Band 22, Heft 3

Generalized Weighted Analog Sampling is a variance-reducing method for solving radiative transport problems that makes use of a biased (though asymptotically unbiased) estimator. The introduction of bias provides a mechanism for combining the best features of unbiased estimators while avoiding their limitations. In this paper we present a new proof that adaptive GWAS estimation based on combining the variance-reducing power of importance sampling with the sampling simplicity of correlated sampling yields geometrically convergent estimates of radiative transport solutions. The new proof establishes a stronger and more general theory of geometric convergence for GWAS.

In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2${1/2}$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity O⁢(ϵ-2)${O(\epsilon^{-2})}$ for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time τ. The main motivations are giving a probabilistic representation of the Sobolev’s solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when τ is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided.

In this short article we suggest randomized scalable stochastic matrix-based algorithms for large linear systems. The idea behind these stochastic methods is a randomized vector representation of matrix iterations. In addition, to minimize the variance, it is suggested to use stochastic and double stochastic matrices for efficient randomized calculation of matrix iterations and a random gradient based search strategy. The iterations are performed by sampling random rows and columns only, thus avoiding not only matrix matrix but also matrix vector multiplications. Further improvements of the methods can be obtained through projections by a random gaussian matrix.