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This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.
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This content will become publicly available on October 28, 2026
Mass conservative limiting and applications to the approximation of the steady-state radiation transport equations
A limiting technique for scalar transport equations is presented. The originality of the method is that it does not require solving nonlinear optimization problems nor does it rely on the construction of a low-order approximation. The method has minimal complexity and is numerically demonstrated to maintain high-order accuracy. The performance of the method is illustrated on the radiation transport equation.
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- Award ID(s):
- 2110868
- PAR ID:
- 10640480
- Publisher / Repository:
- 2024 Elsevier Inc., Journal of Computational Physics 521 (2025) 113531
- Date Published:
- Journal Name:
- Journal of computational physics
- ISSN:
- 0021-9991
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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