Approximation Of Differential Equations Research Articles

Cascade-refine model for cephalometric landmark detection in high-resolution orthodontic images

Landmark detection has been well developed by deep learning methods, and the cascade-connected neural network (CCNN) stands out as a widely used deep learning landmark detection method. CCNNs consist of several stacked network backbones, where the predictions of the previous network backbone are used as the input of the following one. Due to GPU memory bottlenecks, CCNNs have two limitations. First, the network backbones of CCNNs have limited numbers and depths; thus, the learning ability of CCNNs is limited. Second, CCNNs are usually trained in low-resolution images. However, the neighboring pixels in high-resolution images are usually vital for landmark detection, especially for cephalometric landmark detection. This paper interprets CCNNs as the discrete approximation of ordinary differential equations. Relying on this explanation, we further propose a novel model, called the cascade-refine model, which takes advantage of CCNNs and makes it possible to overcome the limitations of number and depth by sharing parameters among stacked network backbones. Moreover, the proposed model obeys the rule of coarse-to-fine architectures, where a global module is used to generate the coarse landmark locations, and a local module is adopted to tune the pixel error of latent landmark locations in the region of interest. The proposed model is trained in an end-to-end manner. Thus, the neighboring detailed high-resolution pixels are directly exploited for cephalometric landmark detection. The proposed cascade-refine model outperforms state-of-the-art methods on the public Automatic Cephalometric X-ray Landmark Detection Challenge 2015 dataset. We also built two private cephalometric X-ray datasets, and experimental results on both datasets demonstrate the good performance of the proposed model.

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form [0,T]times mathbb {R}^{d}ni (t,x){kern -.5pt}mapsto {kern -.5pt} tx{kern -.5pt}in {kern -.5pt} mathbb {R}^{d} where T{kern -.5pt}in {kern -.5pt} (0,infty ), d{kern -.5pt}in {kern -.5pt} mathbb {N}, which we both develop within this article.

Modification of theory A.R. Rzhanitsyn in analysis of multilayer composite beams

The article proposes the development of a numerical method for calculating multilayer beams, based on the theory of composite rods by A.R. Rzhanitsyn. The modification of this theory is to simplify the calculation model for a determined class of structures. It is considered multilayer beams composed of same layers of rectangular cross section, with the same physical and mechanical characteristics. The stiffness of all connecting seams is taken equal. In the research the hypothesis of a functional relationship between shear forces in the seams of the structure is taken. This allows the authors to significantly reduce the dimension of the system of resolving differential equations, from n + 2 equations to three for any finite number of layers. Where n -is the number of seams, and, accordingly, the number of shear forces to find in the seams according to the A.R Rzhanitsyn model, n + 1 is the number of layers. A comparison of three models of the above dependence is given. The numerical methodology is based on the approximation of differential equations by difference equations of the method of successive approximations (MSA). This methodology has proven itself well in the calculation of beams, plates, shells for the action of static loads, in calculations in a dynamic setting and for stability, on an elastic foundation. Including multilayer beams and plates. It allows to take into account the finite discontinuities of the load parameters, stiffness parameters of the structure and foundation. The described methodology can find application in the practice of design organizations and enter the educational courses of higher educational institutions of the construction profile.

On approximations for reflected SDEs and SPDEs with Neumann boundary conditions

In this paper we study an approximation for reflected stochastic differential equations in non-smooth domains by replacing the driving Brownian motion with its smooth approximation and prove that the solutions of approximating equations converge in the mean square to the solution of the Stratonovich reflected stochastic differential equation. We also study the smooth approximation for Neumann-type semilinear stochastic partial differential equations, which is completed through establishing the convergence in $ L^2 $ of backward doubly stochastic differential equations coupled with reflected SDEs.

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Half-sweep Modified SOR Approximation of A Two-dimensional Nonlinear Parabolic Partial Differential Equation

The sole subject of this numerical analysis was the half-sweep modified successive over-relaxation approach (HSMSOR), which takes the form of an iterative formula. This study computed a class of two-dimensional nonlinear parabolic partial differential equations subject to Dirichlet boundary conditions numerically using the implicit-type finite difference scheme. The computational cost optimization was considered by converting the traditional implicit finite difference approximation into the half-sweep finite difference approximation. The implementation required inner-outer iteration cycles, the second-order Newton method, and a linearization technique. The created HSMSOR is utilized to obtain an approximation of the linearized equations system through the inner iteration cycle. In contrast, the problem's numerical solutions are obtained using the outer iteration cycle. The study examined the local truncation error and the stability, convergence, and method analysis. Results from three initial-boundary value issues showed that the proposed method had competitive computational costs compared to the existing method.

Explicit numerical approximations for McKean-Vlasov neutral stochastic differential delay equations

This paper studies the numerical methods to approximate the solutions for a sort of McKean-Vlasov neutral stochastic differential delay equations (MV-NSDDEs) that the growth of the drift coefficients is super-linear. First, we obtain that the solution of MV-NSDDE exists and is unique. Then, we use a stochastic particle method, which is on the basis of the results about the propagation of chaos for a stochastic particle system, to deal with the approximation of the law. Furthermore, we construct the tamed Euler-Maruyama numerical scheme with respect to the corresponding particle system and obtain the rate of convergence. Combining propagation of chaos and the convergence rate of the numerical solution to the particle system, we get the convergence error between the numerical solution and exact solution of the original MV-NSDDE in the stepsize and number of particles.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order . Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Runge Kutta Optimization for Fixed Size Multimodal Test Functions

In this study, it is aimed to increase the success of the Runge Kutta (RUN) algorithm, which is used in the solution of many optimization problems in the literature, on fixed-size test functions by changing the parameter values. Optimization can be defined as making a system most efficient at the least possible cost under certain constraints. For this process, many optimization algorithms have been designed in the literature and used to obtain the best solutions for certain problems. The most important parts in solving these problems are modeling the problem correctly, determining the parameters and constraints of the problem, and finally choosing a suitable meta-heuristic algorithm for the solution of the objective function. Not every algorithm is suitable for every problem structure. Therefore, in this study, the suitability of the RUN algorithm for the solution of fixed-size functions will be evaluated. Theoretically, Runge-Kutta methods used in numerical analysis are an important type of the family of closed and open iterative methods for solution approximations of ordinary differential equations. The RUN algorithm is also designed with inspiration from these methods. In order to evaluate the performance of the RUN algorithm on fixed-size functions in the study, 10 fixed-size multimodal test functions (Shekel's Foxholes, Kowalik, Six-Hump Camel-Back, Branin, Goldstein-Price, Hartman3, Hartman6, Shekel5, Shekel7, Shekel10) have been found in the literature before was selected. Solutions for each of the selected functions are obtained by changing the parameter values of the RUN algorithm. The obtained solution values were evaluated by comparing the solutions obtained with Slime Mold Algorithm (SMA) and Hunger Games Search (HGS) algorithms.

Effective approximation for a nonlocal stochastic Schrödinger equation with oscillating potential

We study the effective approximation for a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behavior as the scaling parameter tends to 0. This is motivated by data assimilation with non-Gaussian uncertainties. The nonlocal operator in this stochastic partial differential equation is the generator of a non-Gaussian Lévy-type process (i.e., a class of anomalous diffusion processes), with non-integrable jump kernel. With the help of a two-scale convergence technique, we establish effective approximation for this nonlocal stochastic partial differential equation. More precisely, we show that a nonlocal stochastic Schrödinger equation has a nonlocal effective equation. We show that it approximates the original stochastic Schrödinger equation weakly in a Sobolev-type space and strongly in \(L^2\) space. In particular, this effective approximation holds when the nonlocal operator is the fractional Laplacian.

New Stability Results of an ABC Fractional Differential Equation in the Symmetric Matrix-Valued FBS

By using a class of aggregation control functions, we introduce the concept of multiple-HU-OS1-stability and get an optimum approximation for a nonlinear single fractional differential equation (NS-ABC-FDE) with a Mittag–Leffler kernel. We apply an alternative fixed-point theorem to prove the existence of a unique solution and the multiple-HU-OS1-stability for the NS-ABC-FDE in the symmetric matrix-valued FBS. Finally, with an example, we show the application of the obtained results.

Numerical methods and hypoexponential approximations for gamma distributed delay differential equations.

Gamma distributed delay differential equations (DDEs) arise naturally in many modelling applications. However, appropriate numerical methods for generic gamma distributed DDEs have not previously been implemented. Modellers have therefore resorted to approximating the gamma distribution with an Erlang distribution and using the linear chain technique to derive an equivalent system of ordinary differential equations (ODEs). In this work, we address the lack of appropriate numerical tools for gamma distributed DDEs in two ways. First, we develop a functional continuous Runge-Kutta (FCRK) method to numerically integrate the gamma distributed DDE without resorting to Erlang approximation. We prove the fourth-order convergence of the FCRK method and perform numerical tests to demonstrate the accuracy of the new numerical method. Nevertheless, FCRK methods for infinite delay DDEs are not widely available in existing scientific software packages. As an alternative approach to solving gamma distributed DDEs, we also derive a hypoexponential approximation of the gamma distributed DDE. This hypoexponential approach is a more accurate approximation of the true gamma distributed DDE than the common Erlang approximation but, like the Erlang approximation, can be formulated as a system of ODEs and solved numerically using standard ODE software. Using our FCRK method to provide reference solutions, we show that the common Erlang approximation may produce solutions that are qualitatively different from the underlying gamma distributed DDE. However, the proposed hypoexponential approximations do not have this limitation. Finally, we apply our hypoexponential approximations to perform statistical inference on synthetic epidemiological data to illustrate the utility of the hypoexponential approximation.

The Existence, Uniqueness, and Carathéodory’s Successive Approximation of Fractional Neutral Stochastic Differential Equation

The existence, uniqueness, and Carathe´odory’s successive approximation of the fractional neutral stochastic differential equation (FNSDE) in Hilbert space are considered in this paper. First, we give the Carathe´odory’s approximation solution for the FNSDE with variable time delays. We then establish the boundedness and continuity of the mild solution and Carathe´odory’s approximation solution, respectively. We prove that the mean-square error between the exact solution and the approximation solution depends on the supremum of time delay. Next, we give the Carathe´odory’s approximation solution for the general FNSDE without delay. Under uniform Lipschitz condition and linear growth condition, we show that the proof of the convergence of the Carathe´odory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution. Furthermore, under the non-Lipschitz condition, which is weaker than Lipschitz one, we establish the existence and uniqueness theorem of the solution for the FNSDE based on the Carathe´odory’s successive approximation. Finally, a simulation is given to demonstrate the effectiveness of the proposed methods.

Summation-by-parts finite-difference shallow water model on the cubed-sphere grid. Part I: Non-staggered grid

Summation-by-Parts Finite Differences (SBP-FD) is an approach for approximation of differential operators satisfying a discrete analogue of integration by parts analytic property. SBP-FD allows one to build provably stable high-order spatial approximations. SBP-FD methods are widely used for approximation of partial differential equations in multiblock domains with logically-rectangular curvilinear mesh inside. The gnomonic cubed-sphere grid is an example of such a domain. However, applications of the SBP-FD approximation for the cubed-sphere grid in meteorological context are not a widespread research area.We present a SBP-FD based shallow water model using a non-staggered grid. The model is total-energy conserving, mass-conservative and has discrete analogues of other mimetic properties such as curl-free gradient property. The shallow water model is tested with the commonly used Williamson test suite supplemented with the Galewsky barotropic instability case. High-order convergence is shown for tests with analytic solutions. The SBP-FD shallow water model is more accurate than low-order mimetic finite-element counterparts, but slightly less accurate than high-order finite-volume, spectral-elements and discontinuous Galerkin schemes.

A new framework for polynomial approximation to differential equations

In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework.

Successive Approximations for Caputo-Fabrizio Fractional Differential Equations

Abstract In this work we deal with a uniqueness result of solutions for a class of fractional differential equations involving the Caputo-Fabrizio derivative. We provide a result on the global convergence of successive approximations.

Deep learning schemes for parabolic nonlocal integro-differential equations

In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre by Lius and Lius (Comm PDE 32(8):1245–1260, 2007). Based on the work by Hure, Pham and Warin by Hure et al. (Math Comp 89:1547–1579, 2020), we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lévy processes and a new neural network approximation of the nonlocal part to overcome the lack of a suitable good approximation of the nonlocal part of the solution.

Discrete gradient flow approximations of high dimensional evolution partial differential equations via deep neural networks

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as “Nitsche-type” methods. Moreover, inspired by the seminal work of Jordan, Kinderlehrer, and Otto (JKO) Jordan et al. (1998), we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.

A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem

We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep Ritz method applied to the Laplace and the Stokes equations. Further, we develop an a posteriori error estimator for neural network approximations of partial differential equations. The proposed approach is based on the dual weighted residual estimator. It is destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training. The result is equipped with computational examples for Laplace and Stokes problems.

Tensor train based isogeometric analysis for PDE approximation on parameter dependent geometries

This work develops a numerical solver based on the combination of isogeometric analysis (IGA) and the tensor train (TT) decomposition for the approximation of partial differential equations (PDEs) on parameter-dependent geometries. First, the discrete Galerkin operator as well as the solution for a fixed geometry configuration are represented as tensors and the TT format is employed to reduce their computational complexity. Parametric dependencies are included by considering the parameters that control the geometry configuration as additional dimensions next to the physical space coordinates. The parameters are easily incorporated within the TT-IGA solution framework by introducing a tensor product basis expansion in the parameter space. The discrete Galerkin operators are accordingly extended to accommodate the parameter dependence, thus obtaining a single system that includes the parameter dependency. The system is solved directly in the TT format and a low-rank representation of the parameter-dependent solution is obtained. The proposed TT-IGA solver is applied to several test cases which showcase its high computational efficiency and tremendous compression ratios achieved for representing the parameter-dependent IGA operators and solutions.