Numerical solution to a linear equation with tensor product structure |
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| Authors: | Hung‐Yuan Fan Liping Zhang Eric King‐wah Chu Yimin Wei |
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| Affiliation: | 1. Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan;2. Department of Mathematics, Zhejiang University of Technology, Hangzhou, China;3. School of Mathematical Sciences, Monash University, Victoria, Melbourne, Australia;4. Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, China |
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| Abstract: | We consider the numerical solution of a c‐stable linear equation in the tensor product space  , arising from a discretized elliptic partial differential equation in  . Utilizing the stability, we produce an equivalent d‐stable generalized Stein‐like equation, which can be solved iteratively. For large‐scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of  computational complexity, under appropriate assumptions (with ns being the flop count for solving a linear system associated with  ). Illustrative numerical examples will be presented. |
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| Keywords: | Cayley transform elliptic partial differential equation Kronecker product large‐scale problem linear equation Stein equation Sylvester equation |
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, arising from a discretized elliptic partial differential equation in 
. Utilizing the stability, we produce an equivalent d‐stable generalized Stein‐like equation, which can be solved iteratively. For large‐scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of 
computational complexity, under appropriate assumptions (with ns being the flop count for solving a linear system associated with 
). Illustrative numerical examples will be presented.