Geodesic
Low-rank solvers for fractional differential equations
Electronic transactions on numerical analysis, Tome 45 (2016), pp. 107-132
Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology, fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz structure in the constant coefficient case. We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniques can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.
Classification : 65F08, 65F10, 65F50, 92E20, 93C20
Keywords: fractional calculus, fast solvers, Sylvester equations, preconditioning, low-rank methods, tensor equations 
@article{ETNA_2016__45__a20,
     author = {Breiten,  Tobias and Simoncini,  Valeria and Stoll,  Martin},
     title = {Low-rank solvers for fractional differential equations},
     journal = {Electronic transactions on numerical analysis},
     pages = {107--132},
     year = {2016},
     volume = {45},
     zbl = {1338.65071},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2016__45__a20/}
}
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AU  - Simoncini,  Valeria
AU  - Stoll,  Martin
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EP  - 132
VL  - 45
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%A Simoncini,  Valeria
%A Stoll,  Martin
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%J Electronic transactions on numerical analysis
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%P 107-132
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%U http://geodesic.mathdoc.fr/item/ETNA_2016__45__a20/
%G en
%F ETNA_2016__45__a20
Breiten,  Tobias; Simoncini,  Valeria; Stoll,  Martin. Low-rank solvers for fractional differential equations. Electronic transactions on numerical analysis, Tome 45 (2016), pp. 107-132. http://geodesic.mathdoc.fr/item/ETNA_2016__45__a20/