Related papers: Random Gradient-Free Optimization in Infinite Dime…
The notion of generalization in classical Statistical Learning is often attached to the postulate that data points are independent and identically distributed (IID) random variables. While relevant in many applications, this postulate may…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
In this paper, we explore the concept of metric-driven numerical methods as a powerful tool for solving various types of multiscale partial differential equations. Our focus is on computing constrained minimizers of functionals - or,…
We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region,…
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance…
This paper proposes a novel approach to adaptive step sizes in stochastic gradient descent (SGD) by utilizing quantities that we have identified as numerically traceable -- the Lipschitz constant for gradients and a concept of the local…
We introduce a data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves…
In this paper we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of…
We study statistical inverse learning in the context of nonlinear inverse problems under random design. Specifically, we address a class of nonlinear problems by employing gradient descent (GD) and stochastic gradient descent (SGD) with…
Addressing high-dimensional partial differential equations to derive effective actions within the functional renormalization group is formidable, especially when considering various field configurations, including inhomogeneous states, even…
We introduce a derivative-free global optimization algorithm that efficiently computes minima for various classes of one-dimensional functions, including non-convex, and non-smooth functions.This algorithm numerically approximates the…
Many applications in machine learning or signal processing involve nonsmooth optimization problems. This nonsmoothness brings a low-dimensional structure to the optimal solutions. In this paper, we propose a randomized proximal gradient…
Smooth, non-convex optimization problems on Riemannian manifolds occur in machine learning as a result of orthonormality, rank or positivity constraints. First- and second-order necessary optimality conditions state that the Riemannian…
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…
Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently…
Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension…
In this paper, we study numerical approximations for optimal control of a class of stochastic partial differential equations with partial observations. The system state evolves in a Hilbert space, whereas observations are given in…
Physics-informed neural networks (PINNs) have attracted a lot of attention in scientific computing as their functional representation of partial differential equation (PDE) solutions offers flexibility and accuracy features. However, their…
We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems in a reflexive Banach space. We establish strong duality for a very general type of augmented Lagrangian, in which we assume a less…
We address the problem of learning an unknown smooth function and its derivatives from noisy pointwise evaluations under the supremum norm. While classical nonparametric regression provides a strong theoretical foundation, traditional…