Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUnified Analysis of Periodization-Based Sampling Methods for Matern Covariances
71
0
0.0
(
0
)
Added by
Markus Bachmayr
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matern covariances with smoothness index $ u$ and correlation length $lambda$, we analyse the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $ u$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $ u$, $lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterisation of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.
rate research
Read More
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It^o-Taylor expansion and iterated It^o integrals, the proposed scheme approximates the probability measure $mu(X^{n+1}|X^n=x_n)$ by a mixture of Gaussians. The solution at next time step $X^{n+1}$ is then drawn from the Gaussian mixture with complexity linear in the dimension $d$. This provides a new general strategy to construct efficient high weak order numerical schemes for SDEs.
The sampling Kaczmarz-Motzkin (SKM) method is a generalization of the randomized Kaczmarz and Motzkin methods. It first samples some rows of coefficient matrix randomly to build a set and then makes use of the maximum violation criterion within this set to determine a constraint. Finally, it makes progress by enforcing this single constraint. In this paper, on the basis of the framework of the SKM method and considering the greedy strategies, we present two block sampling Kaczmarz-Motzkin methods for consistent linear systems. Specifically, we also first sample a subset of rows of coefficient matrix and then determine an index in this set using the maximum violation criterion. Unlike the SKM method, in the rest of the block methods, we devise different greedy strategies to build index sets. Then, the new methods make progress by enforcing the corresponding multiple constraints simultaneously. Theoretical analyses demonstrate that these block methods converge at least as quickly as the SKM method, and numerical experiments show that, for the same accuracy, our methods outperform the SKM method in terms of the number of iterations and computing time.
Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for large timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.
Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of an ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.
Parametric sensitivity analysis is a critical component in the study of mathematical models of physical systems. Due to its simplicity, finite difference methods are used extensively for this analysis in the study of stochastically modeled reaction networks. Different coupling methods have been proposed to build finite difference estimators, with the split coupling, also termed the stacked coupling, yielding the lowest variance in the vast majority of cases. Analytical results related to this coupling are sparse, and include an analysis of the variance of the coupled processes under the assumption of globally Lipschitz intensity functions [Anderson, SIAM Numerical Analysis, Vol. 50, 2012]. Because of the global Lipschitz assumption utilized in [Anderson, SIAM Numerical Analysis, Vol. 50, 2012], the main result there is only applicable to a small percentage of the models found in the literature, and it was conjectured that similar results should hold for a much wider class of models. In this paper we demonstrate this conjecture to be true by proving the variance of the coupled processes scales in the desired manner for a large class of non-Lipschitz models. We further extend the analysis to allow for time dependence in the parameters. In particular, binary systems with or without time-dependent rate parameters, a class of models that accounts for the vast majority of systems considered in the literature, satisfy the assumptions of our theory.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
