Number Of Spatial Grid Points Research Articles

Complex-valued adaptive-coefficient finite-difference frequency-domain method for wavefield modeling based on the diffusive-viscous wave equation

The diffusive-viscous wave (DVW) equation is an effective model for analyzing seismic low-frequency anomalies and attenuation in porous media. To effectively simulate DVW wavefields, the finite-difference or finite-element method in the time domain is favored, but the time-domain approach proves less efficient with multiple shots or a few frequency components. The finite-difference frequency-domain (FDFD) method featuring optimal or adaptive coefficients is favored in seismic simulations due to its high efficiency. Initially, we develop a real-valued adaptive-coefficient (RVAC) FDFD method for the DVW equation, which ignores the numerical attenuation error and is a generalization of the acoustic adaptive-coefficient FDFD method. To reduce the numerical attenuation error of the RVAC FDFD method, we introduce a complex-valued adaptive-coefficient (CVAC) FDFD method for the DVW equation. The CVAC FDFD method is constructed by incorporating correction terms into the conventional second-order FDFD method. The adaptive coefficients are related to the spatial sampling ratio, number of spatial grid points per wavelength, and diffusive and viscous attenuation coefficients in the DVW equation. Numerical dispersion and attenuation analysis confirm that, with a maximum dispersion error of 1% and a maximum attenuation error of 10%, the CVAC FDFD method only necessitates 2.5 spatial grid points per wavelength. Compared with the RVAC FDFD method, the CVAC FDFD method exhibits enhanced capability in suppressing the numerical attenuation during anelastic wavefield modeling. To validate the accuracy of our method, we develop an analytical solution for the DVW equation in a homogeneous medium. Three numerical examples substantiate the high accuracy of the CVAC FDFD method when using a small number of spatial grid points per wavelength, and this method demands computational time and computer memory similar to those required by the conventional second-order FDFD method. A fluid-saturated model featuring various layer thicknesses is used to characterize the propagation characteristics of DVW.

Exponential-sum-approximation technique for variable-order time-fractional diffusion equations

In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function \(\alpha (t)\in (0,1)\), we develop an exponential-sum-approximation (ESA) technique to approach the VO Caputo fractional derivative. The ESA technique keeps both the quadrature exponents and the number of exponentials in the summation unchanged at different time level. Approximating parameters are properly selected to achieve the efficient accuracy. Compared with the general direct method, the proposed method reduces the storage requirement from \({\mathcal {O}}(n)\) to \({\mathcal {O}}(\log ^2 n)\) and the computational cost from \({\mathcal {O}}(n^2)\) to \(\mathcal {O}(n\log ^2 n)\), respectively, with n being the number of the time levels. When this fast algorithm is exploited to construct a fast ESA scheme for the VO time-fractional diffusion equations, the computational complexity of the proposed scheme is only of \({\mathcal {O}}(mn\log ^2 n)\) with \({\mathcal {O}}(m\log ^2n)\) storage requirement, where m denotes the number of spatial grid points. Theoretically, the unconditional stability and error analysis of the fast ESA scheme are given. The effectiveness of the proposed algorithm is verified by numerical examples.

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation

This work deals with the efficient numerical solution of the time-fractional heat equation discretized on non-uniform temporal meshes. Non-uniform grids are essential to capture the singularities of “typical” solutions of time-fractional problems. We propose an efficient space-time multigrid method based on the waveform relaxation technique, which accounts for the nonlocal character of the fractional differential operator. To maintain an optimal complexity, which can be obtained for the case of uniform grids, we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix (H-matrix) representation. In particular, the proposed method has a computational cost of O(kNMlog⁡(M)), where M is the number of time steps, N is the number of spatial grid points, and k is a parameter which controls the accuracy of the H-matrix approximation. The efficiency and the good convergence of the algorithm, which can be theoretically justified by a semi-algebraic mode analysis, are demonstrated through numerical experiments in both one- and two-dimensional spaces.

Estimation of Spatial Concentration Profiles in a Plug Flow Reactor using Reduced Dimensional Models

Efficient monitoring and control of a plug flow reactor system requires knowledge of temporal variations of spatial profiles of concentrations. However, estimating the state profiles in spatial domain requires discretization of the governing PDEs at a large number of spatial grid points, which results in high order DAE systems. Computational cost associated with solving these DAEs makes them unattractive candidates for development of advanced on-line monitoring and control schemes. In this work, a novel method for reconstructing spatial profiles of state variables is proposed, which exploits the fact that orthogonal collocation based discretization of PDEs employs time-dependent interpolation polynomials as approximate solutions. A reduced-order approximation of PDE system is developed using Lagrange interpolation polynomials and further used to develop a reduced dimensional extended Kalman filter (EKF). States estimated using the reduced-order EKF, in combination with Lagrange polynomials, are used to construct estimates of spatial profiles of state variables. The proposed approach facilitates handling of measurements available from sensors placed at arbitrary spatial locations other than the collocation points. The performance of proposed reduced-order observer is evaluated via simulation studies performed on a benchmark counter-current plug flow reactor system. Analysis of the simulation results reveals that the proposed approach is capable of generating state profile estimates that are comparable to profile estimates obtained using a high dimensional observer with a significant reduction in the computational cost.

Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained.

Numerical Solution of the Time Dependent 3D Schrödinger Equation Describing Tunneling of Atoms from Anharmonic Traps

We present an efficient numerical method for the integration of the 3D Schrödinger equation. A tunneling problem of two interacting bosonic atoms confined in a 1D anharmonic trap has been successfully solved by means of this method. We demonstrate fast convergence of the final results with respect to spatial and temporal grid steps. The computational scheme is based on the operator-splitting technique with the implicit Crank-Nicolson algorithm on spatial sixth-order finite-differences. The computational time is proportional to the number of spatial grid points.

Fast and spectrally accurate evaluation of gyroaverages in non-periodic gyrokinetic-Poisson simulations

We present a fast and spectrally accurate numerical scheme for the evaluation of the gyroaveraged electrostatic potential in non-periodic gyrokinetic-Poisson simulations. Our method relies on a reformulation of the gyrokinetic-Poisson system in which the gyroaverage in Poisson’s equation is computed for the compactly supported charge density instead of the non-periodic, non-compactly supported potential itself. We calculate this gyroaverage with a combination of two Fourier transforms and a Hankel transform, which has the near optimal run-time complexity$O(N_{\unicode[STIX]{x1D70C}}(P+\hat{P})\log (P+\hat{P}))$, where$P$is the number of spatial grid points,$\hat{P}$the number of grid points in Fourier space and$N_{\unicode[STIX]{x1D70C}}$the number of grid points in velocity space. We present numerical examples illustrating the performance of our code and demonstrating geometric convergence of the error.

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

In this article, we study fast discontinuous Galerkin finite element methods to solve a space‐time fractional diffusion‐wave equation. We introduce a piecewise‐constant discontinuous finite element method for solving this problem and derive optimal error estimates. Importantly, a fast solution technique to accelerate Toeplitz matrix‐vector multiplications which arise from discontinuous Galerkin finite element discretization is developed. This fast solution technique is based on fast Fourier transform and it depends on the special structure of coefficient matrices. In each temporal step, it helps to reduce the computational work from required by the traditional methods to log , where is the size of the coefficient matrices (number of spatial grid points). Moreover, the applicability and accuracy of the method are verified by numerical experiments including both continuous and discontinuous examples to support our theoretical analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2043–2061, 2017

A fast solution technique for finite element discretization of the space–time fractional diffusion equation

In this paper, we study fast Galerkin finite element methods to solve a space–time fractional diffusion equation. We develop an optimal piecewise-linear and piecewise-quadratic finite element methods for solving this problem and give optimal error estimates. Furthermore, we develop piecewise-constant discontinuous finite element method for discontinuous problem of this model. Importantly, a fast solution technique to accelerate non-square Toeplitz matrix–vector multiplications which arise from both continuous and discontinuous Galerkin finite element discretization respectively is considered. This fast solution technique is based on fast Fourier transform and depends on the special structure of coefficient matrices and it helps to reduce the computational work from O(N3) required by the traditional methods to O(Nlog2⁡N), where N is the size (number of spatial grid points) of the coefficient matrices for every time step. Moreover, the applicability and accuracy of the method are demonstrated by numerical experiments to support our theoretical analysis.

Multigrid Waveform Relaxation for the Time-Fractional Heat Equation

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(NM\log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semialgebraic mode analysis is also developed to theoretically confirm t...

Fast algorithm for periodic density fitting for Bloch waves

We propose an efficient algorithm for density fitting of Bloch waves for Hamiltonian operators with periodic potential. The algorithm is based on column selection and random Fourier projection of the orbital functions. The computational cost of the algorithm scales as O i NgridN 2 ¯ NgridN K log(N K ) ¢ , where Ngrid is number of spatial grid points, K is the number of sampling k- points in first Brillouin zone, and N is the number of bands under consideration. We validate the algorithm by numerical examples in both two and three dimensions.

CUDA programs for solving the time-dependent dipolar Gross–Pitaevskii equation in an anisotropic trap

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross-Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank-Nicolson method as in the previous version (R. Kishor Kumar et al., Comput. Phys. Commun. 195, 117 (2015)), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12 to 25, depending on the program and input size.

A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation

This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

In this paper, we study the block lower triangular Toeplitz-like with tri-diagonal blocks system which arises from the time-fractional partial differential equation. Existing fast numerical solver (e.g., fast approximate inversion method) cannot handle such linear system as the main diagonal blocks are different. The main contribution of this paper is to propose a fast direct method for solving this linear system, and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system. Our idea is based on the divide-and-conquer strategy and together with the fast Fourier transforms for calculating Toeplitz matrix–vector multiplication. The complexity needs O(MNlog2⁡M) arithmetic operations, where M is the number of blocks (the number of time steps) in the system and N is the size (number of spatial grid points) of each block. Numerical examples from the finite difference discretization of time-fractional partial differential equations are also given to demonstrate the efficiency of the proposed method.

Compression of the electron repulsion integral tensor in tensor hypercontraction format with cubic scaling cost

Electron repulsion integral tensor has ubiquitous applications in electronic structure computations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with O(nN2log⁡N) computational cost, where N is the number of orbital functions and n is the number of spatial grid points that the discretization of each orbital function has. The algorithm is based on a novel strategy of density fitting using a selection of a subset of spatial grid points to approximate the pair products of orbital functions on the whole domain.

Optimized spatial matrix representations of quantum Hamiltonians

We examine the accuracy of several approaches to represent the quantum mechanical Schrödinger, Klein–Gordon and Dirac Hamilton operators by optimized spatial matrices. Two of the approaches are based on periodic and reflecting boundaries and have an error scaling with the number of spatial grid points that is significantly better than the ones based on the usual approaches where the momentum operator is approximated by finite-difference schemes. These N×N matrices are optimum in the sense that their eigenvalues and eigenvectors are exact representations on the spatial grid for the continuous solutions of the corresponding force-free Hamiltonian. As an example, we apply these techniques to compute the vacuum’s polarization charge density from the Dirac and Foldy–Wouthuysen theory.

A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh

Numerical methods for fractional differential equations generate full stiffness matrices, which were traditionally solved via Gaussian type direct solvers that require O(N3) of computational work and O(N2) of memory to store where N is the number of spatial grid points in the discretization. We develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite volume schemes defined on a locally refined composite mesh for fractional differential equations to resolve boundary layers of the solutions. Numerical results are presented to show the utility of the method.

Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions

Numerical methods for space-fractional diffusion equations often generate dense or even full stiffness matrices. Traditionally, these methods were solved via Gaussian type direct solvers, which requires O(N3) of computational work per time step and O(N2) of memory to store where N is the number of spatial grid points in the discretization.In this paper we develop a preconditioned fast Krylov subspace iterative method for the efficient and faithful solution of finite difference methods (both steady-state and time-dependent) space-fractional diffusion equations with fractional derivative boundary conditions in one space dimension. The method requires O(N) of memory and O(Nlog⁡N) of operations per iteration. Due to the application of effective preconditioners, significantly reduced numbers of iterations were achieved that further reduces the computational cost of the fast method. Numerical results are presented to show the utility of the method.

Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires computational work of O(N3) per time step and O(N2) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-, y-, z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a computational work count of O(NlogN) per iteration at each time step and a memory requirement of O(N). We also develop a fast multistep ADI finite difference method, which has a computational work count of O(Nlog2N) per time step and a memory requirement of O(NlogN). Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved computational cost and memory requirement. These numerical experiments show the utility of the fast method.