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We consider the construction of semiimplicit linear multistep methods w...
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A New Block Preconditioner for Implicit RungeKutta Methods for Parabolic PDE
A new preconditioner based on a block LDU factorization with algebraic m...
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Linearized Implicit Methods Based on a SingleLayer Neural Network: Application to KellerSegel Models
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Highorder implicit time integration scheme based on Padé expansions
A singlestep highorder implicit time integration scheme for the soluti...
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Symmetric indefinite triangular factorization revealing the rank profile matrix
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Nearly optimal scaling in the SR decomposition
In this paper we analyze the nearly optimal block diagonal scalings of t...
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Fast Power Series Solution of Large 3D Electrodynamic Integral Equation for PEC Scatterers
This paper presents a new fast power series solution method to solve the...
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Optimal and LowMemory NearOptimal Preconditioning of Fully Implicit RungeKutta Schemes for Parabolic PDEs
RungeKutta (RK) schemes, especially GaussLegendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their Astability and highorder accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to semiimplicit RK (aka diagonally implicit RK or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by blockcirculant preconditioners and Jordan form solution techniques for IRK. We then derive an efficient, nearoptimal singlydiagonal approximate BRSD (SABRSD) by approximating the quasitriangular matrix in real Schur decomposition using an optimized uppertriangular matrix with a single diagonal value. A desirable feature of SABRSD is that it has comparable memory requirements and factorization cost as singly DIRK (SDIRK). We approximately factorize the diagonal blocks in these preconditioners using a nearlinearcomplexity multilevel ILU factorization called HILUCSI, which is significantly more robust and more efficient than ILU(0). We apply the preconditioners in rightpreconditioned GMRES to solve the advectiondiffusion equation in 3D using finite element and finite difference methods. We show that BCSD, BRSD, and BJF significantly outperform other preconditioners in terms of GMRES iterations, and SABRSD is competitive with them and the prior state of the art in terms of computational cost while requiring the least amount of memory.
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