Wavelet Method for Partial Differential Equations and Image Processing

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

Tremendous progress has been achieved in image analysis and processing, and in particular, the use of partial differential equation (PDE) methods in image analysis has proliferated in recent years. However, PDE methods have seen little application to optical image-based structural damage detection for civil systems. This paper applies the Chan-Vese active contour model and its level set representation to detect concrete surface damage, in this case concrete cracks, in optical images. The detected cracks can be further characterized by extracted geometric quantities including their width, length and area of coverage, and associated with engineering design. With regard to crack width extraction, an approximate method is proposed, which relies on solving for a signed distance function. We test these methods by using synthetic images as well as real multi-temporal images from a laboratory experiment. This paper illustrates that using the proposed methods, cracks with complex topographical patterns can be successfully detected with sufficient accuracy.

In the process of acquisition, compression, transmission and storage, digital images are often interfered by imaging equipment and the external environment, resulting in image distortion. Image denoising aims to truly reconstruct the image from the observation data contaminated by noise to provide more accurate and reliable information for the subsequent processing of the image. Therefore, image noise reduction is the most basic and most important link in the field of image processing research. In the past decade or so, the application of theories and methods of partial differential equations (PDEs) in various fields of image processing has attracted more and more attention. This article researches and analyzes the image processing methods based on partial differential equations. First, it introduces the research background of this article, the development history, principles of partial differential equations and the status quo of image processing; then, it summarizes and analyzes the application of partial differential equations to images Classical model of denoising. Through the analysis of the classic model, it can be known that the image noise reduction algorithm based on partial differential equations can be based on the structural characteristics of the image. This method adopts a corresponding smoothing strategy, and it can achieve satisfactory image denoising effects.

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

Some multiscale methods for partial differential equations by J. Xu and A. Zhou Terascale implicit methods for partial differential equations by D. E. Keyes Boundary conditions for fully implicit two-phase flow models by M. Peszynska, E. W. Jenkins, and M. F. Wheeler Phase-field models of solidification by G. B. McFadden Singularity formation in free-surface Stokes flows by Q. Nie, S. Tanveer, T. F. Dupont, and X. Li A note on cache memory methods for multigrid in three dimensions by C. C. Douglas and D. T. Thorne.

The time required to execute real-world scientific computations is a major issue. A single simulation may last hours, days, or even weeks to reach a certain level of accuracy, despite running on large-scale parallel architectures. Strict time limits may often be imposed too -- 60 minutes in the case of the UK Met Office to produce a forecast. In this thesis, it is demonstrated that by raising the level of abstraction, the performance of a class of numerical methods for solving partial differential equations is improvable with minimal user intervention or, in many circumstances, with no user intervention at all. The use of high level languages to express mathematical problems enables domain-specific optimization via compilers. These automated optimizations are proven to be effective in a variety of real-world applications and computational kernels. The focus is on numerical methods based on unstructured meshes, such as the finite element method. The loop nests for unstructured mesh traversal are often irregular (i.e., they perform non-affine memory accesses, such as A[B[i]]), which makes reordering transformations for data locality essentially impossible for low level compilers. Further, the computational kernels are often characterized by complex mathematical expressions, and manual optimization is simply not conceivable. We discuss algorithmic solutions to these problems and present tools for their automation. These tools -- the COFFEE compiler and the SLOPE library -- are currently in use in frameworks for solving partial differential equations.

: This is a grant to cover partial cost of the organization of the International Conference on Scientific Computing, Partial Differential Equations and Image Processing held at IPAM, UCLA on Apr 5-7 2002. The Conference was held in honor of Stanley Osher on the occasion of his 60th birthday, to commemorate his achievement in many of the areas covered by this conference, including high resolution shock capturing schemes, level set method, applications to multi-phase flows, computer vision. TV based image restoration, just to name a few.

This paper generalises the preconditioning techniques, introduced by Evans [2], and defines sparse and compact preconditioned iterative methods for the numerical solution of the linear system Au = b.The difference between the methods is shown to depend on whether a conditioning matrix R consists of components derived from a splitting or factorization of A. Some theoretical results for the iterative schemes are given when A has particular properties such as consistent ordering, irreducibility, diagonal dominance, positive definiteness, etc., when derived from the finite difference discretisation of a 2nd order self-adjoint elliptic partial differential equation. Finally, the application of both forms of preconditioning to the Conjugate Gradient method is presented and computational results compared.

This work considers the partial solutions of partial differential equations for initial/boundary conditions using the Adomian decomposition method. The study formally shows that the partial solutions are always identical for all styles of boundary conditions. We also prove that the partial solution in the t-direction requires less computational work if compared with other partial solutions developed in any space variable direction. In addition, several mathematical models that govern the heat distribution and the wave propagation phenomenas have been tested, and the results obtained have shown that the t-solution minimizes the size of calculations if compared with the traditional techniques.

In this article, we review the development of multigrid methods for partial differential equations over the last 30 years, illuminating, in particular, the software question. With respect to industrial software development, we will distinguish “optimal” multigrid, multigrid “acceleration” and “robust” multigrid. Surprisingly, not geometric multigrid but algebraic multigrid (AMG) finally brought the breakthrough. With the software package SAMG, which is based on block-type AMG, systems of partial differential equations can be treated efficiently also. Finally, we outline how SAMG is used for industrial applications.

In scientific and engineering calculation, the effective solution of partial differential equations (PDEs) has great significance. This paper presents an innovative method based on the combination of a U-Net neural network with Fourier neural operators, aiming to improve the accuracy and efficiency of solving partial differential equations. U-Net neural networks with a unique encoding–decoder structure and hopping connections can efficiently extract and integrate spatial-domain features and accurately describe the spatial structure of PDEs. The Fourier neural operator combines the advantages of neural networks and Fourier transform to learn the mapping relationship of high-dimensional data by training the linear transformation parameters in the detailed frequency domain of the data. Fourier transform is used to explore and process the frequency-domain features and capture the frequency characteristics of the solution of partial differential equations. By combining the two, this method not only retains spatial details but also makes full use of frequency-domain information, which significantly reduces the dependence of the model on large-scale data and improves the generalization ability of the model. The experimental results show that compared with the traditional method, this method performs well in various complex partial differential equation (PDE) solving tasks and achieves high accuracy.

Most instructors manage two opposing currents in classroom dynamics. The first is a desire to describe the subject in its broadest sense including all the subtleties and nuances. It gives the students a road map to the field, and if they can appreciate it at the time, it provides structure to a course that might otherwise appear to be a loosely connected series of topics. The second is the desire to ground the subject with concrete examples that students can grasp at the finest level. The examples link prerequisite knowledge with one of the new subjects and provide students with a foundation upon which to construct a deeper understanding. Few courses draw together so many disparate topics as numerical methods for partial differential equations (PDEs). When first exposed to finite differences for numerical PDEs, students will readily notice the regular structures in matrices, but few texts address these structures systematically and connect them directly to the analytical properties of the solution. There are many good examples, but it's hard to see the broader landscape. In this Education section article, “Functions of Difference Matrices Are Toeplitz Plus Hankel,” authors Gil Strang and Shev MacNamara fill in some of the big picture by presenting some new results and observations about the structure of difference matrices for approximating solutions to PDEs. A common approach in the analysis of numerical methods for PDEs is to discretize the spatial operator and leave time continuous so that one studies the semidiscrete system. The exact solution to these systems will be functions of the underlying difference matrices, hence the structure of functions of difference matrices is a natural point of interest for students, instructors, and researchers. Toeplitz matrices, having entries that are constant along diagonals, arise naturally in difference matrices. As the authors point out, this is to be expected because of the translation-invariance of the differentiation operator. In an unbounded domain, if initial condition $u_0$ leads to a solution $u$, a translation of $u_0$ will lead to a solution that is a translation of $u$. The authors discover that solutions to the semidiscrete systems are Toeplitz plus Hankel. The role of Hankel matrices, matrices that are constant along antidiagonals, is a little more subtle to explain. If one considers the role of boundary conditions for canonical problems like the heat or wave equations, one can see how this might arise. In the case of a single Neumann boundary condition for the heat equation, the classic method of images provides some insight. Shifting the initial condition toward the boundary shifts the image solution toward the boundary outside the domain, which shifts the influence of the image away from the boundary within the domain. Thus, shifting the initial condition has both a Toeplitz part from transition invariance and a Hankel part from the influence of the boundaries. The Toeplitz plus Hankel decomposition paves the way for exact solutions to semidiscrete approximations, allowing better error estimates and a deeper general understanding of spatial discretization. While it makes sense intuitively that functions of difference matrices should be Toeplitz plus Hankel, the authors enthusiastically explore the topic using a variety of methods and techniques from advanced undergraduate or introductory graduate mathematics including spectral decompositions, Riemann sums, and Fourier analysis. In other words, the destination is interesting, but the voyage is even more fun! This paper would serve as an excellent module for a numerical analysis course or an integrative seminar for undergraduates or graduate students interested in numerical methods.

Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.

We develop partial differential equation (PDE) methods to study the dynamics of pattern formation in partial integro-differential equations (PIDEs) defined on a spatially extended domain. Our primary focus is on scalar equations in two spatial dimensions. These models arise in a variety of neuronal modeling problems and also occur in material science. We first derive a PDE which is equivalent to the PIDE. We then find circularly symmetric solutions of the resultant PDE; the linearization of the PDE around these solutions provides a criterion for their stability. When a solution is unstable, our analysis predicts the exact number of peaks that form to comprise a multipeak solution of the full PDE. We illustrate our results with specific numerical examples and discuss other systems for which this technique can be used.

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions