In Section 7.1 we introduced the idea of decomposing the set of 𝑁𝑃 many boundary value problems, for which we knew the Cauchy data, in two subsets of 2𝑁𝑃 complementary well-posed mixed direct problems. The idea now is exploring the fact that these two sets of solutions indexed by 𝐼 and 𝐼𝐼 must be, under ideal conditions, equal for each problem (𝑝), 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,𝑑𝛼,𝑈_𝐼,𝑈_𝐼𝐼=^𝑁𝑃_𝑝=1𝑢_𝐼^(𝑝)(𝛼,⋅)−𝑢^(𝑝)_𝐼𝐼(𝛼,⋅)_𝑉, (95) where 𝑉 can be the norm of continuous functions solutions 𝒞(Ω), or square integrable 𝐿²(Ω) solutions, or even with some first derivative control 𝐻¹(Ω) solutions.

We now posed the optimization problem: in the guess set of parameters 𝛼∈{[𝛼₁,𝛼₂]⊂ℝ^𝑁𝐴}, to find 𝛼 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 𝑝=1,…,𝑁𝑃 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 𝛼. 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 𝑛+1 vertices in the 𝑛 dimensional parameters space. The Nelder-Mead simplex algorithm is based on four operations:(1)reflection with algorithm parameter 𝜌>0; (2)expansion with algorithm parameter 𝜒>1;(3)contraction with algorithm parameter 0<𝛾<1:(i)outside;(ii)inside;(4)shrink with algorithm parameter 0<𝜎<1.Steps (1)--(3) 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 (4), and the entire procedure is repeated. We adopted here the universal typical algorithm parameters values: 𝜌=1, 𝜒=2, 𝛾=1/2, and 𝜎=1/2.

9. Some Numerical Results

The numerical experiment that illustrates this work is a model in which the square (−1,+1)×(−1,+1) has in its interior a small rectangle that has unknown center and unknown edges 𝑎 and 𝑏, which support unknown parameters related with the conductivity, 𝑐, the potential, 𝑎, and the source intensity, 𝑓. Cauchy data are synthetically generated with a problem in which parameters values are known to be equal to 1 in the exterior of the small rectangle, and all equal 2 in the interior. Also the unknown information about the rectangle used are center at the origin and side 𝑎=𝑏=.2. The parameters in operator ℒ are 𝐴𝑚=1,_𝑗𝑘=1+(𝑐−1)𝜒𝑥₀−12𝑥₁,𝑥₀+12𝑥₁×𝜒𝑦₀−12𝑦₁,𝑦₀+12𝑦₁𝛿_𝑗𝑘,𝐴_𝑗()𝜒𝑥=0,𝐴=1+𝑎−1₀−12𝑥₁,𝑥₀+12𝑥₁×𝜒𝑦₀−12𝑦₁,𝑦₀+12𝑦₁𝛿_𝑗𝑘,𝑓(𝑥)=1+𝑓−1𝜒𝑥₀−12𝑥₁,𝑥₀+12𝑥₁×𝜒𝑦₀−12𝑦₁,𝑦₀+12𝑦₁𝛿_𝑗𝑘, (96) and the set of parameters are 𝛼=(𝑥₀,𝑦₀,𝑥₁,𝑦₁,𝑐,𝑎,𝑓)∈ℝ⁷.

The rectangle has boundaries Γ=Γ_𝑦=−1∪Π_(−1,+1)∪Γ_𝑥=+1∪Π_(+1,+1)∪Γ_𝑦=+1∪Π_(−1,+1)∪Γ_𝑥=−1∪Π_(−1,−1) (97) and is counterclockwise oriented.

Cauchy data are synthetically produced by solving a set of 5 Dirichlet direct problems with parameters 𝛼=(0,0,1,1,2,2,2)∈ℝ⁷ with quadratic Lagrange finite elements method:(0)𝑔|_Γ𝑦=−1=0;𝑔|_Γ𝑥=+1=0;𝑔|_Γ𝑦=+1=0;𝑔|_Γ𝑥=−1=0; (1)𝑔|_Γ𝑦=−1=(1−𝑥)(1+𝑥);𝑔|_Γ𝑥=+1=0;𝑔|_Γ𝑦=+1=0;𝑔|_Γ𝑥=−1=0; (2)𝑔|_Γ𝑦=−1=0;𝑔|_Γ𝑥=+1=(1−𝑦)(1+𝑦);𝑔|_Γ𝑦=+1=0;𝑔|_Γ𝑥=−1=0; (3)𝑔|_Γ𝑦=−1=0;𝑔|_Γ𝑥=+1=0;𝑔|_Γ𝑦=+1=(1−𝑥)(1+𝑥);𝑔|_Γ𝑥=−1=0; (4)𝑔|_Γ𝑦=−1=0;𝑔|_Γ𝑥=+1=0;𝑔|_Γ𝑦=+1=0;𝑔|_Γ𝑥=−1=(1−𝑦)(1+𝑦), generating Neumann data {𝑔_𝜈^(𝑝)|_Γ𝑦=−1; 𝑔_𝜈^(𝑝)|_Γ𝑥=+1; 𝑔_𝜈^(𝑝)|_Γ𝑦=+1; 𝑔_𝜈^(𝑝)|_Γ𝑥=−1; 𝑝=0,1,2,3,4}.

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

Figure 1 

Cauchy data Dirichlet problems (0).

Figure 2 

Cauchy data Dirichlet problems (1), (2), (3), and (4).

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 Γ_𝐼=Γ_𝑦=−1∪Γ_𝑦=+1,Γ_𝐼𝐼=Γ_𝑥=−1∪Γ_𝑥=+1. (98) Ten problems formulated with the dissection of these Cauchy data can now be used to evaluate the discrepancy function based on the following sup norm: 𝑑𝛼,𝑈_𝐼,𝑈_𝐼𝐼=ma^𝑁𝑃x_𝑝=1sup_𝑥∈Ω𝑢_𝐼^(𝑝)(𝛼,𝑥)−𝑢^(𝑝)_𝐼𝐼(𝛼,𝑥). (99)

The search starts with random generated initial data in the intervals [0,2]⁷ for the 7 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.