Construction of a Mean Square Error Adaptive Euler--Maruyama Method with Applications in Multilevel Monte Carlo

A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping Euler--Maruyama method for numerical solutions of SDE, and the resulting method is incorporated into a multilevel Monte Carlo (MLMC) method for weak approximations of SDE. This gives an efficient MSE adaptive MLMC method for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC method is shown to outperform the uniform time stepping MLMC method by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).Comment: 43 pages, 12 figure

Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients

We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority. The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances. Numerical calculations demonstrating the effectiveness of the method for one- and two-dimensional model problems arising in groundwater flow are presented. © 2011 Springer-Verlag

How deep are deep Gaussian processes?

© 2018 Matthew M. Dunlop, Mark A. Girolami, Andrew M. Stuart and Aretha L. Teckentrup. Recent research has shown the potential utility of deep Gaussian processes. These deep structures are probability distributions, designed through hierarchical construction, which are conditionally Gaussian. In this paper, the current published body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples generated from a deep Gaussian process have a Markovian structure with respect to the depth parameter, and the effective depth of the resulting process is interpreted in terms of the ergodicity, or non-ergodicity, of the resulting Markov chain. For the classes of deep Gaussian processes introduced, we provide results concerning their ergodicity and hence their effective depth. We also demonstrate how these processes may be used for inference; in particular we show how a Metropolis-within-Gibbs construction across the levels of the hierarchy can be used to derive sampling tools which are robust to the level of resolution used to represent the functions on a computer. For illustration, we consider the effect of ergodicity in some simple numerical examples

Comparative analysis of different xanthan samples by atomic force microscopy

Teckentrup J, Al-Hamood O, Steffens T, et al. Comparative analysis of different xanthan samples by atomic force microscopy. Journal of Biotechnology. 2017;257:2-8.The polysaccharide xanthan which is produced by the gamma-proteobacterium Xanthomonas campestris is used as a food thickening agent and rheologic modifier in numerous food, cosmetics and technical applications. Its great commercial importance stimulated biotechnological approaches to optimize the xanthan production. By targeted genetic modification the metabolism of Xanthomonas can be modified in such a way that the xanthan production efficiency and/or the shear-thickening potency is optimized. Using atomic force microscopy (AFM) the secondary structure of single xanthan polymers produced by the wild type Xanthomonas campestris B100 and several genetically modified variations were analyzed. We found a wide variation of characteristic differences between xanthan molecules produced by different strains. The structures ranged from single-stranded coiled polymers to branched xanthan double-strands. These results can help to get a better understanding of the polymerization- and secretion-machinery that are relevant for xanthan synthesis. Furthermore, we demonstrate that the xanthan secondary structure strongly correlates with its viscosifying properties. Copyright A 2016 Elsevier B.V. All rights reserved