Related papers: Rounding error using low precision approximate ran…
Motivated by problems in contact mechanics, we propose a duality approach for computing approximations and associated a posteriori error bounds to solutions of variational inequalities of the first kind. The proposed approach improves upon…
An improvement on precision of recursive function simulation in IEEE floating point standard is presented. It is shown that the average of rounding towards negative infinite and rounding towards positive infinite yields a better result than…
In this paper, we formulate a new \emph{multiple-correction method}. The goal is to accelerate the rate of convergence. In particular, we construct some sequences to approximate the Euler-Mascheroni and Landau constants, which are faster…
It is a well-known rule of thumb that approximations of stochastic partial differential equations have essentially twice the order of weak convergence compared to the corresponding order of strong convergence. This is already known for many…
We present a new simple method for rounding a semidefinite programming relaxation of a constraint satisfaction problem. We apply it to the problem of approximate angular synchronization. Specifically, we are given directed distances on a…
We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an…
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte…
We discuss numerical approximation methods for Random Time Change equations which possess a deterministic drift part and jump with state-dependent rates. It is first established that solutions to such equations are versions of certain…
We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the…
In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the…
Stochastic collocation methods for approximating the solution of partial differential equations with random input data (e.g., coefficients and forcing terms) suffer from the curse of dimensionality whereby increases in the stochastic…
We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's…
The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a…
Stochastic rounding (SR) is a probabilistic method used to round numbers to floating-point and fixed-point representations. In length $n$ summation, the worst-case error of SR grows as $\sqrt{n}$ with high probability, unlike for standard…
We study the approximation of expectations $\E(f(X))$ for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We…
Many problems in machine learning can be solved by rounding the solution of an appropriate linear program (LP). This paper shows that we can recover solutions of comparable quality by rounding an approximate LP solution instead of the ex-…
In a wide range of statistical learning problems such as ranking, clustering or metric learning among others, the risk is accurately estimated by $U$-statistics of degree $d\geq 1$, i.e. functionals of the training data with low variance…
Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition…
We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of…
In this work, we adapt the {\em micro-macro} methodology to stochastic differential equations for the purpose of numerically solving oscillatory evolution equations. The models we consider are addressed in a wide spectrum of regimes where…