On sparse and compact preconditioned conjugate gradient methods for partial differential equations

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...

We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

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.

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.

This work proposes and analyzes an anisotropic sparse grid stochastic collocation
\nmethod for solving partial differential equations with random coefficients and forcing terms (input
\ndata of the model). The method consists of a Galerkin approximation in the space variables and a
\ncollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis or
\nGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution
\nof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori
\nand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.
\nThese procedures seem to be very effective for the problems under study. The proposed method
\ncombines the advantages of isotropic sparse collocation with those of anisotropic full tensor product
\ncollocation: the first approach is effective for problems depending on random variables which weigh
\napproximately equally in the solution, while the benefits of the latter approach become apparent when
\nsolving highly anisotropic problems depending on a relatively small number of random variables, as
\nin the case where input random variables are Karhunen–Lo`eve truncations of “smooth” random
\nfields. This work also provides a rigorous convergence analysis of the fully discrete problem and
\ndemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in
\nthe preasymptotic regime, with respect to the total number of collocation points. It also shows
\nthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.
\nNumerical examples illustrate the theoretical results and are used to compare this approach with
\nseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensional
\nproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and
\nsuperior to all examined methods.

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.

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

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 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 chapter, applications of wavelet theory to partial differential equations and image processing are discussed.

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.

Helmholtz decomposition-based SPH

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.