Name
Title | ||
|---|---|---|
Adaptive change of basis in entropy-based moment closures for linear kinetic equations |
Abstract | ||
|---|---|---|
Entropy-based (M"N) moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space-time mesh, making it important to solve these optimization problems accurately and efficiently. We present a complete and practical numerical algorithm for solving the dual problem in one-dimensional, slab geometries. The closure is only well-defined on the set of moments that are realizable from a positive underlying distribution, and as the boundary of the realizable set is approached, the dual problem becomes increasingly difficult to solve due to ill-conditioning of the Hessian matrix. To improve the condition number of the Hessian, we advocate the use of a change of polynomial basis, defined using a Cholesky factorization of the Hessian, that permits solution of problems nearer to the boundary of the realizable set. We also advocate a fixed quadrature scheme, rather than adaptive quadrature, since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes. For very ill-conditioned problems, we use regularization to make the optimization algorithm robust. We design a manufactured solution and demonstrate that the adaptive-basis optimization algorithm reduces the need for regularization. This is important since we also show that regularization slows, and even stalls, convergence of the numerical simulation when refining the space-time mesh. We also simulate two well-known benchmark problems. There we find that our adaptive-basis, fixed-quadrature algorithm uses less regularization than alternatives, although differences in the resulting numerical simulations are more sensitive to the regularization strategy than to the choice of basis. |
Year | DOI | Venue |
|---|---|---|
2014 | 10.1016/j.jcp.2013.10.049 | J. Comput. Physics |
Keywords | Field | DocType |
adaptive change,practical numerical algorithm,dual problem,linear kinetic equation,computationally realizable,fixed-quadrature algorithm,adaptive-basis optimization algorithm,optimization problem,regularization strategy,entropy-based moment closure,numerical simulation,realizable set,space-time mesh,convex optimization,kinetic theory,transport | Polynomial basis,Condition number,Mathematical optimization,Adaptive quadrature,Change of basis,Hessian matrix,Regularization (mathematics),Optimization problem,Mathematics,Cholesky decomposition | Journal |
Volume | ISSN | Citations |
258, | 0021-9991 | 4 |
PageRank | References | Authors |
0.49 | 10 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Graham W. Alldredge | 1 | 16 | 1.91 |
| Cory D. Hauck | 2 | 68 | 12.42 |
| O'Leary, Dianne P. | 3 | 1064 | 222.93 |
| André L. Tits | 4 | 429 | 76.27 |