Showing papers on "Numerical analysis published in 2020"

A numerical method for computing the overall response of nonlinear composites with complex microstructure

Abstract: The local and overall responses of nonlinear composites are classically investigated by the Finite Element Method. We propose an alternate method based on Fourier series which avoids meshing and which makes direct use of microstructure images. It is based on the exact expression of the Green function of a linear elastic and homogeneous comparison material. First, the case of elastic nonhomogeneous constituents is considered and an iterative procedure is proposed to solve the Lippman-Schwinger equation which naturally arises in the problem. Then, the method is extended to non-linear constituents by a step-by-step integration in time. The accuracy of the method is assessed by varying the spatial resolution of the microstructures. The flexibility of the method allows it to serve for a large variety of microstructures. (C) 1998 Elsevier Science S.A.

Dedalus: A flexible framework for numerical simulations with spectral methods

Abstract: This paper describes Dedalus, an open-source Python code for simulating partial differential equations from all areas of physics. Dedalus translates plain-text equations into efficient and parallelized solvers using global spectral methods. Here the authors detail the numerical methods enabling this translation and describe the code's design and implementation. They also illustrate its capabilities with diverse examples, including optical network dynamics, magnetized shocks in plasmas, large-scale oceanic flows, low Reynolds number flows, stellar and atmospheric waves, and diamagnetic levitation.

Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model

Abstract: In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo operator (in terms of the power law), the Caputo-Fabrizio operator (with exponential decay law) and the Atangana-Baleanu fractional derivative (based on the Mittag-Liffler law). We design some algorithms for the Schnakenberg model by using the newly proposed numerical methods. In such schemes, it worth mentioning that the classical cases are recovered whenever α = 1 and β = 1 . Numerical results obtained for different fractal-order ( β ∈ ( 0 , 1 ) ) and fractional-order ( α ∈ ( 0 , 1 ) ) are also given to address any point and query that may arise.

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

Abstract: Hybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows.

Numerical methods for nonlocal and fractional models

Abstract: Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

A New Iterative Method for the Numerical Solution of High-Order Non-linear Fractional Boundary Value Problems

Abstract: The boundary value problems (BVPs) have attracted the attention of many scientists from both practical and theoretical points of view, for these problems have remarkable applications in different branches of pure and applied sciences Due to this important property, this research aims to develop an efficient numerical method for solving a class of nonlinear fractional BVPs The proposed method is free from perturbation, discretization, linearization, or restrictive assumptions, and provides the exact solution in the form of a uniformly convergent series Moreover, the exact solution is determined by solving only a sequence of linear BVPs of fractional-order Hence, from practical viewpoint, the suggested technique is efficient and easy to implement To achieve an approximate solution with enough accuracy, we provide an iterative algorithm that is also computationally efficient Finally, four illustrative examples are given verifying the superiority of the new technique compared to the other existing results

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

Abstract: Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.

Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model

Abstract: In this work, we introduce a numerical and analytical study of the Peyrard-Bishop DNA dynamic model equation. This model is studied analytically by hyperbolic and exponential ansatz methods and numerically by finite difference method. A comparison between the results obtained by the analytical methods and the numerical method is investigated. Furthermore, some figures are introduced to show how accurate the solutions will be obtained from the analytical and numerical methods.

Fractional discretization: The African’s tortoise walk

Abstract: We proposed a new way to discretizing a differential or integral equation using a fractional step. The new way has improved the stability and accuracy of numerical methods. We presented some examples with classical and fractional differential and integral equations.

The phase field method for geometric moving interfaces and their numerical approximations

Abstract: This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modelling and their numerical methods in materials science, fluid mechanics, biology and image science.

Numerical Optimization, 2nd edition

Abstract: Practical Methods of OptimizationAn Introduction to Continuous OptimizationPractical Mathematical OptimizationLinear and Nonlinear OptimizationApplied Numerical Methods Using MATLABAlgorithms for OptimizationNumerical MathematicsOptimization Algorithms for Networks and GraphsLinear and Nonlinear OptimizationOptimization Theory and MethodsNumerical Methods and Optimization in FinanceConvex OptimizationPractical Methods for Optimal Control and Estimation Using Nonlinear ProgrammingNumerical Methods in EconomicsNonlinear ProgrammingLectures on Convex OptimizationNumerical Analysis for StatisticiansNumerical OptimizationApplied Optimization with MATLAB ProgrammingDynamic Optimization, Second EditionNumerical OptimizationPractical OptimizationOptimization of Power System OperationNatureInspired Optimization AlgorithmsIntroduction to Optimum DesignNumerical Optimization Techniques for Engineering DesignOptimization Methods in FinanceNumerical OptimizationNumerical Methods and OptimizationNumerical Methods for Least Squares ProblemsOptimizationPyomo – Optimization Modeling in PythonNumerical OptimizationIterative Methods for OptimizationJulia Programming for Operations ResearchAn Introduction to OptimizationConstrained Optimization and Lagrange Multiplier MethodsNumerical AlgorithmsAccuracy and Stability of Numerical AlgorithmsFirst-Order Methods in Optimization

Neural network approximation

Abstract: Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing theoretical explanation for this success is still lacking. Since these applications revolve around approximating an unknown function from data observations, part of the answer must involve the ability of NNs to produce accurate approximations. This article surveys the known approximation properties of the outputs of NNs with the aim of uncovering the properties that are not present in the more traditional methods of approximation used in numerical analysis, such as approximations using polynomials, wavelets, rational functions and splines. Comparisons are made with traditional approximation methods from the viewpoint of rate distortion, i.e. error versus the number of parameters used to create the approximant. Another major component in the analysis of numerical approximation is the computational time needed to construct the approximation, and this in turn is intimately connected with the stability of the approximation algorithm. So the stability of numerical approximation using NNs is a large part of the analysis put forward. The survey, for the most part, is concerned with NNs using the popular ReLU activation function. In this case the outputs of the NNs are piecewise linear functions on rather complicated partitions of the domain of f into cells that are convex polytopes. When the architecture of the NN is fixed and the parameters are allowed to vary, the set of output functions of the NN is a parametrized nonlinear manifold. It is shown that this manifold has certain space-filling properties leading to an increased ability to approximate (better rate distortion) but at the expense of numerical stability. The space filling creates the challenge to the numerical method of finding best or good parameter choices when trying to approximate.

Numerical Investigation on the Swimming of Gyrotactic Microorganisms in Nanofluids through Porous Medium over a Stretched Surface

Abstract: In this article, the effects of swimming gyrotactic microorganisms for magnetohydrodynamics nanofluid using Darcy law are investigated. The numerical results of nonlinear coupled mathematical model are obtained by means of Successive Local Linearization Method. This technique is based on a simple notion of the decoupling systems of equations utilizing the linearization of the unknown functions sequentially according to the order of classifying the system of governing equations. The linearized equations, that developed a sequence of linear differential equations along with variable coefficients, were solved by employing the Chebyshev spectral collocation method. The convergence speed of the SLLM technique can be willingly upgraded by successive applying over relaxation method. The comparison of current study with available published literature has been made for the validation of obtained results. It is found that the reported numerical method is in perfect accord with the said similar methods. The results are displayed through tables and graphs.

Solving Fokker-Planck equation using deep learning

Abstract: The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numerical methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-, two-, and three-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Furthermore, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.

Numerical methods for nonlocal and fractional models

Abstract: Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

A Time Delay Dynamical Model for Outbreak of 2019-nCoV and the Parameter Identification

Abstract: In this paper, we propose a novel dynamical system with time delay to describe the outbreak of 2019-nCoV in China. One typical feature of this epidemic is that it can spread in latent period, which is therefore described by the time delay process in the differential equations. The accumulated numbers of classified populations are employed as variables, which is consistent with the official data and facilitates the parameter identification. The numerical methods for the prediction of outbreak of 2019-nCoV and parameter identification are provided, and the numerical results show that the novel dynamic system can well predict the outbreak trend so far. Based on the numerical simulations, we suggest that the transmission of individuals should be greatly controlled with high isolation rate by the government.

A modified invariant subspace method for solving partial differential equations with non-singular kernel fractional derivatives

Abstract: In this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives.

Parametric finite element approximations of curvature-driven interface evolutions

Abstract: Parametric finite elements lead to very efficient numerical methods for surface evolution equations. We introduce several computational techniques for curvature driven evolution equations based on a weak formulation for the mean curvature. The approaches discussed, in contrast to many other methods, have good mesh properties that avoid mesh coalescence and very nonuniform meshes. Mean curvature flow, surface diffusion, anisotropic geometric flows, solidification, two-phase flow, Willmore and Helfrich flow as well as biomembranes are treated. We show stability results as well as results explaining the good mesh properties.