Showing papers on "Discretization published in 2020"

Conservative Finite-Difference Methods on General Grids

Abstract: INTRODUCTION Governing Equations Elliptic Equations Heat Equation Equation of Gas Dynamic in Lagrangian Form The Main Ideas of Finite-Difference Algorithms 1-D Case 2-D Case Methods of Solution of Systems of Linear Algebraic Equation Methods of Solution of Systems of Nonlinear Equations METHOD OF SUPPORT-OPERATORS Main Stages The Elliptic Equations Gas Dynamic Equations System of Consistent Difference Operators in 1-D Inner Product in Spaces of Difference Functions and Properties of Difference Operators System of Consistent Difference Operators in 2-D THE ELLIPTIC EQUATIONS Introduction Continuum Elliptic Problems with Dirichlet Boundary Conditions Continuum Elliptic Problems with Robin Boundary Conditions One-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Centered Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Two-Dimensional Support Operator Algorithms Nodal Discretization of Scalar Functions and Cell-Valued Discretization of Vector Functions Cell-Valued Discretization of Scalar Functions and Nodal Discretization of Vector Functions Numerical Solution of Test Problems Conclusion Two-Dimensional Support Operator Algorithms Discretization Spaces of Discrete Functions The Prime Operator The Derived Operator Multiplication by a Matrix and the Operator D The Difference Scheme for the Elliptic Operator The Matrix Problem Approximation and Convergence Properties HEAT EQUATION Introduction Finite-Difference Schemes for Heat Equation in 1-D Finite-Difference Schemes for Heat Equation in 2-D LAGRANGIAN GAS DYNAMICS Kinematics of Fluid Motions Integral Form of Gas Dynamics Equations Integral Equations for One Dimensional Case Differential Equations of Gas Dynamics in Lagrangian Form The Differential Equations in 1D. Lagrange Mass Variables The Statements of Gas Dynamics Problems in Lagrange Variables Different Forms of Energy Equation Acoustic Equations Reference Information Characteristic Form of Gas Dynamics Equations Riemann's Invariants Discontinuous Solutions Conservation Laws and Properties of First Order Invariant Operators Finite-Difference Algorithm in 1D Discretization in 1D Discrete Operators in 1D Semi-Discrete Finite-Difference Scheme in 1D Fully Discrete, Explicit, Computational Algorithm Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Stability Conditions Homogeneous Finite-Difference Schemes. Artificial Viscosity Artificial Viscosity in 1D Numerical Example Finite Difference Algorithm in 2D Discretization in 2D Discrete Operators in 2D Semi-Discrete Finite-Difference Scheme in 2D Stability Conditions Finite-Difference Algorithm in 2D Computational Algorithm-New Time Step-Explicit Finite-Difference Scheme Computational Algorithm-New Time Step-Implicit Finite-Difference Scheme Artificial Viscosity in 2D Numerical Example APPENDIX: FORTRAN CODE DIRECTORY General Description of Structure of Directories on the Disk Programs for Elliptic Equations Programs for 1D Equations Programs for 2D Equations Programs for Heat Equations Programs for 1D Equations Programs for 2D Equations Programs for Gas Dynamics Equations Programs for 1D Equations Programs for 2D Equations Bibliography

Neural Operator: Graph Kernel Network for Partial Differential Equations

Abstract: The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.

Improved Binary Grey Wolf Optimizer and Its application for feature selection

Abstract: Grey Wolf Optimizer (GWO) is a new swarm intelligence algorithm mimicking the behaviours of grey wolves. Its abilities include fast convergence, simplicity and easy realization. It has been proved its superior performance and widely used to optimize the continuous applications, such as, cluster analysis, engineering problem, training neural network and etc. However, there are still some binary problems to optimize in the real world. Since binary can only be taken from values of 0 or 1, the standard GWO is not suitable for the problems of discretization. Binary Grey Wolf Optimizer (BGWO) extends the application of the GWO algorithm and is applied to binary optimization issues. In the position updating equations of BGWO, the a parameter controls the values of A and D , and influences algorithmic exploration and exploitation. This paper analyses the range of values of A D under binary condition and proposes a new updating equation for the a parameter to balance the abilities of global search and local search. Transfer function is an important part of BGWO, which is essential for mapping the continuous value to binary one. This paper includes five transfer functions and focuses on improving their solution quality. Through verifying the benchmark functions, the advanced binary GWO is superior to the original BGWO in the optimality, time consumption and convergence speed. It successfully implements feature selection in the UCI datasets and acquires low classification errors with few features.

Model Reduction and Neural Networks for Parametric PDEs

Abstract: We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers' equation.

Multipole Graph Neural Operator for Parametric Partial Differential Equations

Abstract: One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.

A machine learning framework for solving high-dimensional mean field game and mean field control problems

Abstract: Mean field games (MFG) and mean field control (MFC) are critical classes of multiagent models for the efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse of dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-Bellman (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using a Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that are beyond reach with existing numerical methods.

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

Abstract: This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

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 deep energy method for finite deformation hyperelasticity

Abstract: We present a deep energy method for finite deformation hyperelasticitiy using deep neural networks (DNNs). The method avoids entirely a discretization such as FEM. Instead, the potential energy as a loss function of the system is directly minimized. To train the DNNs, a backpropagation dealing with the gradient loss is computed and then the minimization is performed by a standard optimizer. The learning process will yield the neural network's parameters (weights and biases). Once the network is trained, a numerical solution can be obtained much faster compared to a classical approach based on finite elements for instance. The presented approach is very simple to implement and requires only a few lines of code within the open-source machine learning framework such as Tensorflow or Pytorch. Finally, we demonstrate the performance of our DNNs based solution for several benchmark problems, which shows comparable computational efficiency such as FEM solutions.

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.

A Symplectic Instantaneous Optimal Control for Robot Trajectory Tracking With Differential-Algebraic Equation Models

Abstract: Robot trajectory tracking control based on differential-algebraic equation (DAE) models is still a thorny issue, because the DAEs of such systems are inherently complex and unstable, such as the high-index problem. In this paper, a symplectic instantaneous optimal control (IOC) method for robot trajectory tracking with input saturation based on controlled DAEs is proposed. Based on the discrete variational principle and the canonical transformation, a symplectic discretization form for the controlled DAEs is first constructed. Then, the continuous trajectory tracking problem is approximated for a series of IOC problems at every time step, and the linear complementarity problem (LCP) can be derived for solving the IOC problems. Finally, the control inputs can be obtained by solving the corresponding standard LCP. The proposed method provides a unified framework for solving the trajectory tracking control problems of robot multibody dynamic systems. Numerical simulations and virtual experiments are conducted to verify the robustness and the efficiency of the proposed method, i.e., the input saturation constraints are satisfied at the discrete time points, and high accuracy tracking control results can be obtained at low computational cost.

Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

Abstract: In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

The Random Feature Model for Input-Output Maps between Banach Spaces

Abstract: Well known to the machine learning community, the random feature model, originally introduced by Rahimi and Recht in 2008, is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional, function space viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.

A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations

Abstract: This work aims to propose a new analyzing tool, called the fractional iteration algorithm I for finding numerical solutions of nonlinear time fractional-order Cauchy reaction-diffusion model equations. The key property of the suggested technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The proposed approach can be utilized without the use of any transformation, Adomian polynomials, small perturbation, discretization or linearization. The main feature of the fractional iteration algorithm-I is the improvement of an auxiliary parameter that can ensure a rapid convergence. To check the stability, accuracy and speed of the method, obtained results are compared numerically and graphically with the exact solutions and results available in the latest literature. In addition, numerical results are displayed graphically for various cases of the fractional-order α . These results demonstrate the viability of the proposed technique and show that this technique is exceptionally powerful and suitable for solving fractional PDEs.

Newton versus the machine: solving the chaotic three-body problem using deep neural networks

Abstract: Since its formulation by Sir Isaac Newton, the problem of solving the equations of motion for three bodies under their own gravitational force has remained practically unsolved. Currently, the solution for a given initialization can only be found by performing laborious iterative calculations that have unpredictable and potentially infinite computational cost, due to the system’s chaotic nature. We show that an ensemble of converged solutions for the planar chaotic three-body problem obtained using an arbitrarily precise numerical integrator can be used to train a deep artificial neural network (ANN) that, over a bounded time interval, provides accurate solutions at a fixed computational cost and up to 100 million times faster than the numerical integrator. In addition, we demonstrate the importance of training an ANN using converged solutions from an arbitrary precise integrator, relative to solutions computed by a conventional fixed precision integrator, which can introduce errors in the training data, due to numerical round-off and time discretization, that are learned by the ANN. Our results provide evidence that, for computationally challenging regions of phase space, a trained ANN can replace existing numerical solvers, enabling fast and scalable simulations of many-body systems to shed light on outstanding phenomena such as the formation of black hole binary systems or the origin of the core collapse in dense star clusters.

First-order optimization algorithms via inertial systems with Hessian driven damping

Abstract: In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form $$ abla ^2 f (x(t)) \dot{x} (t)$$ . By treating this term as the time derivative of $$ abla f (x (t)) $$ , this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings.

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Abstract: Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Numerical solution of fractional elliptic stochastic PDEs with spatial white noise

Abstract: The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R-d is considered. The differential operator is given by the fractional power L-beta, beta is an element of (0, 1) of an integer-order elliptic differential operator L and is therefore nonlocal. Its inverse L-beta is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator L-beta is approximated by a weighted sum of nonfractional resolvents (I + exp(2yl)L)(-1) at certain quadrature nodes t(j) > 0. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = kappa(2) - Delta, kappa > 0 with homogeneous Dirichlet boundary conditions on the unit cube (0, 1)(d) in d = 1, 2, 3 spatial dimensions for varying beta is an element of (0, 1) attest to the theoretical results.

Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

Abstract: In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order $$\alpha \in (0,1)$$ . First, the well-posedness and (limited) smoothing property are studied, by using the maximal $$L^p$$ regularity of fractional evolution equations and the fractional Gronwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Gronwall’s inequality and maximal $$\ell ^p$$ regularity, we prove that the convergence rates of those time-stepping schemes are $$O(\tau ^\alpha )$$ without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.

A novel Chebyshev-wavelet-based approach for accurate and fast prediction of milling stability

Abstract: Currently, semi-analytical stability analysis methods for milling processes focus on improving prediction accuracy and simultaneously reducing computing time. This paper presents a Chebyshev-wavelet-based method for improved milling stability prediction. When including regenerative effect, the milling dynamics model can be concluded as periodic delay differential equations, and is re-presented as state equation forms via matrix transformation. After divide the period of the coefficient matrix into two subintervals, the forced vibration time interval is mapped equivalently to the definition interval of the second kind Chebyshev wavelets. Thereafter, the explicit Chebyshev–Gauss–Lobatto points are utilized for discretization. To construct the Floquet transition matrix, the state term is approximated by finite series second kind Chebyshev wavelets, while its derivative is acquired with a simple and explicit operational matrix of derivative. Finally, the milling stability can be semi-analytically predicted using Floquet theory. The effectiveness and superiority of the presented approach are verified by two benchmark milling models and comparisons with the representative existing methods. The results demonstrate that the presented approach is highly accurate, fast and easy to implement. Meanwhile, it is shown that the presented approach achieves high stability prediction accuracy and efficiency for both large and low radial-immersion milling operations.

A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method

Abstract: This paper deals with the numerical solutions and convergence analysis for general singular Lane–Emden type models of fractional order, with appropriate constraint initial conditions. A modified reproducing kernel discretization technique is used for dealing with the fractional Atangana–Baleanu–Caputo operator. In this tendency, novel operational algorithms are built and discussed for covering such singular models in spite of the operator optimality used. Several numerical applications using the well-known fractional Lane–Emden type models are examined, to expound the feasibility and suitability of the approach. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features stability for dealing with many fractional models emerging in physics and mathematics, using the new presented derivative.