# Fractal-Based Methods and Inverse Problems for Differential Equations: Current State of the Art

Authors: Herb E. Kunze; Davide La Torre; Franklin Mendivil; Manuel Ruiz Galán; Rachad Zaki
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

We illustrate, in this short survey, the current state of the art of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. We review several methods based on the Collage Theorem and its extensions. We also discuss two innovative applications: the first one is related to a vibrating string model while the second one considers a collage-based approach for solving inverse problems for partial differential equations on a perforated domain.


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## Body

## 1. Inverse Problems for Fixed Point Equations

According to Keller [1], “we call two problemsinverse of one another if the formulation of each involves all or part of the solution of the other. Often, for historical reasons, one of the two problems has been studied extensively for some time, while the other one is newer and not so well understood. In such cases, the former is called thedirect problem, while the latter is theinverse problem.” In practice, a general inverse problem asks us to use observed data to estimate parameters in the functional form of the governing model of the phenomenon under study [2–6].There is a fundamental difference between the direct and the inverse problem; often the direct problem iswell-posed while the corresponding inverse problem isill-posed. Hadamard [7] introduced the concept ofwell-posed problem to describe a mathematical model that has the properties of existence, uniqueness, and stability of the solution. When one of these properties fails to hold, the mathematical model is said to be anill-posed problem. There are several inverse problems in literature that areill-posed whereas the corresponding direct problems arewell-posed. The literature is rich in papers studying ad hoc methods to addressill-posed inverse problems by minimizing a suitable approximation error along with utilizing some regularization techniques [8–13].Many inverse problems may be recast as the approximation of a target elementx in a complete metric space X,d by the fixed point x- of a contraction mapping T:X→X. Thanks to a simple consequence of Banach’s Fixed Point Theorem known as theCollage Theorem, most practical methods of solving the inverse problem for fixed point equations seek an operator T for which thecollage distancedx,Tx is as small as possible.Theorem 1 (“Collage Theorem” [14]).
Let(X,d) be a complete metric space and let T:X→X be a contraction mapping with contraction factor c∈[0,1). Then, for any x∈X,
(1)dx,x-≤11-cdx,Tx,
where x- is the fixed point of T.This vastly simplifies this type of inverse problem as it is much easier to estimatedx,Tx than it is to find the fixed point x- and then compute dx,x-. One now seeks a contraction mapping T that minimizes the so-calledcollage errordx,Tx, in other words, a mapping that sends the target x as close as possible to itself. This is the essence of the method ofcollage coding which has been the basis of most, if not all, fractal image coding and compression methods. Barnsley et al. [14, 15] were the first to see the potential of using the Collage Theorem above for the purpose offractal image approximation andfractal image coding [16]. However, this method ofcollage coding may be applied in other situations where contractive mappings are encountered.We have shown this to be the case for inverse problems involving several families of differential equations and application to different areas: ordinary differential equations [17–19], Urison-type integral equations [20], random differential equations [21, 22], boundary value problems [23–25], parabolic partial differential equations [26], stochastic differential equations [27–29], and others [3, 30]. In practical applications, from a family of contraction mappings Tλ,  λ∈Λ⊂Rn, one wishes to find the parameter λ- for which the approximation error dx,x-λ is as small as possible. In practice, the feasible set is often taken to be Λc=λ∈Rn:0≤cλ≤c<1 which guarantees the contractivity of Tλ for any λ∈Λc. The main difference between this “collage” approach and the one based on the Tikhonov regularization is the following (see [12, 13]): in the collage approach, the constraint λ∈Λc guarantees that Tλ is a contraction and, therefore, it replaces the effect of the regularization term in the Tikhonov approach (see also [6, 8, 11]).The collage-based inverse problem can be formulated as an optimization problem as follows:(2)min⁡λ∈Λcdx,Tλx.
This is a nonlinear and nonsmooth optimization model. However, as the next sections below show, the above model (2) can often be reduced to a quadratic optimization program. Several algorithms can be used to solve it including, for instance, penalization methods and particle swarm ant colony techniques.The paper is organized as follows. Section2 presents the method for the case of differential equations while Section 3 illustrates the case of different families of partial differential equations (PDEs), namely, elliptic, parabolic, and hyperbolic equations. Section 4 presents applications to a randomly forced vibrating string and also to an inverse problem on perforated domains.


## 2. Inverse Problems for IVPs by the Collage Theorem

In [18], and subsequent works [17, 20–22, 25], the authors showed how collage coding could be used to solve inverse problems for systems of differential equations having the form
(3)u˙=ft,u,u(0)=u0,
by reducing the problem to the corresponding Picard integral operator associated with it
(4)Tut=u0+∫0t‍fs,usds.
Let us recall the basic results in the case where f belongs to L2. Let us consider the complete metric space C0,T endowed with the usual d∞ metric and assume that ft,x is Lipschitz in the variable x; that is, there exists a K≥0 such that fs,x1-fs,x2≤Kx1-x2, for all x1,x2∈R. For simplicity, we suppose that x∈R, but the same consideration can be developed for the case of several variables. Under these hypotheses, T is Lipschitz on the space C-δ,δ×-M,M for some δ and M>0.Theorem 2 (see [18]).
The functionT satisfies
(5)Tu-Tv2≤cu-v2
for all u,v∈C-δ,δ×-M,M, where c=δK.Now, letδ′>0 be such that δ′K<1. In order to solve the inverse problem for (4), we take the L2 expansion of the function f. Let ϕi be a basis of functions in L2-δ′,δ′×-M,M and consider
(6)fλs,x=∑i=1+∞‍λiϕis,x.
Each sequence of coefficients λ=λii=1+∞ then defines a Picard operator Tλ. Suppose further that each function ϕis,x is Lipschitz in x with constants Ki.Theorem 3 (see [18]).
LetK,λ∈l2(R). Then,
(7)fλs,x1-fλs,x2≤K2λ2x1-x2
for all s∈-δ′,δ′ and x1,x2∈[-M,M], where K2=∑i=1+∞‍Ki21/2 and λ2=∑i=1+∞‍λi21/2.Given a target solutionx, we now seek to minimize the collage distance u-Tλu2. The square of the collage distance becomes
(8)Δ2λ=u-Tλu22=∫-δδ‍ut-∫0t‍∑i=1+∞‍λiϕis,usds2dt
and the inverse problem can be formulated as
(9)min⁡λ∈ΛΔ(λ),
where Λ=λ∈l2R:λ2K2<1. To solve this problem numerically, let us consider the first n terms of the L2 basis; in this case, the previous problem can be reduced to
(10)min⁡λ∈Λ~Δ~2λ=∫-δδ‍xt-∫0t‍∑i=1n‍λiϕis,xsds2dt,
where Λ~=λ∈Rn:λ2K2<1. This is a classical quadratic optimization problem which can be solved by means of classical numerical methods. A penalized version of (10) is the following:
(11)min⁡λ∈RnΔ~2λ+σ1max⁡0,1-λ22K222+σ2∑i=1n‍max⁡0,-λi2.
Let Δ~min⁡n be the minimum value of Δ~ over Λ~. This is a nonincreasing sequence of numbers (depending on n) and as shown in [16] it is possible to show that liminf⁡n→+∞Δ~min⁡n=0. This means that the distance between the target element and the unknown solution of the differential equation can be made arbitrarily small. In Kunze et al. [21], the authors considered the case of inverse problems for random stochastic differential equations while in Capasso et al. [29] the case of stochastic differential equations is analyzed.Example 4.
Suppose that the stochastic processXt is believed to follow a geometric Brownian motion; then it satisfies the stochastic differential equation
(12)dXt=aXtdt+bXtdWt,
where Wt is a Wiener process and the constants a and b are the percentage drift and the percentage volatility, respectively. We consider the following inverse problem: given realizations/paths Xti,  1≤i≤N, estimate the values a and b. Taking the expectation in (12), we see that EXt satisfies the simple fixed point equation
(13)EXt=TEXt=X0+∫0t‍aEXrdr.
Hence, to solve the inverse problem, we construct the mean of the realizations
(14)Xt*=1N∑i=1N‍Xti
and use collage coding to determine the value of a that minimizes the collage distance d2Xt*,TXt*. We can then estimate the value of b by using the known formula varXt=e2atX02eb2t-1, approximating varXt from the realizations. As an example, we set a=2, b=4, and X0=1 and then generate N paths on [0,1], dividing the interval into M subintervals in order to simulate the Brownian motion on [0,1]. Beginning with these paths, we seek estimates of a and b using collage coding. Figure 1 shows five paths for the Brownian motion and the process Xt. Table 1 presents the numerical results of the example.Table 1
Minimal collage distance parameters for differentN and M, to five decimal places.
N
M
b
a
100
300
3.78599
1.69618
100
600
4.33750
1.78605
100
1000
4.05374
1.85780
300
300
3.34231
1.80282
300
600
3.54219
1.81531
300
1000
3.84973
1.78323Figure 1
Different paths of the Brownian motion withM=1000 and N=300.


## 3. Inverse Problems for BVPs by the Generalized Collage Theorem

### 3.1. Elliptic Equations

Let us consider the following variational equation:(15)a(u,v)=ϕ(v),v∈H,
where ϕv and au,v are linear and bilinear maps, respectively, both defined on a Hilbert space H. Let us denote by 〈·,·〉 the inner product in H, u2=u,u and du,v=u-v, for all u,v∈H. The inverse problem may now be viewed as follows: suppose that we have an observed solution u and a given (restricted) family of bilinear functionals aλu,v, λ∈Rn. We now seek “optimal” values of λ. The existence and uniqueness of solutions to this kind of equation are provided by the classical Lax-Milgram representation theorem. Suppose that we have a “target” element u∈H and a family of bilinear functionals aλ. Then, by the Lax-Milgram theorem, there exists a unique vector uλ∈H such that ϕv=aλuλ,v for all v∈H. We would like to determine if there exists a value of the parameter λ such that uλ=u or, more realistically, such that uλ-u is small enough. The following theorem will be useful for the solution of this problem.Theorem 5 (Generalized Collage Theorem [24]).
Suppose thataλu,v:Λ×H×H→R is a family of bilinear forms for all λ∈Λ and ϕ:H→R is a given linear functional. Let uλ denote the solution of the equation aλu,v=ϕv for all v∈H as guaranteed by the Lax-Milgram theorem. Given a target element u∈H,
(16)u-uλ≤1mλFλ,
where
(17)Fλ=sup⁡v∈H,v=1aλu,v-ϕv.In order to ensure that the approximationuλ is close to a target element u∈H, we can, by the Generalized Collage Theorem, try to make the term Fλ/mλ as close to zero as possible. The appearance of the mλ factor complicates the procedure as does the factor 1/(1-c) in standard collage coding, that is, (1). If inf⁡λ∈Λmλ≥m>0, then the inverse problem can be reduced to the minimization of the function Fλ on the space Λ; that is,
(18)min⁡λ∈ΛF(λ).
The choice of λ according to (18) for minimizing the residual is, in general, not stabilizing (see [8]). However, as the next sections show, under the condition inf⁡λ∈Λmλ≥m>0, our approach is stable. Following our earlier studies of inverse problems using fixed points of contraction mappings, we will refer to the minimization of the functional Fλ as a “generalized collage method.” Such an optimization problem has a solution that can be approximated with a suitable discrete and quadratic program, derived from the application of the Generalized Collage Theorem and an adequate use of an orthonormal basis in the Hilbert space H, as seen in [24].Example 6.
We now present an inverse problem for the two-dimensional steady-state diffusion equation. WithD=0<x,y<1,
(19)-∇·κx,y∇ux,y+qx,yux,y=fx,y,x,y∈D,ux,y=0,x,y∈∂D,
where κx,y and qx,y vary in both x and y. Given ux,y, qx,y, and fx,y on 0,12, we wish to find an approximation of κx,y. As the first example, we set ux,y=sinπxsinπy, qx,y=0, and use the function κx,y=1+6x2y1-y to determine fx,y via (29). Now, given the functions ux,y, fx,y, and qx,y, we seek to approximate κx,y. This inverse problem is treated as Example  3 in [9], using a modified Uzawa algorithm. In Figure 2, we present graphs of our actual κx,y, as well as the results obtained by minimizing (18). For N and M fixed natural numbers, we define hx=1/N and hy=1/M, as well as the N+1M+1 nodes in 0,12 as follows:
(20)xi,yj=ihx,jhy,0≤i≤N,0≤j≤M.
The corresponding finite element basis functions ξijx,y are pyramids with hexagonal bases, such that ξijxi,yj=1 and ξijxk,yl=0 for k≠i, l≠j (see [24] for more details on this).Figure 2
(Left to right, top to bottom) for two-dimensional Example4, the graphs of our actual κx,y, and the collage-coded approximations of κ with N=M=3 through N=M=9.Next, we perturb the target functionux,y, leaving fx,y and qx,y exact. Table 2 presents the L2 error u-unoisy between the true solution u and the noised target unoisy and the resulting error κ-κcollage between the true κ and the collage-coded approximation κcollage for numerous cases of N and M. Note that κ2=1.38082 and u2=0.5.Table 2
Numerical results for the inverse problem with different levels of noise.
N
=
M
‖
κ
-
κ
collage
‖
2
u
noisy
=
u
‖
u
-
u
noisy
‖
2
=
0.025
‖
u
-
u
noisy
‖
2
=
0.05
3
0.06306
0.09993
0.17050
4
0.03480
0.07924
0.15561
5
0.02246
0.07275
0.15128
6
0.01564
0.07118
0.15065
7
0.01160
0.07051
0.15039
8
0.00902
0.07008
0.15014
9
0.00733
0.06981
0.14996As the second example, let us follow Example 7 of [10], and we set ux,y=sinπxsinπy, qx,y=4+cos⁡πxy, and κx,y=1+x2+xy/1000. With these choices, we determine the function fx,y. The inverse problem is to estimate κx,y when given ux,y, fx,y, and qx,y. In Figure 3, we present graphs of the results obtained by minimizing (18) with N=M=3 through N=M=5.Figure 3
(Left to right) for two-dimensional Example6, the graphs of our actual κx,y, and the collage-coded approximations of κ with N=M=3,4,5.These results have been extended to a wider class of elliptic equations problems in [31, 32], by considering not only Hilbert but also reflexive Banach spaces, and even replacing the primal variational formulation of such a problem, (15), with a more general constrained variational one. Let us mention that this kind of formulation arises, for instance, when the boundary constraints are weakly imposed.Theorem 7 (see [32, 33]).
LetX, Y, Z, and W be reflexive Banach spaces, let ϕ:Y→R and ψ:W→R be bounded and linear functionals, and let Λ be a nonempty set and assume that for each λ∈Λ, αλ and βλ are positive real numbers and aλ:X×Y→R, bλ:Y×Z→R, and cλ:X×W→R are bounded and bilinear forms. If one writes
(21)Kbλ≔y∈Y:bλy,z=0,∀z∈Z,Kcλ≔{x∈X:cj(x,w)=0,∀w∈W}
and suppose that
(22)x∈Kcλ,aλ(x,y)=0,∀y∈Kbλ⟹x=0,y∈Kbλ⟹αλy≤sup⁡x∈Kcλ,x=1aλx,y,w∈W⟹βλw≤sup⁡x∈X,x=1cλ(x,w),
then the constrained variational equation
(23)x∈Xwithy∈Kbλ⟹aλx,y=ϕyw∈W⟹cλx,w=ψw
admits a unique solution xλ∈X and in addition for all x0∈X there holds
(24)xλ-x0≤1αλsup⁡y∈Y,y=1ϕy-aλx0,y+1βλ1+aλαλsup⁡w∈W,w=1ψw-cλx0,w.Ifinf⁡λ∈Λαλ>0, inf⁡λ∈Λβλ>0, and sup⁡λ∈Λaλ<∞, then the corresponding inverse problem leads to the minimization problem
(25)min⁡λ∈Λsup⁡y∈Y,y=1ϕy-aλx0,yyyyyy+sup⁡w∈W,w=1ψw-cλx0,wsup⁡y∈Y,y=1ϕy-aλx0,y,
for which an approximated quadratic program follows from the preceding collage-type result and some properties of Schauder bases in the involved reflexive Banach spaces (see [32] for the details).


### 3.2. Parabolic Equations

Suppose that we have a given Hilbert spaceH and let us consider the following abstract formulation of a parabolic equation:
(26)ddtu,v=ψv+au,v,u0=f,
where ψ:H→R is a linear functional, a:H×H→R is a bilinear form, and f∈H is an initial condition. The aim of the inverse problem for the above system of (29) consists of getting an approximation of the coefficients and parameters starting from a sample of observations of a target u∈H. To do this, let us consider a family of bilinear functionals aλ and let uλ be the solution to
(27)ddtuλ,v=ψv+aλuλ,vu0=f.We would like to determine if there exists a value of the parameterλ such that uλ=u or, more realistically, such that uλ-u is small enough. To this end, Theorem 8 states that the distance between the target solution u and the solution uλ of (27) can be reduced by minimizing a functional which depends on parameters.Theorem 8.
Letu:0,T→L2D be the target solution which satisfies the initial condition in (29) and suppose that d/dtu exists and belongs to H. Suppose that aλu,v:Λ×H×H→R is a family of bilinear forms for all λ∈Λ. One has the following result:
(28)∫0T‍u-uλHdt≤1mλ2∫0T‍sup⁡v=1ddtu,vggggggggggg-ψv-aλu,vsup⁡v=1ddtu,v2dt,
where uλ is the solution of (27) s.t. uλ0=u0 and uλT=uT.Example 9.
Let us consider the following equation:(29)ut=kxuxx+gx,t,0<x<1,u0=0,u1=0,
where gx,t=tx1-x, subject to ux,0=10sinπx and u0,t=u1,t=0. We set kx=1+3x+2x2, solve the problem for ux,t, and sample the solution at N2 uniformly positioned grid points for x,t∈0,12 to generate a collection of targets. Given this data and gx,t, we then seek an estimation of kx in the form kx=k0+k1x+k2x2. The results we obtain through the generalized collage method are summarized in Table 3. As for the elliptic case, the table shows that the method subject to noisy perturbations is stable.Table 3
Collage coding results for the parabolic equation in Example9.
Noiseε
N
k
0
k
1
k
2
0
10
0.87168
2.90700
0.21353
0
20
0.93457
2.97239
1.49201
0
30
0.94479
2.98304
1.76421
0
40
0.94347
2.97346
1.85572
0.01
10
0.87573
2.82810
0.33923
0.01
20
0.92931
2.91536
1.32864
0.01
30
0.92895
2.84553
0.59199
0.10
10
0.90537
1.97162
0.59043
0.10
20
0.77752
0.92051
−0.77746
0.10
30
0.60504
−0.12677
−0.14565


### 3.3. Hyperbolic Equations

Let us now consider the following weakly formulated hyperbolic equation:(30)d2dt2u,v=ψv+au,v,u0=f,ddtu0=g,
where ψ:H→R is a linear functional, a:H×H→R is a bilinear form, and f, g∈H define the initial conditions. As in the previous sections, the aim of the inverse problem for the above system of equations consists of reconstructing the coefficients starting from a sample of observations of a target u∈H. We consider a family of bilinear functionals aλ and let uλ be the solution to
(31)d2dt2uλ,v=ψv+aλuλ,v,u0=f,ddtu(0)=g.
We would like to determine if there exists a value of the parameter λ such that uλ=u or, more realistically, such that uλ-u is small enough. Theorem 10 states that the distance between the target solution u and the solution uλ of (31) can be reduced by minimizing a functional which depends on parameters.Theorem 10.
Letu:0,T→L2D be the target solution which satisfies the initial condition in (30) and suppose that d2/dt2u exists and belongs to H. Suppose that there exists a family of mλ>0 such that aλv,v≥mλv2 for all v∈H. One has the following result:
(32)∫0T‍u-uλ2dt≤1mλ2∫0T‍sup⁡v=1d2dt2u,v-ψv-au,v2dt,
where uλ is the solution of (31) s.t. u0=uλ0 and uT=uλT.Example 11.
We adjust Example9, considering
(33)utt-kxuxx=gx,t,
where gx,t=tx1-x, subject to ux,0=sinπx and utx,0=0 and u0,t=u1,t=0. We set kx=1+3x+2x2 and construct target data as in Example 9 and then seek to recover kx=k0+k1x+k2x2 given this data and gx,t. The results we get from the generalized collage method are summarized in Table 4, which again proves the stability of the method under noise.Table 4
Collage coding results for the hyperbolic equation in Example11.
Noiseε
N
k
0
k
1
k
2
0
10
0.87168
2.90700
0.21353
0
20
0.93457
2.97239
1.49201
0
30
0.94479
2.98304
1.76421
0
40
0.94347
2.97346
1.85572
0.01
10
0.87573
2.82810
0.33923
0.01
20
0.92931
2.91536
1.32864
0.01
30
0.92895
2.84553
0.59199
0.10
10
0.90537
1.97162
0.59043
0.10
20
0.77752
0.92051
−0.77746
0.10
30
0.60504
−0.12677
−0.14565


## 3.1. Elliptic Equations

Let us consider the following variational equation:(15)a(u,v)=ϕ(v),v∈H,
where ϕv and au,v are linear and bilinear maps, respectively, both defined on a Hilbert space H. Let us denote by 〈·,·〉 the inner product in H, u2=u,u and du,v=u-v, for all u,v∈H. The inverse problem may now be viewed as follows: suppose that we have an observed solution u and a given (restricted) family of bilinear functionals aλu,v, λ∈Rn. We now seek “optimal” values of λ. The existence and uniqueness of solutions to this kind of equation are provided by the classical Lax-Milgram representation theorem. Suppose that we have a “target” element u∈H and a family of bilinear functionals aλ. Then, by the Lax-Milgram theorem, there exists a unique vector uλ∈H such that ϕv=aλuλ,v for all v∈H. We would like to determine if there exists a value of the parameter λ such that uλ=u or, more realistically, such that uλ-u is small enough. The following theorem will be useful for the solution of this problem.Theorem 5 (Generalized Collage Theorem [24]).
Suppose thataλu,v:Λ×H×H→R is a family of bilinear forms for all λ∈Λ and ϕ:H→R is a given linear functional. Let uλ denote the solution of the equation aλu,v=ϕv for all v∈H as guaranteed by the Lax-Milgram theorem. Given a target element u∈H,
(16)u-uλ≤1mλFλ,
where
(17)Fλ=sup⁡v∈H,v=1aλu,v-ϕv.In order to ensure that the approximationuλ is close to a target element u∈H, we can, by the Generalized Collage Theorem, try to make the term Fλ/mλ as close to zero as possible. The appearance of the mλ factor complicates the procedure as does the factor 1/(1-c) in standard collage coding, that is, (1). If inf⁡λ∈Λmλ≥m>0, then the inverse problem can be reduced to the minimization of the function Fλ on the space Λ; that is,
(18)min⁡λ∈ΛF(λ).
The choice of λ according to (18) for minimizing the residual is, in general, not stabilizing (see [8]). However, as the next sections show, under the condition inf⁡λ∈Λmλ≥m>0, our approach is stable. Following our earlier studies of inverse problems using fixed points of contraction mappings, we will refer to the minimization of the functional Fλ as a “generalized collage method.” Such an optimization problem has a solution that can be approximated with a suitable discrete and quadratic program, derived from the application of the Generalized Collage Theorem and an adequate use of an orthonormal basis in the Hilbert space H, as seen in [24].Example 6.
We now present an inverse problem for the two-dimensional steady-state diffusion equation. WithD=0<x,y<1,
(19)-∇·κx,y∇ux,y+qx,yux,y=fx,y,x,y∈D,ux,y=0,x,y∈∂D,
where κx,y and qx,y vary in both x and y. Given ux,y, qx,y, and fx,y on 0,12, we wish to find an approximation of κx,y. As the first example, we set ux,y=sinπxsinπy, qx,y=0, and use the function κx,y=1+6x2y1-y to determine fx,y via (29). Now, given the functions ux,y, fx,y, and qx,y, we seek to approximate κx,y. This inverse problem is treated as Example  3 in [9], using a modified Uzawa algorithm. In Figure 2, we present graphs of our actual κx,y, as well as the results obtained by minimizing (18). For N and M fixed natural numbers, we define hx=1/N and hy=1/M, as well as the N+1M+1 nodes in 0,12 as follows:
(20)xi,yj=ihx,jhy,0≤i≤N,0≤j≤M.
The corresponding finite element basis functions ξijx,y are pyramids with hexagonal bases, such that ξijxi,yj=1 and ξijxk,yl=0 for k≠i, l≠j (see [24] for more details on this).Figure 2
(Left to right, top to bottom) for two-dimensional Example4, the graphs of our actual κx,y, and the collage-coded approximations of κ with N=M=3 through N=M=9.Next, we perturb the target functionux,y, leaving fx,y and qx,y exact. Table 2 presents the L2 error u-unoisy between the true solution u and the noised target unoisy and the resulting error κ-κcollage between the true κ and the collage-coded approximation κcollage for numerous cases of N and M. Note that κ2=1.38082 and u2=0.5.Table 2
Numerical results for the inverse problem with different levels of noise.
N
=
M
‖
κ
-
κ
collage
‖
2
u
noisy
=
u
‖
u
-
u
noisy
‖
2
=
0.025
‖
u
-
u
noisy
‖
2
=
0.05
3
0.06306
0.09993
0.17050
4
0.03480
0.07924
0.15561
5
0.02246
0.07275
0.15128
6
0.01564
0.07118
0.15065
7
0.01160
0.07051
0.15039
8
0.00902
0.07008
0.15014
9
0.00733
0.06981
0.14996As the second example, let us follow Example 7 of [10], and we set ux,y=sinπxsinπy, qx,y=4+cos⁡πxy, and κx,y=1+x2+xy/1000. With these choices, we determine the function fx,y. The inverse problem is to estimate κx,y when given ux,y, fx,y, and qx,y. In Figure 3, we present graphs of the results obtained by minimizing (18) with N=M=3 through N=M=5.Figure 3
(Left to right) for two-dimensional Example6, the graphs of our actual κx,y, and the collage-coded approximations of κ with N=M=3,4,5.These results have been extended to a wider class of elliptic equations problems in [31, 32], by considering not only Hilbert but also reflexive Banach spaces, and even replacing the primal variational formulation of such a problem, (15), with a more general constrained variational one. Let us mention that this kind of formulation arises, for instance, when the boundary constraints are weakly imposed.Theorem 7 (see [32, 33]).
LetX, Y, Z, and W be reflexive Banach spaces, let ϕ:Y→R and ψ:W→R be bounded and linear functionals, and let Λ be a nonempty set and assume that for each λ∈Λ, αλ and βλ are positive real numbers and aλ:X×Y→R, bλ:Y×Z→R, and cλ:X×W→R are bounded and bilinear forms. If one writes
(21)Kbλ≔y∈Y:bλy,z=0,∀z∈Z,Kcλ≔{x∈X:cj(x,w)=0,∀w∈W}
and suppose that
(22)x∈Kcλ,aλ(x,y)=0,∀y∈Kbλ⟹x=0,y∈Kbλ⟹αλy≤sup⁡x∈Kcλ,x=1aλx,y,w∈W⟹βλw≤sup⁡x∈X,x=1cλ(x,w),
then the constrained variational equation
(23)x∈Xwithy∈Kbλ⟹aλx,y=ϕyw∈W⟹cλx,w=ψw
admits a unique solution xλ∈X and in addition for all x0∈X there holds
(24)xλ-x0≤1αλsup⁡y∈Y,y=1ϕy-aλx0,y+1βλ1+aλαλsup⁡w∈W,w=1ψw-cλx0,w.Ifinf⁡λ∈Λαλ>0, inf⁡λ∈Λβλ>0, and sup⁡λ∈Λaλ<∞, then the corresponding inverse problem leads to the minimization problem
(25)min⁡λ∈Λsup⁡y∈Y,y=1ϕy-aλx0,yyyyyy+sup⁡w∈W,w=1ψw-cλx0,wsup⁡y∈Y,y=1ϕy-aλx0,y,
for which an approximated quadratic program follows from the preceding collage-type result and some properties of Schauder bases in the involved reflexive Banach spaces (see [32] for the details).


## 3.2. Parabolic Equations

Suppose that we have a given Hilbert spaceH and let us consider the following abstract formulation of a parabolic equation:
(26)ddtu,v=ψv+au,v,u0=f,
where ψ:H→R is a linear functional, a:H×H→R is a bilinear form, and f∈H is an initial condition. The aim of the inverse problem for the above system of (29) consists of getting an approximation of the coefficients and parameters starting from a sample of observations of a target u∈H. To do this, let us consider a family of bilinear functionals aλ and let uλ be the solution to
(27)ddtuλ,v=ψv+aλuλ,vu0=f.We would like to determine if there exists a value of the parameterλ such that uλ=u or, more realistically, such that uλ-u is small enough. To this end, Theorem 8 states that the distance between the target solution u and the solution uλ of (27) can be reduced by minimizing a functional which depends on parameters.Theorem 8.
Letu:0,T→L2D be the target solution which satisfies the initial condition in (29) and suppose that d/dtu exists and belongs to H. Suppose that aλu,v:Λ×H×H→R is a family of bilinear forms for all λ∈Λ. One has the following result:
(28)∫0T‍u-uλHdt≤1mλ2∫0T‍sup⁡v=1ddtu,vggggggggggg-ψv-aλu,vsup⁡v=1ddtu,v2dt,
where uλ is the solution of (27) s.t. uλ0=u0 and uλT=uT.Example 9.
Let us consider the following equation:(29)ut=kxuxx+gx,t,0<x<1,u0=0,u1=0,
where gx,t=tx1-x, subject to ux,0=10sinπx and u0,t=u1,t=0. We set kx=1+3x+2x2, solve the problem for ux,t, and sample the solution at N2 uniformly positioned grid points for x,t∈0,12 to generate a collection of targets. Given this data and gx,t, we then seek an estimation of kx in the form kx=k0+k1x+k2x2. The results we obtain through the generalized collage method are summarized in Table 3. As for the elliptic case, the table shows that the method subject to noisy perturbations is stable.Table 3
Collage coding results for the parabolic equation in Example9.
Noiseε
N
k
0
k
1
k
2
0
10
0.87168
2.90700
0.21353
0
20
0.93457
2.97239
1.49201
0
30
0.94479
2.98304
1.76421
0
40
0.94347
2.97346
1.85572
0.01
10
0.87573
2.82810
0.33923
0.01
20
0.92931
2.91536
1.32864
0.01
30
0.92895
2.84553
0.59199
0.10
10
0.90537
1.97162
0.59043
0.10
20
0.77752
0.92051
−0.77746
0.10
30
0.60504
−0.12677
−0.14565


## 3.3. Hyperbolic Equations

Let us now consider the following weakly formulated hyperbolic equation:(30)d2dt2u,v=ψv+au,v,u0=f,ddtu0=g,
where ψ:H→R is a linear functional, a:H×H→R is a bilinear form, and f, g∈H define the initial conditions. As in the previous sections, the aim of the inverse problem for the above system of equations consists of reconstructing the coefficients starting from a sample of observations of a target u∈H. We consider a family of bilinear functionals aλ and let uλ be the solution to
(31)d2dt2uλ,v=ψv+aλuλ,v,u0=f,ddtu(0)=g.
We would like to determine if there exists a value of the parameter λ such that uλ=u or, more realistically, such that uλ-u is small enough. Theorem 10 states that the distance between the target solution u and the solution uλ of (31) can be reduced by minimizing a functional which depends on parameters.Theorem 10.
Letu:0,T→L2D be the target solution which satisfies the initial condition in (30) and suppose that d2/dt2u exists and belongs to H. Suppose that there exists a family of mλ>0 such that aλv,v≥mλv2 for all v∈H. One has the following result:
(32)∫0T‍u-uλ2dt≤1mλ2∫0T‍sup⁡v=1d2dt2u,v-ψv-au,v2dt,
where uλ is the solution of (31) s.t. u0=uλ0 and uT=uλT.Example 11.
We adjust Example9, considering
(33)utt-kxuxx=gx,t,
where gx,t=tx1-x, subject to ux,0=sinπx and utx,0=0 and u0,t=u1,t=0. We set kx=1+3x+2x2 and construct target data as in Example 9 and then seek to recover kx=k0+k1x+k2x2 given this data and gx,t. The results we get from the generalized collage method are summarized in Table 4, which again proves the stability of the method under noise.Table 4
Collage coding results for the hyperbolic equation in Example11.
Noiseε
N
k
0
k
1
k
2
0
10
0.87168
2.90700
0.21353
0
20
0.93457
2.97239
1.49201
0
30
0.94479
2.98304
1.76421
0
40
0.94347
2.97346
1.85572
0.01
10
0.87573
2.82810
0.33923
0.01
20
0.92931
2.91536
1.32864
0.01
30
0.92895
2.84553
0.59199
0.10
10
0.90537
1.97162
0.59043
0.10
20
0.77752
0.92051
−0.77746
0.10
30
0.60504
−0.12677
−0.14565


## 4. Application

### 4.1. The Vibrating String Driven by a Stochastic Process

Before stating and solving two inverse problems, we begin by giving the details and motivations for the specific model we are interested in studying. We consider the following system of coupled differential equations. The first one is a stochastic differential equation and the second one is a hyperbolic partial differential equation. On a domainD⊂Rd, we have the equations
(34)dXt=∫D‍gut,ydμtyXtdt+XtdBt,Xt=0=X0,(35)d2dt2ut,y+∇yκ1y∇yut,y=κ2yδXty,t,y∈0,T×D,u(0,y)=ϕ1(y),∂u∂n(t,y)=ϕ2(t,y),(t,y)∈[0,T]×∂D,
where μt is the law of Xt and δXt is the Dirac delta “function” at the point Xt.For instance, imagine we have a flexible string directed along thex-axis, with the string kept stretched by a constant horizontal tension, and forced to vibrate perpendicularly to the x-axis under random force Fx,t. If u denotes the displacement of a point x at time t, it is well known that u satisfies the following equation:
(36)∂2∂t2u-∂2∂x2u=Fx,t,
where x∈D and t>0. In our model, we suppose that Fy,t=κ2yδXty, where Xt is a stochastic process which is a solution of the stochastic differential equation (34). In other words, the hyperbolic equation has a forcing term that is driven by this stochastic process. The random vibration on an infinite string has received recent attention (see [34, 35]). Figure 4 presents some snapshots of the displacement for the related finite string problem.Figure 4
Snapshots of a randomly forced vibrating string, with time increasing from left to right and top to bottom.In the next sections, we present a solution method for solving two different parameter identification problems for this system of coupled differential equations: one forκ1 and one for g. Both of these two methods are based on the numerical schemes which have been presented in the previous sections.Before we begin the analysis, a few words about (35) are in order, since this equation contains the generalized function δXt. That is, δXty has a meaning only when it is integrated with respect to a test function θy. Thus, the meaning of (35) is that for each θ∈H1D we have
(37)∫D‍θyd2dt2ut,y+∇yκ1y∇yut,ydy=∫D‍θyκ2yδXtydy=θXtκ2Xt.


#### 4.1.1. A Parameter Identification Model forκ1

For this parameter identification problem, we seek to estimateκ1 given κ2, g, and the observations of Xt. From this data, we recover the density fXt of the process Xt. Averaging equation (35) allows one to get a simpler model for the parameter identification problem (see [36]). This replaces the quantity δXt with its expectation EδXty. Since Xt is absolutely continuous, then EδXty=fXty, the density of the distribution of Xt. The previous model can therefore be rewritten in an averaged form as
(38)dXt=∫D‍gu~t,yfXt(y)dyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(39)d2dt2u~t,y+∇κ1y∇yu~t,y=κ2yfXty,t,y∈0,T×Du~(0,y)=ϕ1(y),∂u~∂n(t,y)=ϕ2(t,y),(t,y)∈[0,T]×∂D.
Note that the averaged equation, (39), has solutions in the usual sense. For this particular case, we have Eut,y=u~t,y. To see this, just notice that the operator d2/dt2-Δy is linear, so
(40)Ed2dt2u+∇y(κ1y∇yu)=d2dt2Eu+∇yκ1y∇yEu.
Furthermore,
(41)E∫D‍θyκ2yδXtydy=∫D‍θyκ2yfXtydy,
since κ1 and κ2 are deterministic functions. Thus, E(u) is the solution to the deterministic PDE
(42)d2dt2Eut,y-κ1yΔyEut,y=κ2yfXty,t,y∈0,T×D,Eu0,y=ϕ1y,∂Eu∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
However, this is clearly the same PDE for which u~ is the solution. Thus, we must have Eu=u~. This inverse problem can be solved using the techniques illustrated in a previous section on hyperbolic differential equations.


#### 4.1.2. A Parameter Identification Problem forg

For this parameter identification problem, we assume thatκ1 and κ2 are known. The aim of the inverse problem consists of recovering the functional form of g and the density fXt of the process Xt starting from a sample of observations of the random process ut,y,ω, say ut,y,ω1,…,ut,y,ωn. We do this in several steps. Let us denote by u~=Eut,y,·. The first step is to use ut,y and (35) to obtain an estimate of the distribution of Xt. Let us consider the averaged equation
(43)dXt=∫D‍gu~t,yfXtydyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(44)d2dt2u~t,y-κ1yΔyu~t,y=κ2yfXty,t,y∈0,T×D,u~0,y=ϕ1y,∂u~∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
Dividing the left side of the PDF in (44) by κ2, we get fXt. Let us go back to the stochastic differential equation (43), and let us take the expectation of both sides; we get
(45)dEXt=∫D‍gu~t,yfXtydyEXtdt,EXt=0=EX0.
We remember that the following relationship holds between the expectation of a random variable and its density:
(46)EXt=∫Rd‍yfXtydy.
Thus, we can obtain EXt, which now can be used for solving the inverse problem for (45) via the method illustrated in Section 2. We recover the function
(47)Φt=∫D‍g(u~(t,y))fXt(y)dy.
The last step involves the analysis of (47). The only unknown in this model is g. Taking the L2 expansions of Φ and g with respect to the same L2 orthonormal basis ϕi, we get
(48)∑i≥0‍biϕi=∑i≥0‍ai∫D‍ϕiu~t,yfXtydy,
and then
(49)bj=∑i≥0‍ai∫R‍∫D‍ϕu~s,yfXs(y)dyϕj(s)ds,
for j≥0, a linear system in ai, the solution of which is our final step.


### 4.2. Inverse Problems on Perforated Domains

A porous medium (or perforated domain) is a material characterized by a partitioning of the total volume into a solid portion often called the “matrix” and a pore space usually referred to as “holes.” Mathematically speaking, these holes can be either materials different from those of the matrix or real physical holes. When formulating differential equations over porous media, the term “porous” implies that the state equation is written in the matrix only while boundary conditions should be imposed on the whole boundary of the matrix, including the boundary of the holes. Examples of this are Stokes or Navier-Stokes equations that are usually written only in the fluid part while the rocks play the role of “mathematical” holes. Porous media are encountered everywhere in real life and the concept of porous media is essential in many areas of applied sciences and engineering including petroleum engineering, chemical engineering, civil engineering, aerospace engineering, soil science, geology, and material science.Since porosity in materials can take different forms and appear in varying degrees, solving differential equations over porous media is often a complicated task. Indeed, the size of holes and their distribution within a material play an important role in its characterization, and simulations conducted over porous media that include a large number of matrix-holes interfaces present real challenges. This is due to the need for a very fine discretization mesh which often requires a significant computational time and might even sometimes be irrelevant. This major difficulty is usually overcome by using the mathematical theory of “homogenization,” where the heterogeneous material is replaced by a fictitious homogeneous one through a delicate approach that is not simply an averaging procedure. Several techniques are currently in use in homogenization including the multiple scale method, the method of oscillating test functions of Tartar, the two-scale convergence method, and, most recently, the periodic unfolding method.In the case of porous media, or heterogeneous media in general, characterizing the properties of the material is a tricky process and can be done on different levels, mainly the microscopic and macroscopic scales, where the microscopic scale describes the heterogeneities and the macroscopic one describes the global behavior of the composite. To provide a numerical example of an inverse problem on a perforated domain, we setΩ=0,12 and for 0<ε≤0.1 define Nε=1/10ε and
(50)Ωε=⋃i,j=1Nε‍Bεi-12ε,j-12ε,
a domain with Nε2 uniformly distributed holes of radius ε. Choosing Kx,y=Ktruex,y=10+2x+3y, we consider the steady-state diffusion problem
(51)∇·Kx,y∇ux,y=x2+y2,x,y∈Ω∖Ωε,ux,y=0,∂Ω,∂u∂nx,y=0,∂Ωε.
Suppose that, for a certain ϵ, we sample the solution uϵλ of the above model on the perforated domain Ωε. We aim at estimating λ starting from this data and by using the model which is obtained when ϵ=0 instead. In fact, it is much easier to solve an inverse problem on the initial domain Ω instead of on the perforated domain Ωϵ.For a fixed value ofε, we solve the diffusion problem numerically and sample the solution at M×M uniformly distributed points strictly inside Ω. If such a point lies inside a hole, we obtain no information at the point. Using the M2 (or fewer) data points, we use the Generalized Collage Theorem to solve the related inverse problem, seeking a diffusivity function of the form K(x,y)=λ0+λ1x+λ2y.For this example, we use the particular valuesε=0.1, 0.025, and 0.01, corresponding to Nε=1, 4, and 10. The level curves are illustrated in Figure 5. The results for M=9, 49, and 99 are given in Table 5. In Figure 6, we illustrate the meshes used for numerically solving each of the problems.Table 5
Results for the inverse problem. True values are(λ0,λ1,λ2)=(10,2,3).
ε
N
ε
M
Recovered parameters
λ
0
λ
1
λ
2
0
.
1
1
9
13.2068
-
0.5921
0.6250
49
13.2428
-
0.5837
0.6346
49
13.2419
-
0.5798
0.6398
0.025
4
9
9.8434
1.8148
2.8119
49
9.9758
1.6894
2.6875
99
9.9787
1.6838
2.6820
0.01
10
9
9.9811
1.6221
2.6199
49
10.0069
1.6041
2.6014
99
10.0069
1.6039
2.6014Figure 5
Level curves of solutions in the example, withε=0.1, 0.025, and 0.01.Figure 6
Meshes for each problem.


## 4.1. The Vibrating String Driven by a Stochastic Process

Before stating and solving two inverse problems, we begin by giving the details and motivations for the specific model we are interested in studying. We consider the following system of coupled differential equations. The first one is a stochastic differential equation and the second one is a hyperbolic partial differential equation. On a domainD⊂Rd, we have the equations
(34)dXt=∫D‍gut,ydμtyXtdt+XtdBt,Xt=0=X0,(35)d2dt2ut,y+∇yκ1y∇yut,y=κ2yδXty,t,y∈0,T×D,u(0,y)=ϕ1(y),∂u∂n(t,y)=ϕ2(t,y),(t,y)∈[0,T]×∂D,
where μt is the law of Xt and δXt is the Dirac delta “function” at the point Xt.For instance, imagine we have a flexible string directed along thex-axis, with the string kept stretched by a constant horizontal tension, and forced to vibrate perpendicularly to the x-axis under random force Fx,t. If u denotes the displacement of a point x at time t, it is well known that u satisfies the following equation:
(36)∂2∂t2u-∂2∂x2u=Fx,t,
where x∈D and t>0. In our model, we suppose that Fy,t=κ2yδXty, where Xt is a stochastic process which is a solution of the stochastic differential equation (34). In other words, the hyperbolic equation has a forcing term that is driven by this stochastic process. The random vibration on an infinite string has received recent attention (see [34, 35]). Figure 4 presents some snapshots of the displacement for the related finite string problem.Figure 4
Snapshots of a randomly forced vibrating string, with time increasing from left to right and top to bottom.In the next sections, we present a solution method for solving two different parameter identification problems for this system of coupled differential equations: one forκ1 and one for g. Both of these two methods are based on the numerical schemes which have been presented in the previous sections.Before we begin the analysis, a few words about (35) are in order, since this equation contains the generalized function δXt. That is, δXty has a meaning only when it is integrated with respect to a test function θy. Thus, the meaning of (35) is that for each θ∈H1D we have
(37)∫D‍θyd2dt2ut,y+∇yκ1y∇yut,ydy=∫D‍θyκ2yδXtydy=θXtκ2Xt.


### 4.1.1. A Parameter Identification Model forκ1

For this parameter identification problem, we seek to estimateκ1 given κ2, g, and the observations of Xt. From this data, we recover the density fXt of the process Xt. Averaging equation (35) allows one to get a simpler model for the parameter identification problem (see [36]). This replaces the quantity δXt with its expectation EδXty. Since Xt is absolutely continuous, then EδXty=fXty, the density of the distribution of Xt. The previous model can therefore be rewritten in an averaged form as
(38)dXt=∫D‍gu~t,yfXt(y)dyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(39)d2dt2u~t,y+∇κ1y∇yu~t,y=κ2yfXty,t,y∈0,T×Du~(0,y)=ϕ1(y),∂u~∂n(t,y)=ϕ2(t,y),(t,y)∈[0,T]×∂D.
Note that the averaged equation, (39), has solutions in the usual sense. For this particular case, we have Eut,y=u~t,y. To see this, just notice that the operator d2/dt2-Δy is linear, so
(40)Ed2dt2u+∇y(κ1y∇yu)=d2dt2Eu+∇yκ1y∇yEu.
Furthermore,
(41)E∫D‍θyκ2yδXtydy=∫D‍θyκ2yfXtydy,
since κ1 and κ2 are deterministic functions. Thus, E(u) is the solution to the deterministic PDE
(42)d2dt2Eut,y-κ1yΔyEut,y=κ2yfXty,t,y∈0,T×D,Eu0,y=ϕ1y,∂Eu∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
However, this is clearly the same PDE for which u~ is the solution. Thus, we must have Eu=u~. This inverse problem can be solved using the techniques illustrated in a previous section on hyperbolic differential equations.


### 4.1.2. A Parameter Identification Problem forg

For this parameter identification problem, we assume thatκ1 and κ2 are known. The aim of the inverse problem consists of recovering the functional form of g and the density fXt of the process Xt starting from a sample of observations of the random process ut,y,ω, say ut,y,ω1,…,ut,y,ωn. We do this in several steps. Let us denote by u~=Eut,y,·. The first step is to use ut,y and (35) to obtain an estimate of the distribution of Xt. Let us consider the averaged equation
(43)dXt=∫D‍gu~t,yfXtydyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(44)d2dt2u~t,y-κ1yΔyu~t,y=κ2yfXty,t,y∈0,T×D,u~0,y=ϕ1y,∂u~∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
Dividing the left side of the PDF in (44) by κ2, we get fXt. Let us go back to the stochastic differential equation (43), and let us take the expectation of both sides; we get
(45)dEXt=∫D‍gu~t,yfXtydyEXtdt,EXt=0=EX0.
We remember that the following relationship holds between the expectation of a random variable and its density:
(46)EXt=∫Rd‍yfXtydy.
Thus, we can obtain EXt, which now can be used for solving the inverse problem for (45) via the method illustrated in Section 2. We recover the function
(47)Φt=∫D‍g(u~(t,y))fXt(y)dy.
The last step involves the analysis of (47). The only unknown in this model is g. Taking the L2 expansions of Φ and g with respect to the same L2 orthonormal basis ϕi, we get
(48)∑i≥0‍biϕi=∑i≥0‍ai∫D‍ϕiu~t,yfXtydy,
and then
(49)bj=∑i≥0‍ai∫R‍∫D‍ϕu~s,yfXs(y)dyϕj(s)ds,
for j≥0, a linear system in ai, the solution of which is our final step.


## 4.1.1. A Parameter Identification Model forκ1

For this parameter identification problem, we seek to estimateκ1 given κ2, g, and the observations of Xt. From this data, we recover the density fXt of the process Xt. Averaging equation (35) allows one to get a simpler model for the parameter identification problem (see [36]). This replaces the quantity δXt with its expectation EδXty. Since Xt is absolutely continuous, then EδXty=fXty, the density of the distribution of Xt. The previous model can therefore be rewritten in an averaged form as
(38)dXt=∫D‍gu~t,yfXt(y)dyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(39)d2dt2u~t,y+∇κ1y∇yu~t,y=κ2yfXty,t,y∈0,T×Du~(0,y)=ϕ1(y),∂u~∂n(t,y)=ϕ2(t,y),(t,y)∈[0,T]×∂D.
Note that the averaged equation, (39), has solutions in the usual sense. For this particular case, we have Eut,y=u~t,y. To see this, just notice that the operator d2/dt2-Δy is linear, so
(40)Ed2dt2u+∇y(κ1y∇yu)=d2dt2Eu+∇yκ1y∇yEu.
Furthermore,
(41)E∫D‍θyκ2yδXtydy=∫D‍θyκ2yfXtydy,
since κ1 and κ2 are deterministic functions. Thus, E(u) is the solution to the deterministic PDE
(42)d2dt2Eut,y-κ1yΔyEut,y=κ2yfXty,t,y∈0,T×D,Eu0,y=ϕ1y,∂Eu∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
However, this is clearly the same PDE for which u~ is the solution. Thus, we must have Eu=u~. This inverse problem can be solved using the techniques illustrated in a previous section on hyperbolic differential equations.


## 4.1.2. A Parameter Identification Problem forg

For this parameter identification problem, we assume thatκ1 and κ2 are known. The aim of the inverse problem consists of recovering the functional form of g and the density fXt of the process Xt starting from a sample of observations of the random process ut,y,ω, say ut,y,ω1,…,ut,y,ωn. We do this in several steps. Let us denote by u~=Eut,y,·. The first step is to use ut,y and (35) to obtain an estimate of the distribution of Xt. Let us consider the averaged equation
(43)dXt=∫D‍gu~t,yfXtydyXtdt+XtdBt,Xt=0=X0,
coupled with the deterministic PDE
(44)d2dt2u~t,y-κ1yΔyu~t,y=κ2yfXty,t,y∈0,T×D,u~0,y=ϕ1y,∂u~∂nt,y=ϕ2t,y,t,y∈0,T×∂D.
Dividing the left side of the PDF in (44) by κ2, we get fXt. Let us go back to the stochastic differential equation (43), and let us take the expectation of both sides; we get
(45)dEXt=∫D‍gu~t,yfXtydyEXtdt,EXt=0=EX0.
We remember that the following relationship holds between the expectation of a random variable and its density:
(46)EXt=∫Rd‍yfXtydy.
Thus, we can obtain EXt, which now can be used for solving the inverse problem for (45) via the method illustrated in Section 2. We recover the function
(47)Φt=∫D‍g(u~(t,y))fXt(y)dy.
The last step involves the analysis of (47). The only unknown in this model is g. Taking the L2 expansions of Φ and g with respect to the same L2 orthonormal basis ϕi, we get
(48)∑i≥0‍biϕi=∑i≥0‍ai∫D‍ϕiu~t,yfXtydy,
and then
(49)bj=∑i≥0‍ai∫R‍∫D‍ϕu~s,yfXs(y)dyϕj(s)ds,
for j≥0, a linear system in ai, the solution of which is our final step.


## 4.2. Inverse Problems on Perforated Domains

A porous medium (or perforated domain) is a material characterized by a partitioning of the total volume into a solid portion often called the “matrix” and a pore space usually referred to as “holes.” Mathematically speaking, these holes can be either materials different from those of the matrix or real physical holes. When formulating differential equations over porous media, the term “porous” implies that the state equation is written in the matrix only while boundary conditions should be imposed on the whole boundary of the matrix, including the boundary of the holes. Examples of this are Stokes or Navier-Stokes equations that are usually written only in the fluid part while the rocks play the role of “mathematical” holes. Porous media are encountered everywhere in real life and the concept of porous media is essential in many areas of applied sciences and engineering including petroleum engineering, chemical engineering, civil engineering, aerospace engineering, soil science, geology, and material science.Since porosity in materials can take different forms and appear in varying degrees, solving differential equations over porous media is often a complicated task. Indeed, the size of holes and their distribution within a material play an important role in its characterization, and simulations conducted over porous media that include a large number of matrix-holes interfaces present real challenges. This is due to the need for a very fine discretization mesh which often requires a significant computational time and might even sometimes be irrelevant. This major difficulty is usually overcome by using the mathematical theory of “homogenization,” where the heterogeneous material is replaced by a fictitious homogeneous one through a delicate approach that is not simply an averaging procedure. Several techniques are currently in use in homogenization including the multiple scale method, the method of oscillating test functions of Tartar, the two-scale convergence method, and, most recently, the periodic unfolding method.In the case of porous media, or heterogeneous media in general, characterizing the properties of the material is a tricky process and can be done on different levels, mainly the microscopic and macroscopic scales, where the microscopic scale describes the heterogeneities and the macroscopic one describes the global behavior of the composite. To provide a numerical example of an inverse problem on a perforated domain, we setΩ=0,12 and for 0<ε≤0.1 define Nε=1/10ε and
(50)Ωε=⋃i,j=1Nε‍Bεi-12ε,j-12ε,
a domain with Nε2 uniformly distributed holes of radius ε. Choosing Kx,y=Ktruex,y=10+2x+3y, we consider the steady-state diffusion problem
(51)∇·Kx,y∇ux,y=x2+y2,x,y∈Ω∖Ωε,ux,y=0,∂Ω,∂u∂nx,y=0,∂Ωε.
Suppose that, for a certain ϵ, we sample the solution uϵλ of the above model on the perforated domain Ωε. We aim at estimating λ starting from this data and by using the model which is obtained when ϵ=0 instead. In fact, it is much easier to solve an inverse problem on the initial domain Ω instead of on the perforated domain Ωϵ.For a fixed value ofε, we solve the diffusion problem numerically and sample the solution at M×M uniformly distributed points strictly inside Ω. If such a point lies inside a hole, we obtain no information at the point. Using the M2 (or fewer) data points, we use the Generalized Collage Theorem to solve the related inverse problem, seeking a diffusivity function of the form K(x,y)=λ0+λ1x+λ2y.For this example, we use the particular valuesε=0.1, 0.025, and 0.01, corresponding to Nε=1, 4, and 10. The level curves are illustrated in Figure 5. The results for M=9, 49, and 99 are given in Table 5. In Figure 6, we illustrate the meshes used for numerically solving each of the problems.Table 5
Results for the inverse problem. True values are(λ0,λ1,λ2)=(10,2,3).
ε
N
ε
M
Recovered parameters
λ
0
λ
1
λ
2
0
.
1
1
9
13.2068
-
0.5921
0.6250
49
13.2428
-
0.5837
0.6346
49
13.2419
-
0.5798
0.6398
0.025
4
9
9.8434
1.8148
2.8119
49
9.9758
1.6894
2.6875
99
9.9787
1.6838
2.6820
0.01
10
9
9.9811
1.6221
2.6199
49
10.0069
1.6041
2.6014
99
10.0069
1.6039
2.6014Figure 5
Level curves of solutions in the example, withε=0.1, 0.025, and 0.01.Figure 6
Meshes for each problem.



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*Source: 737694-2014-11-20.xml*