Published by De Gruyter February 12, 2022

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

Tianyi Shi , Harbir Antil and Drew P. Kouri

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2021-0118

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Abstract

Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces. However, this nonlocality makes it challenging to design efficient solvers for such problems. In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains. We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs. In addition, we introduce both serial and parallel domain decomposition solvers. We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.

Keywords: Ultraspherical polynomial; Fractional PDEs; Tensor Methods; Domain Decomposition; 3D Problems; Optimal Control

MSC 2010: 49J20; 49K20; 35S15; 65R20; 65N30

Dedicated to our Friend Francisco-Javier Sayas

Funding source: Air Force Office of Scientific Research

Award Identifier / Grant number: FA9550-19-1-0036

Award Identifier / Grant number: F4FGA09135G001

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2110263

Award Identifier / Grant number: DMS-1913004

Funding source: National Nuclear Security Administration

Award Identifier / Grant number: DE-NA-0003525

Funding statement: This work is partially supported by the Air Force Office of Scientific Research under award numbers FA9550-19-1-0036 and F4FGA09135G001 and NSF grants DMS-2110263 and DMS-1913004. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under grant DE-NA-0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

A Visualization of Matrices in Ultraspherical Discretization

In this appendix, we provide the readers with some details of the matrices we use during discretization in Sections 3 and 4.

𝐷 is a diagonal matrix representing second derivative of ( 1 - ρ 2 ) ⁢ C ~ ( 3 / 2 ) ⁢ ( ρ ) , with diagonal elements D j , j = - ( j ⁢ ( j + 3 ) + 2 ) .
𝑀 is a symmetric penta-diagonal matrix with 0 super- and subdiagonals representing multiplication of 1 - ρ 2 in C ~ ( 3 / 2 ) basis, with elements
M j , j = 2 ⁢ ( j + 1 ) ⁢ ( j + 2 ) ( 2 ⁢ j + 1 ) ⁢ ( 2 ⁢ j + 5 ) , M j , j + 1 = 0 , M j , j + 2 = - 1 ( 2 ⁢ j + 3 ) ⁢ ( 2 ⁢ j + 5 ) ⁢ ( j + 4 ) ! ⁢ ( 2 ⁢ j + 3 ) j ! ⁢ ( 2 ⁢ j + 7 ) .
B 1 and B 2 are two banded matrices representing multiplications in different ultraspherical polynomial basis, where the bandwidth is determined by the degree of the polynomial approximation in the respective basis. In addition, B 1 can be shown to be a Toeplitz-plus-Hankel-plus-rank-1 operator [26], and both B 1 and B 2 satisfy three-term recurrence [30, Chapter 6].
D 2 represents second derivative of Chebyshev basis, with the form
D 2 = [ 0 0 2 3 4 ⋱ ] .
Here, elements of D 2 on the diagonal and first upper top diagonal are all 0.
S 2 , 0 converts Chebyshev basis to C ( 2 ) basis, and can be calculated by S 2 , 0 = S 1 ⁢ S 0 , where
S 0 = [ 1 0 - 1 / 2 1 / 2 0 - 1 / 2 1 / 2 0 ⋱ 1 / 2 ⋱ ⋱ ] , S 1 = [ 1 0 - 1 / 3 1 / 2 0 - 1 / 4 1 / 3 0 ⋱ 1 / 4 ⋱ ⋱ ] .

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Received: 2021-06-24

Revised: 2021-12-02

Accepted: 2022-01-04

Published Online: 2022-02-12

Published in Print: 2022-10-01

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

Abstract

A Visualization of Matrices in Ultraspherical Discretization

References

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