TY - JOUR
T1 - On HMM-like integrators and projective integration methods for systems with multiple time scales
AU - Vanden-Eijnden, Eric
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2007
Y1 - 2007
N2 - HMM-like multiscale integrators and projective integration methods are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMM-integrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin's artificial compressibility method and the Car-Parrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are.
AB - HMM-like multiscale integrators and projective integration methods are two different types of multiscale integrators which have been introduced to simulate efficiently systems with widely disparate time scales. The original philosophies of these methods, reviewed here, were quite different. Recently, however, projective integration methods seem to have evolved in a way that make them increasingly similar to HMM-integrators and quite different from what they were originally. Nevertheless, the strategy of extrapolation which was at the core of the original projective integration methods has its value and should be extended rather than abandoned. An attempt in this direction is made here and it is shown how the strategy of extrapolation can be generalized to stochastic dynamical systems with multiple time scales, in a way reminiscent of Chorin's artificial compressibility method and the Car-Parrinello method used in molecular dynamics. The result is a seamless integration scheme, i.e. one that does not require knowing explicitly what the slow and fast variables are.
KW - Averaging theorems
KW - HMM
KW - Multiscale integrators
KW - Projective integration methods
KW - Stiff ODEs
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U2 - 10.4310/CMS.2007.v5.n2.a14
DO - 10.4310/CMS.2007.v5.n2.a14
M3 - Article
AN - SCOPUS:34547284818
VL - 5
SP - 495
EP - 505
JO - Communications in Mathematical Sciences
JF - Communications in Mathematical Sciences
SN - 1539-6746
IS - 2
ER -