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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 737694, 11 pages
http://dx.doi.org/10.1155/2014/737694
Review Article

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

1Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
2Department of Economics, Management, and Quantitative Methods, University of Milan, 20122 Milan, Italy
3Department of Applied Mathematics and Sciences, Khalifa University, P.O. Box 127788, Abu Dhabi, UAE
4Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada B4P 2R6
5Department of Applied Mathematics, University of Granada, 18071 Granada, Spain

Received 18 March 2014; Accepted 30 July 2014; Published 20 November 2014

Academic Editor: Asier Ibeas

Copyright © 2014 Herb E. Kunze et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Inverse Problems for Fixed Point Equations

According to Keller [1], “we call two problems inverse 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 the direct problem, while the latter is the inverse 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 [26].

There is a fundamental difference between the direct and the inverse problem; often the direct problem is well-posed while the corresponding inverse problem is ill-posed. Hadamard [7] introduced the concept of well-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 an ill-posed problem. There are several inverse problems in literature that are ill-posed whereas the corresponding direct problems are well-posed. The literature is rich in papers studying ad hoc methods to address ill-posed inverse problems by minimizing a suitable approximation error along with utilizing some regularization techniques [813].

Many inverse problems may be recast as the approximation of a target element in a complete metric space by the fixed point of a contraction mapping . Thanks to a simple consequence of Banach’s Fixed Point Theorem known as the Collage Theorem, most practical methods of solving the inverse problem for fixed point equations seek an operator for which the collage distance is as small as possible.

Theorem 1 (“Collage Theorem” [14]). Let be a complete metric space and let be a contraction mapping with contraction factor . Then, for any , where is the fixed point of .

This vastly simplifies this type of inverse problem as it is much easier to estimate than it is to find the fixed point and then compute . One now seeks a contraction mapping that minimizes the so-called collage error , in other words, a mapping that sends the target as close as possible to itself. This is the essence of the method of collage 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 of fractal image approximation and fractal image coding [16]. However, this method of collage 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 [1719], Urison-type integral equations [20], random differential equations [21, 22], boundary value problems [2325], parabolic partial differential equations [26], stochastic differential equations [2729], and others [3, 30]. In practical applications, from a family of contraction mappings ,  , one wishes to find the parameter for which the approximation error is as small as possible. In practice, the feasible set is often taken to be which guarantees the contractivity of for any . 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 guarantees that 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: 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. Section 2 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, 2022, 25], the authors showed how collage coding could be used to solve inverse problems for systems of differential equations having the form by reducing the problem to the corresponding Picard integral operator associated with it Let us recall the basic results in the case where belongs to . Let us consider the complete metric space endowed with the usual metric and assume that is Lipschitz in the variable ; that is, there exists a such that , for all . For simplicity, we suppose that , but the same consideration can be developed for the case of several variables. Under these hypotheses, is Lipschitz on the space for some and .

Theorem 2 (see [18]). The function satisfies for all , where .

Now, let be such that . In order to solve the inverse problem for (4), we take the expansion of the function . Let be a basis of functions in and consider Each sequence of coefficients then defines a Picard operator . Suppose further that each function is Lipschitz in with constants .

Theorem 3 (see [18]). Let . Then, for all and , where and .

Given a target solution , we now seek to minimize the collage distance . The square of the collage distance becomes and the inverse problem can be formulated as where . To solve this problem numerically, let us consider the first terms of the basis; in this case, the previous problem can be reduced to where . 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: Let be the minimum value of over . This is a nonincreasing sequence of numbers (depending on ) and as shown in [16] it is possible to show that . 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 process is believed to follow a geometric Brownian motion; then it satisfies the stochastic differential equation where is a Wiener process and the constants and are the percentage drift and the percentage volatility, respectively. We consider the following inverse problem: given realizations/paths ,  , estimate the values and . Taking the expectation in (12), we see that satisfies the simple fixed point equation Hence, to solve the inverse problem, we construct the mean of the realizations and use collage coding to determine the value of that minimizes the collage distance . We can then estimate the value of by using the known formula , approximating from the realizations. As an example, we set , , and and then generate paths on , dividing the interval into subintervals in order to simulate the Brownian motion on . Beginning with these paths, we seek estimates of and using collage coding. Figure 1 shows five paths for the Brownian motion and the process . Table 1 presents the numerical results of the example.

tab1
Table 1: Minimal collage distance parameters for different and , to five decimal places.
737694.fig.001
Figure 1: Different paths of the Brownian motion with and .

3. Inverse Problems for BVPs by the Generalized Collage Theorem

3.1. Elliptic Equations

Let us consider the following variational equation: where and are linear and bilinear maps, respectively, both defined on a Hilbert space . Let us denote by the inner product in , and , for all . The inverse problem may now be viewed as follows: suppose that we have an observed solution and a given (restricted) family of bilinear functionals , . We now seek “optimal” values of . 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 and a family of bilinear functionals . Then, by the Lax-Milgram theorem, there exists a unique vector such that for all . We would like to determine if there exists a value of the parameter such that or, more realistically, such that is small enough. The following theorem will be useful for the solution of this problem.

Theorem 5 (Generalized Collage Theorem [24]). Suppose that is a family of bilinear forms for all and is a given linear functional. Let denote the solution of the equation for all as guaranteed by the Lax-Milgram theorem. Given a target element , where

In order to ensure that the approximation is close to a target element , we can, by the Generalized Collage Theorem, try to make the term as close to zero as possible. The appearance of the factor complicates the procedure as does the factor in standard collage coding, that is, (1). If , then the inverse problem can be reduced to the minimization of the function on the space ; that is, The choice of according to (18) for minimizing the residual is, in general, not stabilizing (see [8]). However, as the next sections show, under the condition , 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 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 , as seen in [24].

Example 6. We now present an inverse problem for the two-dimensional steady-state diffusion equation. With , where and vary in both and . Given , , and on , we wish to find an approximation of . As the first example, we set , , and use the function to determine via (29). Now, given the functions , , and , we seek to approximate . This inverse problem is treated as Example  3 in [9], using a modified Uzawa algorithm. In Figure 2, we present graphs of our actual , as well as the results obtained by minimizing (18). For and fixed natural numbers, we define and , as well as the nodes in as follows: The corresponding finite element basis functions are pyramids with hexagonal bases, such that and for , (see [24] for more details on this).

737694.fig.002
Figure 2: (Left to right, top to bottom) for two-dimensional Example 4, the graphs of our actual , and the collage-coded approximations of with through .

Next, we perturb the target function , leaving and exact. Table 2 presents the error between the true solution and the noised target and the resulting error between the true and the collage-coded approximation for numerous cases of and . Note that and .

tab2
Table 2: Numerical results for the inverse problem with different levels of noise.

As the second example, let us follow Example 7 of [10], and we set , , and . With these choices, we determine the function . The inverse problem is to estimate when given , , and . In Figure 3, we present graphs of the results obtained by minimizing (18) with through .

737694.fig.003
Figure 3: (Left to right) for two-dimensional Example 6, the graphs of our actual , and the collage-coded approximations of with .

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]). Let , , , and be reflexive Banach spaces, let and be bounded and linear functionals, and let be a nonempty set and assume that for each , and are positive real numbers and , , and are bounded and bilinear forms. If one writes and suppose that then the constrained variational equation admits a unique solution and in addition for all there holds

If , , and , then the corresponding inverse problem leads to the minimization problem 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 space and let us consider the following abstract formulation of a parabolic equation: where is a linear functional, is a bilinear form, and 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 . To do this, let us consider a family of bilinear functionals and let be the solution to

We would like to determine if there exists a value of the parameter such that or, more realistically, such that is small enough. To this end, Theorem 8 states that the distance between the target solution and the solution of (27) can be reduced by minimizing a functional which depends on parameters.

Theorem 8. Let be the target solution which satisfies the initial condition in (29) and suppose that exists and belongs to . Suppose that is a family of bilinear forms for all . One has the following result: where is the solution of (27) s.t. and .

Example 9. Let us consider the following equation: where , subject to and . We set , solve the problem for , and sample the solution at uniformly positioned grid points for to generate a collection of targets. Given this data and , we then seek an estimation of in the form . 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.

tab3
Table 3: Collage coding results for the parabolic equation in Example 9.

3.3. Hyperbolic Equations

Let us now consider the following weakly formulated hyperbolic equation: where is a linear functional, is a bilinear form, and , 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 . We consider a family of bilinear functionals and let be the solution to We would like to determine if there exists a value of the parameter such that or, more realistically, such that is small enough. Theorem 10 states that the distance between the target solution and the solution of (31) can be reduced by minimizing a functional which depends on parameters.

Theorem 10. Let be the target solution which satisfies the initial condition in (30) and suppose that exists and belongs to . Suppose that there exists a family of such that for all . One has the following result: where is the solution of (31) s.t. and .

Example 11. We adjust Example 9, considering where , subject to and and . We set and construct target data as in Example 9 and then seek to recover given this data and . The results we get from the generalized collage method are summarized in Table 4, which again proves the stability of the method under noise.

tab4
Table 4: Collage coding results for the hyperbolic equation in Example 11.

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 domain , we have the equations where is the law of and is the Dirac delta “function” at the point .

For instance, imagine we have a flexible string directed along the -axis, with the string kept stretched by a constant horizontal tension, and forced to vibrate perpendicularly to the -axis under random force . If denotes the displacement of a point at time , it is well known that satisfies the following equation: where and . In our model, we suppose that , where 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.

737694.fig.004
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 for and one for . 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 . That is, has a meaning only when it is integrated with respect to a test function . Thus, the meaning of (35) is that for each we have

4.1.1. A Parameter Identification Model for

For this parameter identification problem, we seek to estimate given , , and the observations of . From this data, we recover the density of the process . Averaging equation (35) allows one to get a simpler model for the parameter identification problem (see [36]). This replaces the quantity with its expectation . Since is absolutely continuous, then , the density of the distribution of . The previous model can therefore be rewritten in an averaged form as coupled with the deterministic PDE Note that the averaged equation, (39), has solutions in the usual sense. For this particular case, we have . To see this, just notice that the operator is linear, so Furthermore, since and are deterministic functions. Thus, is the solution to the deterministic PDE However, this is clearly the same PDE for which is the solution. Thus, we must have . 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 for

For this parameter identification problem, we assume that and are known. The aim of the inverse problem consists of recovering the functional form of and the density of the process starting from a sample of observations of the random process , say . We do this in several steps. Let us denote by . The first step is to use and (35) to obtain an estimate of the distribution of . Let us consider the averaged equation coupled with the deterministic PDE Dividing the left side of the PDF in (44) by , we get . Let us go back to the stochastic differential equation (43), and let us take the expectation of both sides; we get We remember that the following relationship holds between the expectation of a random variable and its density: Thus, we can obtain , which now can be used for solving the inverse problem for (45) via the method illustrated in Section 2. We recover the function The last step involves the analysis of (47). The only unknown in this model is . Taking the expansions of and with respect to the same orthonormal basis , we get and then for , a linear system in , 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 set and for define and a domain with uniformly distributed holes of radius . Choosing , we consider the steady-state diffusion problem Suppose that, for a certain , we sample the solution of the above model on the perforated domain . We aim at estimating starting from this data and by using the model which is obtained when instead. In fact, it is much easier to solve an inverse problem on the initial domain instead of on the perforated domain .

For a fixed value of , we solve the diffusion problem numerically and sample the solution at uniformly distributed points strictly inside . If such a point lies inside a hole, we obtain no information at the point. Using the (or fewer) data points, we use the Generalized Collage Theorem to solve the related inverse problem, seeking a diffusivity function of the form .

For this example, we use the particular values , , and , corresponding to , , and . The level curves are illustrated in Figure 5. The results for , , and are given in Table 5. In Figure 6, we illustrate the meshes used for numerically solving each of the problems.

tab5
Table 5: Results for the inverse problem. True values are .
737694.fig.005
Figure 5: Level curves of solutions in the example, with , , and .
737694.fig.006
Figure 6: Meshes for each problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

H. Kunze and F. Mendivil were partially supported by NSERC in the form of a discovery grant.

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