Related papers: Certification and the Potential Energy Landscape
A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the…
The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do not mean the accuracy within the discrete setting, which…
Automated theorem provers (ATPs) can disprove conjectures by saturating a set of clauses, but the resulting saturated sets are opaque certificates. In the unit equational fragment, a saturated set can in fact be read as a convergent rewrite…
This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite…
In this short note we prove that, if (C[a,b],{A_n}) is an approximation scheme and (A_n) satisfies de La Vall\'ee-Poussin Theorem, there are instances of continuous functions on [a,b], real analytic on (a,b], which are poorly approximable…
Quadratic-bilinear (QB) systems arise in many areas of science and engineering. In this paper, we present a scalable approach for designing locally stabilizing state-feedback control laws and certifying the local stability of QB systems.…
We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the…
Conformal field theory, describing systems with scaling symmetry, plays a crucial role throughout physics. We describe a quantum algorithm to simulate the dynamics of conformal field theories, including the action of local conformal…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
In the image processing pipeline of almost every digital camera there is a part dedicated to computational color constancy i.e. to removing the influence of illumination on the colors of the image scene. Some of the best known illumination…
Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to…
We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of…
We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a…
Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…
The region of attraction characterizes well-behaved and safe operation of a nonlinear system and is hence sought after for verification. In this paper, a framework for probabilistic region of attraction estimation is developed that combines…
Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear…
Point source localisation is generally modelled as a Lasso-type problem on measures. However, optimisation methods in non-Hilbert spaces, such as the space of Radon measures, are much less developed than in Hilbert spaces. Most numerical…
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the…
We describe an effective landscape introduced in [1] for the analysis of Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph Coloring. This geometric construction reexpresses these problems in the more familiar terms…
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been…