Related papers: New Weak Error bounds and expansions for Optimal Q…

Quasi-Monte Carlo cubature methods often sample the integrand using Sobol' (or other digital) sequences to obtain higher accuracy than IID sampling. An important question is how to conservatively estimate the error of a digital sequence…

We study the approximation of $\mathbb{E}f(X_T)$ by a Monte Carlo algorithm, where $X$ is the solution of a stochastic differential equation and $f$ is a given function. We introduce a new variance reduction method, which can be viewed as a…

We prove lower bounds for the error of optimal cubature formulae for $d$-variate functions from Besov spaces of mixed smoothness $B^{\alpha}_{p,\theta}({\mathbb G}^d)$ in the case $0 < p, \theta \le \infty$ and $\alpha > 1/p$, where…

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study…

We study a one-dimensional linear dispersive equation of differential order $\kappa \geq 2$ with concentrated potential of extension $\varepsilon$ with $0 < \varepsilon \ll 1$, featuring a competition between weak dispersion of strength…

We provide lower error bounds for randomized algorithms that approximate integrals of functions depending on an unrestricted or even infinite number of variables. More precisely, we consider the infinite-dimensional integration problem on…

This work investigates the optimal error estimate of the fully discrete scheme for the variable-exponent subdiffusion model under the nonuniform temporal mesh. We apply the perturbation method to reformulate the original model into its…

We derive sharper probabilistic concentration bounds for the Monte Carlo Empirical Rademacher Averages (MCERA), which are proved through recent results on the concentration of self-bounding functions. Our novel bounds are characterized by…

Quantum mechanics for many-body systems may be reduced to the evaluation of integrals in 3N dimensions using Monte-Carlo, providing the Quantum Monte Carlo ab initio methods. Here we limit ourselves to expectation values for trial…

In this paper, by using tools of second-order variational analysis, we study the popular forward-backward splitting method with Beck-Teboulle's line-search for solving convex optimization problem where the objective function can be split…

We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the…

We consider the fundamental problem of estimating a discrete distribution on a domain of size $K$ with high probability in Kullback-Leibler divergence. We provide upper and lower bounds on the minimax estimation rate, which show that the…

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An…

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…

Gauss--Christoffel quadrature is a fundamental method for numerical integration, and its convergence analysis is closely related to the decay of Chebyshev expansion coefficients. Classical estimates, including those due to Trefethen, are…

We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the…

This paper considers stochastic weakly convex optimization without the standard Lipschitz continuity assumption. Based on new adaptive regularization (stepsize) strategies, we show that a wide class of stochastic algorithms, including the…

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz…

We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence…

We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a…