A Simple Solver for the Fractional Laplacian in Multiple Dimensions

Minden, Victor; Ying, Lexing

doi:10.1137/18m1170406

Cited by 49 publications

(34 citation statements)

References 61 publications

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“…Del Teso, Endal and Jakobsen (2018, 2019) extended the aforementioned approach to higher-dimensional problems, and improved the convergence rate by using adapted vanishing viscosity approximation. Minden and Ying (2018) discussed the discretization in two and three dimensions by applying the 'window' function w(z) := w(|z|) such that 1 − w(z) = O(|z| p ) as z → 0 for some positive integer p. Then the following splitting is applied:…”

Section: Quadrature Rule-based Finite Difference Methodsmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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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.

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Section: Quadrature Rule-based Finite Difference Methodsmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

et al. 2020

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“…For forward problems, there have been some works on solving 3D space-fractional ADEs. In [36,37,38,39], only the 3D extension of the Riesz space-fractional derivatives is considered, which differs from the directional fractional Laplacian of our interest; in [40], Riesz fractional Laplacian is first regularized by singularity subtraction and then approximated by using trapezoidal rule. Specifically, in [40], equispaced nodes are required in order to make fast Fourier transform available; however, the method may be difficult to extend to non-equispaced nodes which have to be considered for a BB forcing.…”

Section: Introductionmentioning

confidence: 99%

“…In [36,37,38,39], only the 3D extension of the Riesz space-fractional derivatives is considered, which differs from the directional fractional Laplacian of our interest; in [40], Riesz fractional Laplacian is first regularized by singularity subtraction and then approximated by using trapezoidal rule. Specifically, in [40], equispaced nodes are required in order to make fast Fourier transform available; however, the method may be difficult to extend to non-equispaced nodes which have to be considered for a BB forcing. For inverse problems, little is known about 3D space-time-fractional ADEs with a fractional Laplacian, despite some existing works on 1D and 2D problems [41,42,43,44,45,46].…”

Section: Introductionmentioning

confidence: 99%

fPINNs: Fractional Physics-Informed Neural Networks

Pang¹,

Lu²,

Karniadakis³

2019

SIAM J. Sci. Comput.

663

286

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Physics-informed neural networks (PINNs), introduced in [1], are effective in solving integerorder partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. Here we extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we study systematically their convergence, hence explaining both of fPINNs and PINNs for first time. Specifically, we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. This approach bypasses the difficulties stemming from the fact that automatic differentiation is not applicable to fractional operators because the standard chain rule in integer calculus is not valid in fractional calculus. To discretize the fractional operators, we employ the Grünwald-Letnikov (GL) formula in one-dimensional fractional ADEs and the vector GL formula in conjunction with the directional fractional Laplacian in two-and three-dimensional fractional ADEs. We first consider the one-dimensional fractional Poisson equation and compare the convergence of the fPINNs against the finite difference method (FDM). We present the solution convergence using both the mean L 2 error as well as the standard deviation due to sensitivity to NN parameter initializations. Using different GL formulas we observe first-, second-, and third-order convergence rates for small size of training sets but the error saturates for larger training sets. We explain these results by analyzing the four sources of numerical errors due to discretization, sampling, NN approximation, and optimization. The total error decays monotonically (below 10 −5 for third order GL formula) but it saturates beyond that point due to the optimization error. We also analyze the relative balance between discretization and sampling errors and observe that the sampling size and the number of discretization points (auxiliary points) should be comparable to achieve the highest accuracy. As we increase the depth of the NN up to certain value, the mean error decreases and the standard deviation increases whereas the width has essentially no effect unless its value is either too small or too large. We next consider time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the WB forcing, our results are similar to the aforementioned cases, however,...

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“…The resulting discrete operator they produce is fully dense due to the non local nature of the underlying equations and hence requires O(N 2 ) memory and O(N 2 ) operations for the core matrix vector multiplication operation. For simple cartesian geometries with constant coefficients, translation invariance characteristics can be exploited [26] to reduce the storage and operator application costs. However, the general setting inevitably leads to a quadratic growth in computational resource requirements.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical matrix approximations for space-fractional diffusion equations

Boukaram

Lucchesi

Turkiyyah

et al. 2020

Computer Methods in Applied Mechanics and Engineering

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Space fractional diffusion models generally lead to dense discrete matrix operators, which lead to substantial computational challenges when the system size becomes large. For a state of size N , full representation of a fractional diffusion matrix would require O(N 2) memory storage requirement, with a similar estimate for matrix-vector products. In this work, we present H 2 matrix representation and algorithms that are amenable to efficient implementation on GPUs, and that can reduce the cost of storing these operators to O(N) asymptotically. Matrix-vector multiplications can be performed in asymptotically linear time as well. Performance of the algorithms is assessed in light of 2D simulations of space fractional diffusion equation with constant diffusivity. Attention is focused on smooth particle approximation of the governing equations, which lead to discrete operators involving explicit radial kernels. The algorithms are first tested using the fundamental solution of the unforced space fractional diffusion equation in an unbounded domain, and then for the steady, forced, fractional diffusion equation in a bounded domain. Both matrix-inverse and pseudotransient solution approaches are considered in the latter case. Our experiments show that the construction of the fractional diffusion matrix, the matrix-vector multiplication, and the generation of an approximate inverse pre-conditioner all perform very well on a single GPU on 2D problems with N in the range 10 5-10 6. In addition, the tests also showed that, for the entire range of parameters and fractional orders considered, results obtained using the H 2 approximations were in close agreement with results obtained using dense operators, and exhibited the same spatial order of convergence. Overall, the present experiences showed that the H 2 matrix framework promises to provide practical means to handle large-scale space fractional diffusion models in several space dimensions, at a computational cost that is asymptotically similar to the cost of handling classical diffusion equations.

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A Simple Solver for the Fractional Laplacian in Multiple Dimensions

Cited by 49 publications

References 61 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

fPINNs: Fractional Physics-Informed Neural Networks

Hierarchical matrix approximations for space-fractional diffusion equations

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