Related papers: Improving solution accuracy and convergence for st…
We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an…
Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. In particular, although ODEs are differentiable and would…
A technique is introduced for estimating unknown parameters when time series of only one variable from a multivariate nonlinear dynamical system is given. The technique employs a combination of two different control methods, a linear…
Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a…
Predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences. Under suitable conditions on the nature of the system, of the perturbation, and of the observables of interest, response theories…
In this paper we study the problem of convergence and generalization error bound of stochastic momentum for deep learning from the perspective of regularization. To do so, we first interpret momentum as solving an $\ell_2$-regularized…
The stochastic time-fractional equation $\partial_t \psi -\Delta\partial_t^{1-\alpha} \psi = f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error…
Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical…
The modelling of small-scale processes is a major source of error in climate models, hindering the accuracy of low-cost models which must approximate such processes through parameterization. Red noise is essential to many operational…
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution…
Conformal Prediction offers a powerful framework for quantifying uncertainty in machine learning models, enabling the construction of prediction sets with finite-sample validity guarantees. While easily adaptable to non-probabilistic…
The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a…
In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an It\^o process. The introduced formula essentially generalizes both the…
This paper studies an optimal control problem for continuous-time stochastic systems subject to reachability objectives specified in a subclass of metric interval temporal logic specifications, a temporal logic with real-time constraints.…
Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such…
The technique of stochastic solutions, previously used for deterministic equations, is here proposed as a solution method for partial differential equations driven by distribution-valued noises.
In this paper, we introduce various mechanisms to obtain accelerated first-order stochastic optimization algorithms when the objective function is convex or strongly convex. Specifically, we extend the Catalyst approach originally designed…
It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct…
Stochastic-gradient-based optimization has been a core enabling methodology in applications to large-scale problems in machine learning and related areas. Despite the progress, the gap between theory and practice remains significant, with…
The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming…