Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1502.07186

Parallel optimized sampling for stochastic equations

Bogdan Opanchuk·Simon Kiesewetter·Peter D. Drummond
View original source ↗Read source document ↗Export metadata ↓

Abstract

Stochastic equations play an important role in computational science, due to their ability to treat a wide variety of complex statistical problems. However, current algorithms are strongly limited by their sampling variance, which scales proportionate to 1/N_S for N_S samples. In this paper, we obtain a new class of variance reduction methods for treating stochastic equations, called parallel optimized sampling. The objective of parallel optimized sampling is to reduce the sampling variance in the observables of an ensemble of stochastic trajectories. This is achieved through calculating a finite set of observables - typically statistical moments - in parallel, and minimizing the errors compared to known values. The algorithm is both numerically efficient and unbiased. Importantly, it does not increase the errors in higher order moments, and generally reduces such errors as well. The same procedure is applied both to initial ensembles and to changes in a finite time-step. Results of these methods show that errors in initially optimized moments can be reduced to the machine precision level, typically around 10^(-16) in current hardware. For nonlinear stochastic equations, sampled moment errors during time-evolution are larger than this, due to error propagation effects. Even so, we provide evidence for error reductions of up to two orders of magnitude in a nonlinear equation example, for low order moments, which is a large practical benefit. The sampling variance typically scales as 1/N_S, but with the advantage of a very much smaller prefactor than for standard, non-optimized methods.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bogdan Opanchuk, Simon Kiesewetter, Peter D. Drummond. 2016-04-04. Parallel optimized sampling for stochastic equations. https://doi.org/10.1137/15m1040098

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗