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We describe a basic framework for studying dynamic scaling that has roots in dynamical systems and probability theory. Within this framework, we study Smoluchowski's coagulation equation for the three simplest rate kernels $K(x,y)=2$, $x+y$…
In this paper we consider a general, challenging distributed optimization set-up arising in several important network control applications. Agents of a network want to minimize the sum of local cost functions, each one depending on a local…
We develop a new randomized iterative algorithm---stochastic dual ascent (SDA)---for finding the projection of a given vector onto the solution space of a linear system. The method is dual in nature: with the dual being a non-strongly…
In this work, we develop analysis and algorithms for a class of (stochastic) bilevel optimization problems whose lower-level (LL) problem is strongly convex and linearly constrained. Most existing approaches for solving such problems rely…
We discuss a unified approach to stochastic optimization of pseudo-Boolean objective functions based on particle methods, including the cross-entropy method and simulated annealing as special cases. We point out the need for auxiliary…
Numerical algorithms for the integration of stochastic differential equations in the presence of white noise are introduced and compared. Algorithms for the integration of stochastic correlated forces are also briefly reviewed. Finally, a…
We consider self-similar solutions to Smoluchowski's coagulation equation for kernels $K=K(x,y)$ that are homogeneous of degree zero and close to constant in the sense that \[ -\eps \leq K(x,y)-2 \leq \eps…
We develop a hierarchical functional derivative method to investigate the reduced dynamics of a quantum dissipative system within the framework of a stochastic decoupling description. Keeping only the lowest order truncation of the…
Stochastic processes find applications in modelling systems in a variety of disciplines. A large number of stochastic models considered are Markovian in nature. It is often observed that higher order Markov processes can model the data…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
We present an intensity speckle simulation algorithm based on stochastic differential equations. Intensity speckles are generated with a negative exponential distribution and an exponential auto-correlation decay. The mean of the…
Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state…
The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition…
Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in…
In this paper, we aim to solve a distributed optimization problem with affine coupling constraints in a multi-agent network, where the cost function of the agents is composed of smooth and possibly non-smooth parts. To solve this problem,…
Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
This paper presents two novel ensemble domain decomposition methods for fast-solving the Stokes-Darcy coupled models with random hydraulic conductivity and body force. To address such random systems, we employ the Monte Carlo (MC) method to…
We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal…
Almost sure convergence rates for linear algorithms $h_{k+1} = h_k +\frac{1}{k^\chi} (b_k-A_kh_k)$ are studied, where $\chi\in(0,1)$, $\{A_{k}\}_{k=1}^\infty$ are symmetric, positive semidefinite random matrices and $\{b_{k}\}_{k=1}^\infty$…