Abstract:
We consider random scalar, nonlinear hyperbolic conservation laws in spatial
dimension d ? 1 with bounded random flux functions.
There exists a unique random entropy solution (i.e., a strongly measurable mapping
from a probability space into C([0, T];L1(Rd))) with finite second moments.
We present a convergence analysis of a Multi-Level Monte-Carlo Front-Tracking
(MLMCFT) algorithm.
It is based on “pathwise" application of the Front-Tracking Method for deterministic
SCLs.
We compare the MLMCFT algorithms to Multi-Level Monte-Carlo Finite-
Volume methods. Due to the absence of a CFL time step restriction in the pathwise
front tracking scheme, we can prove favourable complexity estimates: in spatial dimension
d ? 2, the mean field of the random entropy solution can be approximated
numerically with (up to logarithmic terms) the same complexity as the solution of
one instance of the deterministic problem, on the same mesh.
We then present results on large scale simulations of MLMC for linear acoustic
wave propagation in heterogeneous media with log-gaussian random coefficients.
Here, conventional explicit timestepping schemes encounter the CFL constraint
which, due to the lognormal gaussian constitutive parameter, is random.
A probabilistic complexity analysis is presented. Implementation with a novel
adaptive load balancing algorithm achieves near linear strong scaling.
Joint work with S. Mishra, N. Risebro, J. Sukys and F. Weber. |