Related papers: Random Fixed Points, Limits and Systemic risk
The Fast Fourier Transform is extended to functions on finite graphs whose edges are identified with intervals of finite length. Spectral and pseudospectral methods are developed to solve a wide variety of time dependent partial…
We explain how to find the asymptotic form of fixed point solutions in functional truncations, in particular $f(R)$ approximations. We find that quantum fluctuations do not decouple at large $R$, typically leading to elaborate asymptotic…
Networked discrete dynamical systems are often used to model the spread of contagions and decision-making by agents in coordination games. Fixed points of such dynamical systems represent configurations to which the system converges. In the…
The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…
One of the oldest and most studied subject in scientific computing is algorithms for solving partial differential equations (PDEs). A long list of numerical methods have been proposed and successfully used for various applications. In…
In this work we study of the dynamics of large size random neural networks. Different methods have been developed to analyse their behavior, most of them rely on heuristic methods based on Gaussian assumptions regarding the fluctuations in…
We consider an elliptic partial differential equation in non-divergence form with a random diffusion matrix and random forcing term. To address this, we propose a mixed-type continuous finite element discretization in the physical domain,…
In this paper, a class of convex feasibility problems (CFPs) are studied for multi-agent systems through local interactions. The objective is to search a feasible solution to the convex inequalities with some set constraints in a…
For optimal power flow problems with chance constraints, a particularly effective method is based on a fixed point iteration applied to a sequence of deterministic power flow problems. However, a priori, the convergence of such an approach…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
Assume that a stochastic processes can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation"…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…
We propose a computationally efficient alternative to generalized random forests (GRFs) for estimating heterogeneous effects in large dimensions. While GRFs rely on a gradient-based splitting criterion, which in large dimensions is…
We analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point…
Mixtures of $r$ independent distributions for two discrete random variables can be represented by matrices of nonnegative rank $r$. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry…
Formal analysis of the emergent structural properties of dynamic networks is largely uncharted territory. We focus here on the properties of forward reachable sets (FRS) as a function of the underlying degree distribution and edge duration.…
Using the recently developed theory of rigorously validated numerics, we address the Phase-Field-Crystal (PFC) model at the microscopic (atomistic) level. We show the existence of critical points and local minimizers associated with…
We consider an edge-weighted uniform random graph with a given degree sequence (Repeated Configuration Model) which is a useful approximation for many real-world networks. It has been observed that the vertices which are separated from the…
The power flow (PF) problem is a fundamental problem in power system engineering. Many popular solvers face challenges, such as convergence issues. One can try to rewrite the PF problem into a fixed point equation, which can be solved…
The paper provides a framework for the assessment and optimization of the total risk of complex distributed systems. The framework takes into account the risk of each agent, which may arise from heterogeneous sources, as well as the risk…