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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...

The companion volume of this book, ‘Analytic methods for partial differential equations’, is concerned with solution of partial differential equations using classical methods which result in analytic solutions. These equations result when almost any physical situation is modelled, ranging from fluid mechanics problems, through electromagnetic problems to models of the economy. Some specific and fundamental problems were highlighted in the earlier volume, namely the three major classes of linear second order partial differential equations. The heat equation, the wave equation and Laplace’s equation will form a basis for study from a numerical point of view for the same reason as they did in the analytic case. That is, the three equations are the canonical forms to which any quasi-linear second order equation may be reduced using the characteristic transformation.KeywordsPartial Differential EquationCanonical FormSpectral RadiusLinear Algebraic EquationBackground MathematicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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.

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.

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.

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.

Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs

Hamiltonian discretization of boundary control systems

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.

Interface solutions of partial differential equations with point singularity

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The critical issues of the optimization design of centrifugal pump are introduced in detail, and the main obstacle is the complicated implicit relation between the pump hydraulic performance and its interior passage geometry shape. The parametric design methods of centrifugal impeller proposed previously are introduced, which include the method based on NURBS surface, the method based on free-form deformation (FFD) and the method based on partial differential equation (PDE). To reduce the computational cost, the partial differential equation method is used to parametric design of the centrifugal impeller, and the boundary condition was parameterized. The parameters a(u,v) are consider as constant. The response surface methodology (RSM) is proposed to optimize the design of pump impeller. The objective function can be approximated based on trial results. Analyze the response surface model, and we find that the 2nd order polynomial cannot be used to simulate the complicated relation between the design object and the control variables. So the partial differential equations were proposed to construct the hypersurface response of objective function. The boundary-value problem of hyperspace is numerically solved. According to the optimization analysis, we can get the optimum design of pump impeller. The calculation case shows that the proposed theory and method are rational.

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