Related papers: Optimally accurate operators for partial different…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this…
This paper focuses on investigating an inexact stochastic model-based optimization algorithm that integrates preconditioning techniques for solving stochastic composite optimization problems. The proposed framework unifies and extends the…
Deterministic neural operators perform well on many PDEs but can struggle with the approximation of high-frequency wave phenomena, where strong input-to-output sensitivity makes operator learning challenging, and spectral bias blurs…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element…
In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear…
This tutorial introduces a systematic approach for addressing the key question of quantum metrology: For a generic task of sensing an unknown parameter, what is the ultimate precision given a constrained set of admissible strategies. The…
In this paper, a topology optimization framework utilizing automatic differentiation is presented as an efficient way for solving 2D density-based topology optimization problem by calculating gradients through the fully differentiable…
We investigate an approach for the numerical solution of differential equations which is based on the perfect discretization of actions. Such perfect discretizations show up at the fixed points of renormalization group transformations. This…
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to…
Reducing the dimensionality and uncertainty of design spaces is a key prerequisite for shape optimisation in computationally intensive fluid problems. However, running these analyses at an offline stage itself poses a computationally…
We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. Focusing on a stochastic query model that provides noisy evaluations of the operator, we analyze a variance-reduced stochastic…
For the first time we introduce an error estimator for the numerical approximation of the equations describing the dynamics of sea ice. The idea of the estimator is to identify different error contributions coming from spatial and temporal…
Recent results in quantitative homogenisation of the wave equation with rapidly oscillating coefficients are discussed from the operator-theoretic perspective, which views the solution as the result of applying the operator of hyperbolic…
We study solution techniques for a linear-quadratic optimal control problem involving fractional powers of elliptic operators. These fractional operators can be realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem…
This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena.…
The manuscript presents a new technique for computing the exponential of skew-Hermitian operators. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many…
We propose a time-adaptive predictor/multi-corrector method to solve hyperbolic partial differential equations, based on the generalized-$\alpha$ scheme that provides user-control on the numerical dissipation and second-order accuracy in…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…