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Showing papers on "Numerical analysis published in 2017"


Journal ArticleDOI
15 Jun 2017
TL;DR: In this article, a new algorithm for solving parabolic partial differential equations and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of BSDE.
Abstract: We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation, the Hamilton–Jacobi–Bellman equation, and a nonlinear pricing model for financial derivatives.

408 citations


Journal ArticleDOI
TL;DR: In this paper, a micromechanical model based on the agglomeration of these nanoparticles is considered, where the strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.
Abstract: By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix, CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

199 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented a novel numerical method to simulate crack growth in 3D, directly from the Computer-Aided Design (CAD) geometry of the component, without any mesh generation.

172 citations


Journal ArticleDOI
TL;DR: This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques.
Abstract: Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

170 citations


Journal ArticleDOI
TL;DR: This study investigates the desirability of applying a truncated Newton method to FWI and suggests that the inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves.
Abstract: Full waveform inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing the distance between predicted and recorded data. The predicted data are computed as the solution of a wave-propagation problem. Conventional numerical methods for the solution of FWI problems are gradient-based methods, such as the preconditioned steepest descent, the nonlinear conjugate gradient, or more recently the $l$-BFGS quasi-Newton algorithm. In this study, we investigate the desirability of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves. For multiparameter reconstruction, the inverse Hessian operator also offers the possibility of better removing trade-offs due to coupling effects between parameter classes. The truncated Newto...

138 citations


Posted Content
TL;DR: In this paper, a linear multi-step architecture (LM-architecture) is proposed for deep neural networks, which is inspired by the linear mult-step method solving ordinary differential equations.
Abstract: In our work, we bridge deep neural network design with numerical differential equations. We show that many effective networks, such as ResNet, PolyNet, FractalNet and RevNet, can be interpreted as different numerical discretizations of differential equations. This finding brings us a brand new perspective on the design of effective deep architectures. We can take advantage of the rich knowledge in numerical analysis to guide us in designing new and potentially more effective deep networks. As an example, we propose a linear multi-step architecture (LM-architecture) which is inspired by the linear multi-step method solving ordinary differential equations. The LM-architecture is an effective structure that can be used on any ResNet-like networks. In particular, we demonstrate that LM-ResNet and LM-ResNeXt (i.e. the networks obtained by applying the LM-architecture on ResNet and ResNeXt respectively) can achieve noticeably higher accuracy than ResNet and ResNeXt on both CIFAR and ImageNet with comparable numbers of trainable parameters. In particular, on both CIFAR and ImageNet, LM-ResNet/LM-ResNeXt can significantly compress ($>50$\%) the original networks while maintaining a similar performance. This can be explained mathematically using the concept of modified equation from numerical analysis. Last but not least, we also establish a connection between stochastic control and noise injection in the training process which helps to improve generalization of the networks. Furthermore, by relating stochastic training strategy with stochastic dynamic system, we can easily apply stochastic training to the networks with the LM-architecture. As an example, we introduced stochastic depth to LM-ResNet and achieve significant improvement over the original LM-ResNet on CIFAR10.

125 citations


Journal ArticleDOI
TL;DR: In this article, the bending, buckling and buckling of embedded nano-sandwich plates are investigated based on refined zigzag theory (RZT), sinusoidal shear deformation theory (SSDT), first order shear deformability theory (FSDT), and classical plate theory (CPT).

124 citations


Journal ArticleDOI
TL;DR: In this paper, a second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities was presented, which combines a standard second-order Crank-Nicolson method for the Navier-stokes equations and a modification to the Crank Nicolson algorithm for the cahn-hilliard equation.
Abstract: In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. The scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in $$\ell ^\infty \left( 0,T;L^\infty \right) $$ and the discrete chemical potential bounded in $$\ell ^\infty \left( 0,T;L^2\right) $$ , for any time and space step sizes, in two and three dimensions, and for any finite final time T. We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.

123 citations


Journal ArticleDOI
TL;DR: This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients, and upper bound for the error of operational matrix of the fractional integration is given.
Abstract: In this research, a Bernoulli wavelet operational matrix of fractional integration is presented Bernoulli wavelets and their properties are employed for deriving a general procedure for forming this matrix The application of the proposed operational matrix for solving the fractional delay differential equations is explained Also, upper bound for the error of operational matrix of the fractional integration is given This operational matrix is utilized to transform the problem to a set of algebraic equations with unknown Bernoulli wavelet coefficients Several numerical examples are solved to demonstrate the validity and applicability of the presented technique

122 citations


Journal ArticleDOI
TL;DR: In this article, a modified weighted shifted Grunwald-Letnikov (WSGL) formula was proposed to solve multi-term fractional ordinary and partial differential equations, and the linear stability and second-order convergence for both smooth and non-smooth solutions when the regularity of the solutions is known.

119 citations


Book
13 Dec 2017
TL;DR: This book discusses differential Equations, a large-scale version of single-step computer programming, and some of the techniques used to develop it.
Abstract: Differential Equations Classification of Differential Equations Linear Equations Non-Linear Equations Existence and Uniqueness of Solutions Numerical Methods Computer Programming First Ideas and Single-Step Methods Analytical and Numerical Solutions A First Example The Taylor Series Method Runge-Kutta Methods Second and Higher Order Equations Error Considerations Definitions Local Truncation Error for the Taylor Series Method Local Truncation Error for the Runge-Kutta Method Local Truncation and Global Errors Local Error and LTE Runge-Kutta Methods Error Criteria A Third Order Formula Fourth Order Formulae Fifth and Higher Order Formulae Rationale for Higher Order Formulae Computational Examples Step-Size Control Steplength Prediction Error Estimation Local Extrapolation Error Estimation with RK Methods More Runge-Kutta Pairs Application of RK Embedding Dense Output Construction of Continuous Extensions Choice of Free Parameters Higher-Order Formulae Computational Aspects of Dense Output Inverse Interpolation Stability and Stiffness Absolute Stability Non-Linear Stability Stiffness Improving the Stability of RK Methods Multistep Methods The Linear Multistep Process Selection of Parameters A Third Order Implicit Formula A Third Order Explicit Formula Predictor-Corrector Schemes Error Estimation A Predictor-Corrector Program Multistep Formulae from Quadrature Quadrature Applied to Differential Equations The Adams-Bashforth Formulae The Adams-Moulton Formulae Other Multistep Formulae Varying the Step Size Numerical Results Stability of Multistep Methods Some Numerical Experiments Zero-Stability Weak Stability Theory Stability Properties of Some Formulae Stability of Predictor-Corrector Pairs Methods for Stiff Systems Differentiation Formulae Implementation of BDF Schemes A BDF Program Implicit Runge-Kutta Methods A Semi-Implicit RK Program Variable Coefficient Multistep Methods Variable Coefficient Integrators Practical Implementation Step-Size Estimation A Modified Approach An Application of STEP90 Global Error Estimation Classical Extrapolation Solving for the Correction An Example of Classical Extrapolation The Correction Technique Global Embedding A Global Embedding Program Second Order Equations Transformation of the RK Process A Direct Approach to the RKNG Processes The Special Second Order Problem Dense Output for RKN Methods Multistep Methods Partial Differential Equations Finite Differences Semi-Discretization of the Heat Equation Highly Stable Explicit Schemes Equations with Two Space Dimensions Non-Linear Equations Hyperbolic Equations Appendix A: Programs for Single Step Methods A Variable Step Taylor Method An Embedded Runge-Kutta Program A Sample RK Data File An Alternative Runge-Kutta Scheme Runge-Kutta with Dense Output A Sample Continuous RK Data File Appendix B: Multistep Programs A Constant Steplength Program A Variable Step Adams PC Scheme A Variable Coefficient Multistep Package Appendix C: Programs for Stiff Systems A BDF Program A Diagonally Implicit RK Program Appendix D: Global Embedding Programs The Gem Global Embedding Code The GEM90 Package with Global Embedding A Driver Program for GEM90 Appendix E: A Runge-Kutta Nystroem Program Bibliography Index Each chapter also includes an introduction and a section of exercise problems.

Journal ArticleDOI
TL;DR: In this article, numerical methods for solving the fractional-in-space Allen-Cahn equation with small perturbation parameters and strong nonlinearity were considered, and the numerical solutions satisfy discrete maximum principle under reasonable time step constraint.
Abstract: We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only $$\mathcal {O}(N\log N)$$ computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.

Journal ArticleDOI
TL;DR: The first numerical computation of two-loop amplitudes based on the unitarity method is presented, and the four-gluon process in the leading-color approximation is computed.
Abstract: We present the first numerical computation of two-loop amplitudes based on the unitarity method. As a proof of principle, we compute the four-gluon process in the leading-color approximation. We discuss the new method, analyze its numerical properties, and apply it to reconstruct the analytic form of the amplitudes. The numerical method is universal, and can be automated to provide multiscale two-loop computations for phenomenologically relevant signatures at hadron colliders.

Journal ArticleDOI
TL;DR: By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, it is proved the fully discrete system is uniquely solvable.
Abstract: In this paper, a class of nonlinear Riesz space-fractional Schrodinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.

Journal ArticleDOI
TL;DR: In this article, the generalized finite difference method (GFDM) is applied to the heat source recovery problem in steady-state heat conduction problems, and the authors show that the proposed algorithm is accurate, computationally efficient and numerically stable for numerical solution of inverse heat source problems.

Journal ArticleDOI
TL;DR: In this article, the effect of Brownian motion and thermophoresis phenomenon and Lewis number on MHD nanofluid flow along with the heat transfer between two parallel plates was examined.

Journal ArticleDOI
TL;DR: A new arbitrary high order accurate semi-implicit spacetime discontinuous Galerkin (DG) method for the solution of the two and three dimensional compressible Euler and NavierStokes equations on staggered unstructured curved meshes is proposed and is able to deal with all Mach number flows.

Journal ArticleDOI
TL;DR: In this paper, the numerical solutions of conformable fractional-order linear and nonlinear equations are obtained by employing the constructed conformable Adomian decomposition method (CADM).
Abstract: In this paper, the numerical solutions of conformable fractional-order linear and nonlinear equations are obtained by employing the constructed conformable Adomian decomposition method (CADM). We found that CADM is an effective method for numerical solution of conformable fractional-order differential equations. Taking the conformable fractional-order simplified Lorenz system as an example, the numerical solution and chaotic behaviors of the conformable fractional-order simplified Lorenz system are investigated. It is found that rich dynamics exist in the conformable fractional-order simplified Lorenz system, and the minimum order for chaos is even less than 2. The results are validated by means of bifurcation diagram, Lyapunov characteristic exponents and phase portraits.

Journal ArticleDOI
TL;DR: In this paper, the steady boundary layer flow and heat transfer properties of a thin film second-grade fluid through a porous medium past a stretching sheet concerning the effect of viscous dissipation were investigated.
Abstract: This article inquires into the steady boundary layer flow and heat transfer properties of a thin film second-grade fluid through a porous medium past a stretching sheet concerning the effect of viscous dissipation. The aim of the study is to discuss the impacts of film thickness and porosity in the presence of constant reference temperature which completely affect the flow pattern and bring changes in the cooling/heating. The basic governing equations of the problem have been modeled in terms of suitable similarity transformations which result in nonlinear ordinary differential equations with physical conditions. Solution has been obtained by using HAM (Homotopy Analysis Method) which is frequently used for solving nonlinear differential equations encountered in various applied sciences and is found quite useful. Favorable comparison with previously published research papers is performed to show the correlations for the present work. Skin friction coefficient and Nusselt number are presented through tables which describe the verification for the achieved results showing that the thin liquid film results from this study are in close agreement with the results reported in the literature. The physical influences of all the emerging parameters on velocity and temperature fields have been studied graphically and illustrated clearly. The authentication of the present work has been achieved by evaluating the comparison of HAM solution with the numerical method solution. Results achieved by HAM and residual errors are also discussed numerically and graphically.

Journal ArticleDOI
TL;DR: In this paper, a variational differential quadrature (VDQ) method is proposed to discretize the energy functional in the structural mechanics, which is based on the accurate and direct discretization of energy functional.

Journal ArticleDOI
TL;DR: The schemes are based on the second order Crank–Nicolson method for time discretization, projection method for Navier–Stokes equations, as well as several implicit–explicit treatments for phase field equations.
Abstract: In this paper, we propose several second order in time, fully discrete, linear and nonlinear numerical schemes for solving the phase field model of two-phase incompressible flows, in the framework of finite element method. The schemes are based on the second order Crank---Nicolson method for time discretization, projection method for Navier---Stokes equations, as well as several implicit---explicit treatments for phase field equations. The energy stability and unique solvability of the proposed schemes are proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes.

Journal ArticleDOI
TL;DR: A novel finite volume method (FVM) for a distributed-order space-fractional diffusion equation (FDE) is proposed and it is proved that the Crank–Nicolson scheme with FVM is unconditionally stable and convergent with second order accuracy in both time and space.
Abstract: In recent years, considerable attention has been devoted to distributed-order differential equations mainly because they appear to be more effective for modelling complex processes which obey a mixture of power laws or flexible variations in space In this paper, we propose a novel finite volume method (FVM) for a distributed-order space-fractional diffusion equation (FDE) Firstly, we use the mid-point quadrature rule to transform the space distributed-order diffusion equation into a multi-term fractional equation Secondly, the transformed multi-term fractional equation is solved by discretising in space using the finite volume method and then in time using the Crank–Nicolson scheme Thirdly, we prove that the Crank–Nicolson scheme with FVM is unconditionally stable and convergent with second order accuracy in both time and space Finally, two numerical examples are presented to show the effectiveness of the numerical method These methods and techniques can also be used to solve other types of fractional partial differential equations

Journal ArticleDOI
TL;DR: Weak Formulation Isogeometric Analysis (WFIGA) as mentioned in this paper was proposed to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature.

Journal ArticleDOI
TL;DR: A class ofumerical homogenization methods that are very closely related to the method of M{\aa}lqvist and Peterseim, but do not make explicit or implicit use of a scale separation.
Abstract: Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of M{\aa}lqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of M{\aa}lqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.

Journal ArticleDOI
TL;DR: The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator to convert the differential equation for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.
Abstract: In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

Journal ArticleDOI
TL;DR: The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method on the distributed order time-fractional diffusion-wave equation and propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques.
Abstract: In the current decade, the meshless methods have been developed for solving partial differential equations. The meshless methods may be classified in two basic parts: 1.The meshless methods based on the strong form2.The meshless methods based on the weak form The element-free Galerkin (EFG) method is a meshless method based on the global weak form. The test and trial functions in element-free Galerkin are shape functions of moving least squares (MLS) approximation. Also, the traditional MLS shape functions have not the ź-Kronecker property. Recently, a new class of MLS shape functions has been presented. These are well-known as the interpolating MLS (IMLS) shape functions. The IMLS shape functions have the ź-Kronecker property; thus the essential boundary conditions can be applied directly. The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method. To this end, we apply the mentioned technique on the distributed order time-fractional diffusion-wave equation. For comparing the numerical results, we propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques. Also, we investigate the uniqueness, existence and stability analysis of the new schemes and we obtain an error estimate for the full-discrete schemes. The time-fractional derivative has been described in Caputo's sense. Numerical examples demonstrate the theoretical results and the efficiency of the proposed schemes.

Journal ArticleDOI
TL;DR: In this paper, the capability of using a higher order dynamic mode decomposition (HODMD) algorithm both to identify flow patterns and to extrapolate a transient solution to the attractor region was shown.
Abstract: This article shows the capability of using a higher order dynamic mode decomposition (HODMD) algorithm both to identify flow patterns and to extrapolate a transient solution to the attractor region. Numerical simulations are carried out for the three-dimensional flow around a circular cylinder, and both standard dynamic mode decomposition (DMD) and higher order DMD are applied to the non-converged solution. The good performance of HODMD is proved, showing that this method guesses the converged flow patterns from numerical simulations in the transitional region. The solution obtained can be extrapolated to the attractor region. This fact sheds light on the capability of finding real flow patterns in complex flows and, simultaneously, reducing the computational cost of the numerical simulations or the required quantity of data collected in experiments.

Journal ArticleDOI
TL;DR: The unconditional stability and the convergence estimate of the new scheme have been concluded, and results of Galerkin FEM are evaluated with other numerical methods.
Abstract: Our main aim in the current paper is to find a numerical plan for 2D Rayleigh–Stokes model with fractional derivative on irregular domains such as circular, L-shaped and a unit square with a circular and square hole. The employed fractional derivative is the Riemann–Liouville sense. Also, by integrating the equation corresponding to the time variable and then using the Galerkin FEM for the space direction, we obtain a full discrete scheme. The unconditional stability and the convergence estimate of the new scheme have been concluded. Finally, we evaluate results of Galerkin FEM with other numerical methods.

Journal ArticleDOI
TL;DR: In this article, a self-learning Monte Carlo (SLMC) method is proposed to generate new candidate configurations in the Markov chain based on the self-learned bosonic effective model.
Abstract: We develop the self-learning Monte Carlo (SLMC) method, a general-purpose numerical method recently introduced to simulate many-body systems, for studying interacting fermion systems. Our method uses a highly efficient update algorithm, which we design and dub ``cumulative update'', to generate new candidate configurations in the Markov chain based on a self-learned bosonic effective model. From a general analysis and a numerical study of the double exchange model as an example, we find that the SLMC with cumulative update drastically reduces the computational cost of the simulation, while remaining statistically exact. Remarkably, its computational complexity is far less than the conventional algorithm with local updates.

Journal ArticleDOI
TL;DR: This paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method.
Abstract: This paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method. For a general family of elliptic...