1,173 research outputs found
A Nonlinear Multigrid Steady-State Solver for Microflow
We develop a nonlinear multigrid method to solve the steady state of
microflow, which is modeled by the high order moment system derived recently
for the steady-state Boltzmann equation with ES-BGK collision term. The solver
adopts a symmetric Gauss-Seidel iterative scheme nested by a local Newton
iteration on grid cell level as its smoother. Numerical examples show that the
solver is insensitive to the parameters in the implementation thus is quite
robust. It is demonstrated that expected efficiency improvement is achieved by
the proposed method in comparison with the direct time-stepping scheme
Stochastic subspace correction in Hilbert space
We consider an incremental approximation method for solving variational
problems in infinite-dimensional Hilbert spaces, where in each step a randomly
and independently selected subproblem from an infinite collection of
subproblems is solved. we show that convergence rates for the expectation of
the squared error can be guaranteed under weaker conditions than previously
established in [Constr. Approx. 44:1 (2016), 121-139]. A connection to the
theory of learning algorithms in reproducing kernel Hilbert spaces is revealed.Comment: 15 page
Optimal Energy Estimation in Path-Integral Monte Carlo Simulations
We investigate the properties of two standard energy estimators used in
path-integral Monte Carlo simulations. By disentangling the variance of the
estimators and their autocorrelation times we analyse the dependence of the
performance on the update algorithm and present a detailed comparison of
refined update schemes such as multigrid and staging techniques. We show that a
proper combination of the two estimators leads to a further reduction of the
statistical error of the estimated energy with respect to the better of the two
without extra cost.Comment: 45 pp. LaTeX, 22 Postscript Figure
Multilevel Monte Carlo finite element methods for stochastic elliptic variational inequalities
Multilevel Monte Carlo finite element methods (MLMC-FEMs) for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of “pathwise” weak formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for the classical MC method and novel MLMC-FEMs. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FEMs then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments
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