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.
According to Keller [
There is a fundamental difference between the direct and the inverse problem; often the direct problem is
Many inverse problems may be recast as the approximation of a target element
Let
This vastly simplifies this type of inverse problem as it is much easier to estimate
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 [
The collage-based inverse problem can be formulated as an optimization problem as follows:
The paper is organized as follows. Section
In [
The function
Now, let
Let
Given a target solution
Suppose that the stochastic process
Minimal collage distance parameters for different
|
|
|
|
---|---|---|---|
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 | 1.78323 |
Different paths of the Brownian motion with
Let us consider the following variational equation:
Suppose that
In order to ensure that the approximation
We now present an inverse problem for the two-dimensional steady-state diffusion equation. With
(Left to right, top to bottom) for two-dimensional Example
Next, we perturb the target function
Numerical results for the inverse problem with different levels of noise.
|
|
||
---|---|---|---|
|
|
|
|
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 | 0.14996 |
As the second example, let us follow Example 7 of [
(Left to right) for two-dimensional Example
These results have been extended to a wider class of elliptic equations problems in [
Let
If
Suppose that we have a given Hilbert space
We would like to determine if there exists a value of the parameter
Let
Let us consider the following equation:
Collage coding results for the parabolic equation in Example
Noise |
|
|
|
|
---|---|---|---|---|
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 |
Let us now consider the following weakly formulated hyperbolic equation:
Let
We adjust Example
Collage coding results for the hyperbolic equation in Example
Noise |
|
|
|
|
---|---|---|---|---|
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 |
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
For instance, imagine we have a flexible string directed along the
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
Before we begin the analysis, a few words about (
For this parameter identification problem, we seek to estimate
For this parameter identification problem, we assume that
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
For a fixed value of
For this example, we use the particular values
Results for the inverse problem. True values are
|
|
|
Recovered parameters | ||
---|---|---|---|---|---|
|
|
| |||
|
|
9 |
|
|
|
49 |
|
|
| ||
49 |
|
|
| ||
|
|||||
|
|
9 |
|
|
|
49 |
|
|
| ||
99 |
|
|
| ||
|
|||||
|
|
9 |
|
|
|
49 |
|
|
| ||
99 |
|
|
|
Level curves of solutions in the example, with
Meshes for each problem.
The authors declare that there is no conflict of interests regarding the publication of this paper.
H. Kunze and F. Mendivil were partially supported by NSERC in the form of a discovery grant.