TY - JOUR
T1 - Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems
AU - Luo, Xiaopeng
AU - Xu, Xin
AU - Rabitz, Herschel
N1 - Funding Information:
Acknowledgements X.L. and X.X. acknowledge support from the National Science Foundation (Grant No. CHE-1763198), and H.R. acknowledges support from the Templeton Foundation (Grant No. 52265).
Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
PY - 2019/6/15
Y1 - 2019/6/15
N2 - In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to 30 show that resulting approximations are of very high quality.
AB - In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to 30 show that resulting approximations are of very high quality.
KW - Finite difference method
KW - Hermite-HDMR approximation
KW - High-dimensional Dirichlet problems
KW - Meshless method
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U2 - 10.1007/s10910-019-01031-2
DO - 10.1007/s10910-019-01031-2
M3 - Article
AN - SCOPUS:85066129969
SN - 0259-9791
VL - 57
SP - 1652
EP - 1669
JO - Journal of Mathematical Chemistry
JF - Journal of Mathematical Chemistry
IS - 6
ER -