  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.
