A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations

doi:10.1016/j.neucom.2012.10.007

Neurocomputing

Volume 104, 15 March 2013, Pages 138-145

https://doi.org/10.1016/j.neucom.2012.10.007 Get rights and content

Abstract

A wavelet multiscale iterative regularization method is proposed for the parameter estimation problems of partial differential equations. The wavelet analysis is introduced and a wavelet multiscale method is constructed based on the idea of hierarchical approximation. The inverse problem is decomposed into a sequence of inverse problems which rely on the scale variables and are solved successively according to the size of scale from the longest to the shortest. By combining multiscale approximations, the problem of local minimization is overcomed, and the computational cost is reduced. At each scale, based on the wavelet approximation, the problem of inverting the parameter is transformed into the problem of estimating the finite wavelet coefficients in the scale space. A novel iterative regularization method is constructed. The efficiency of the method is illustrated by solving the coefficient inverse problems of one- and two-dimensional elliptical partial differential equations.

Introduction

Inverse problems of partial differential equations (PDEs) originate from various practical problems, and belong to the category of cross subjects. Its meaningful to carry out studies on the theory of inverse problems and their applications [1], [2], [3]. Generally speaking, to solve inverse problems is very difficult, the main reason lies in:

Ill-posedness: The solution does not depend continuously on the observed data. A minor disturbance of the observed data may cause large change on the solution of the inverse problems.
Nonlinearity: Nonlinear dependence of the observation with respect to the parameter to be estimated causes the presence of numerous local minimum, a good initial estimate is crucial for any numerical methods.
Large computational cost: An inversion process needs tremendous forward computations, the computational cost is usually very large.

Therefore, the key problem of an inverse problem is how to quickly find a stable solution in a wide range.

In order to overcome the three main difficulties, we investigate an inversion strategy, called wavelet multiscale iterative regularization method, for the general parameter estimation problem of partial differential equation in time–space domain. This strategy synthesizes the property of easy computation of the iterative method with the advantages of minor dependence on the initial model, fast convergence, and stability of results of multiscale inversion. We first decompose the parameter to be inverted by a wavelet multiscale decomposition algorithm, then the original problems are transformed into a sequence of inverse problems depending on the scale variables. At each scale, based on the wavelet approximation, the problem of estimating the parameter in a differential equation is reduced to the problem of estimating the finite coefficients in the approximation spaces. Therefore, when the scale is coarsened for one step, the scope of the inverse problem will be reduced to one-half of it. Consequently the number of unknown coefficients is greatly reduced, and the speed of computation is quickened. In the whole computation from the coarsest scale to the finest one the terminative value of a former scale is taken as an initial estimate for the next finer scale. In this way, the risk of trapping in a local minimum is reduced, and the computational cost may be saved as well. Besides, in order to overcome the ill-posedness, Tikhonov regularization and the perturbation method are combined into a new and stable iterative method, which may be used as an optimistic algorithm to search out the global optimal solution at each scale.

The rest of this paper is organized as follows. Related works about the inverse problems of partial differential equations are reviewed in Section 2. The relevant properties of wavelet are summarized in Section 3. We present the wavelet multiscale inversion method and the iterative method for the general parameter estimation problem of differential equations in Section 4. In order to test our method's effectiveness, numerical simulations are carried out for inverse problems of recovering coefficients in elliptic differential equations in Section 5. Finally, in Section 6 some conclusions are given.

Section snippets

Related works

Multiscale inversion is a fairly developed inversion strategy to accelerate convergence, enhance stability of inversion, overcome disturbance of local minima, and by which one can search out the global minimum [4]. The main essence of multiscale inversion is to decompose the original problem into different scales or frequency bands, and to carry out inversion beginning at the longest scale. Since at a longer scale, the objective function shows stronger convexity and has less minima. Therefore,

Wavelet

In this section we summarize the main properties of wavelet as far as they are needed in the following section. For a more detailed introduction to the theory of wavelet we refer to [6], [12], [23].

Model and wavelet multiscale inversion strategy

Consider the following partial differential equation defined in the time–space domain: $L (c (x), t) u (x, t) = 0, x \in Ω, 0 \leq t \leq T,$ where $x = (x_{1}, x_{2}, \dots, x_{p})^{T}$ , $Ω \subset R^{p}$ , $\partial Ω$ is the boundary of $Ω$ , $u (x, t)$ is a sufficiently smooth function defined in $Ω \times (0, T)$ . L is a differential operator. The initial condition is $Eu (x, 0) = g (x), x \in Ω .$ The boundary condition is $Bu (x, t) = f (x, t), x \in \partial Ω, 0 \leq t \leq T,$ where E, B are the initial condition operator and boundary condition operator, respectively. If c(x), g(x), $f (x, t)$ are known, (4.1), (4.2), (4.3)

Numerical examples

To show the feasibility of our approach and illustrate its numerical behavior, we consider the elliptical differential equation $\{\begin{matrix} - \nabla \cdot (c (x) \nabla u) = f & in Ω \subset R^{d}, \\ u = 0 & on \partial Ω, \end{matrix}$ where $f \in L^{2} (Ω)$ . The observations of the solution u are used to recover the coefficient c(x).

Our task is to determine an approximation of parameter c(x) from a measurement $u^{δ}$ in the model problem (5.1) for d=1,2. Since the model problem is only well-posed for $c (x) \geq \underset{̲}{γ} > 0$ , and if $f \in H^{- 1} (Ω)$ for almost all $x \in Ω$ , we denote $X = {c (x) \in L^{\infty} (Ω) : 0 < \underset{̲}{γ} \leq c (x) \leq \bar{γ},$

Conclusions

For the general parameter estimation problem of partial differential equations, a wavelet multiscale iterative regularization method is proposed in this paper. Numerical simulations for the inversion problem of one- and two-dimensional elliptical equation showed the method's effectiveness in the aspects of stability, global convergence, noise suppression and fast computation. Wavelet multiscale method widens the scope of research of multiscale inversion. To conduct wavelet multiscale

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (Grant nos. 41074088 and 61173035), the Doctoral Science Technology Foundation of Liaoning Province (Grant no. 20091007), the China Postdoctoral Science Foundation (Grant no. 20110491533), the Fundamental Research Funds for the Central Universities (Grant no. 2012TD027), the Program for New Century Excellent Talents in University (Grant no. NCET-11-0861).

Hongsun Fu received MS and PhD at Harbin Institute of Technology, China. She is an Associate Professor of the Department of Mathematics, Dalian Maritime University. She serves as a referee for the inverse problems in Science & Engineering.

References (26)

B. Choi et al.
Comparison of generalization ability on solving differential equations using backpropagation and reformulated radial basis function networks
Neurocomputing
(2009)
D. Luengo et al.
Novel fast random search clustering algorithm for mixing matrix identification in mimo linear blind inverse problems with sparse inputs
Neurocomputing
(2012)
D. Benaouda et al.
Wavelet-based nonlinear multiscale decomposition model for electricity load forecasting
Neurocomputing
(2006)
M. Mu et al.
Shift and gray scale invariant features for palmprint identification using complex directional wavelet and local binary pattern
Neurocomputing
(2011)
W. Dahmen
Wavelet methods for PDEs—some recent developments
J. Comput. Appl. Math.
(2001)
J. Fröhlich et al.
An adaptive wavelet—vaguelette algorithm for the solution of pdes
J. Comput. Phys.
(1997)
V. Comincioli et al.
A wavelet-based method for numerical solution of nonlinear evolution equations
Appl. Numer. Math.
(2000)
D. Donoho
Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition
Appl. Comput. Harmonic Anal.
(1995)
R. Ghanem et al.
A wavelet-based approach for model and parameter identification of non-linear systems
Int. J. Non-lin. Mech.
(2001)
X. Lu et al.
Adaptive wavelet-Galerkin methods for limited angle tomography
Image Vis. Comput.
(2010)

A. Qian

A new wavelet method for solving an ill-posed problem

Appl. Math. Comput.

(2008)

V. Isakov

(2006)

I. Daubechies

Ten Lectures on Wavelet

(1992)

Cited by (5)

A wavelet multiscale method for the inverse problem of a nonlinear convection–diffusion equation
2018, Journal of Computational and Applied Mathematics
Citation Excerpt :
Recently, there exists a significant amount of research in the literature on the wavelet multiresolution method for the parameter identification inverse problem. Successful applications of this method include the parameter identification inverse problem of an elliptical equation [29,30], the parameter identification inverse problem of a two-dimensional acoustic wave equation [31,32], the parameter identification inverse problem of a nonlinear diffusion equation [33], the parameter identification inverse problem of Maxwell equations [34], the parameter identification inverse problem of fluid-saturated porous media [35,36], the parameter identification inverse problem of the electrical capacitance tomography model [37] and the parameter identification inverse problem of the diffuse optical tomography model [38]. All these works showed the effectiveness of wavelet multiscale method on the parameter identification inverse problem.
This paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. The forward problem is discretized using finite-difference methods and the inverse problem is formulated as a minimization problem with regularization terms. In order to overcome disturbance of local minimum, a wavelet multiscale method is applied to solve this parameter identification inverse problem. This method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub-inverse problems relying on scale variables, and successively solving these sub-inverse problems according to the size of scale from the smallest to the largest. The stable and fast regularized Gauss–Newton method is applied to each scale. Numerical simulations show that the proposed algorithm is widely convergent, computationally efficient, and has the anti-noise and de-noising abilities.
A wavelet multi-scale method for the inverse problem of diffuse optical tomography
2015, Journal of Computational and Applied Mathematics
Citation Excerpt :
Regarding the first three listed methods, some problems remain to be overcome such as the stability with respect to the initial guesses and the blurring effect of the reconstructed images due to the need for relatively strong regularization tools [17,14]. A relatively new method has emerged in the field of inversion, namely the wavelet multi-scale method [18–25]. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one.
This paper deals with the estimation of optical property distributions of participating media from a set of light sources and sensors located on the boundaries of the medium. This is the so-called diffuse optical tomography problem. Such a non-linear ill-posed inverse problem is solved through the minimization of a cost function which depends on the discrepancy, in a least-square sense, between some measurements and associated predictions. In the present case, predictions are based on the diffuse approximation model in the frequency domain while the optimization problem is solved by the L-BFGS algorithm. To cope with the local convergence property of the optimizer and the presence of numerous local minima in the cost function, a wavelet multi-scale method associated with the L-BFGS method is developed, implemented, and validated. This method relies on a reformulation of the original inverse problem into a sequence of sub-inverse problems of different scales using wavelet transform, from the largest scale to the smallest one. Numerical results show that the proposed method brings more stability with respect to the ordinary L-BFGS method and enhances the reconstructed images for most of initial guesses of optical properties.
An efficient parameter estimation method for nonlinear high-order systems via surrogate modeling and cuckoo search
2020, Soft Computing
Research on Numerically Solving the Inverse Problem Based on L1 Norm
2020, IOP Conference Series: Materials Science and Engineering
Estimation of a Permeability Field within the Two-Phase Porous Media Flow Using Nonlinear Multigrid Method
2017, Mathematical Problems in Engineering

Bo Han received MS and PhD at the Harbin Institute of Technology, China. He is a Professor of the Department of Mathematics, Harbin Institute of Technology. He is an Executive Director of Society for Industrial and Applied Mathematics in China. He is also the President of Society for Industrial and Applied Mathematics in Heilongjiang Province, China. He serves as a Referee for the Mathematical Reviews of Chinese Mathematical Society. Also he serves as a Referee for International Journals: Multiscale Modeling and Simulation (SIAM); International Journal on Numerical Analysis and Modeling; Selected Publications of Chinese Universities: Mathematics; Journal on Information and Computational Science; Communications in Computational Physics; Journal of Computational Mathematics.

Hongbo Liu received his three level educations (BSc, MSc, PhD) at the Dalian University of Technology, China. Currently he is a Professor at School of Information Engineering and Science, Dalian Maritime University, with an affiliate appointment in the Computer Science and Biomedical Engineering Division at the Dalian University of Technology. His research interests are in system modeling and optimization involving soft computing, probabilistic modeling, cognitive computing, machine learning, data mining, etc. He participates and organizes actively International Conference and Workshop and International Journals/Publications.

View full text

A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations

Abstract

Introduction

Section snippets

Related works

Wavelet

Model and wavelet multiscale inversion strategy

Numerical examples

Conclusions

Acknowledgments

Neurocomputing

Neurocomputing

Neurocomputing

Neurocomputing

J. Comput. Appl. Math.

J. Comput. Phys.

Appl. Numer. Math.

Appl. Comput. Harmonic Anal.

Int. J. Non-lin. Mech.

Image Vis. Comput.

Appl. Math. Comput.

Ten Lectures on Wavelet