Related papers: Bounds on mean energy in the Kuramoto-Sivashinsky …
Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although…
For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an…
We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schr\"odinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous…
We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by…
We give improved lower bounds for the number of solutions of some $S$-unit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method…
To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first order partial…
In this article we introduce several kinds of easily implementable explicit schemes, which are amenable to Khasminski's techniques and are particularly suitable for highly nonlinear stochastic differential equations (SDEs). We show that…
Initial-boundary value problems for the $n$-dimensional ($n$ is a natural number from the interval [2,7]) Kuramoto-Sivashinsky equation posed on smooth bounded domains in $\mathbb{R}^n$ were considered. The existence and uniqueness of…
Computational approaches to PDE-constrained optimization under uncertainty may involve finite-dimensional approximations of control and state spaces, sample average approximations of measures of risk and reliability, smooth approximations…
Two-time-scale Stochastic Approximation (SA) is an iterative algorithm with applications in reinforcement learning and optimization. Prior finite time analysis of such algorithms has focused on fixed point iterations with mappings…
We describe a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our…
Simulations of chaotic systems can only produce high-fidelity trajectories if the initial and boundary conditions are well specified. When these conditions are unknown but measurements are available, variational state estimation can…
We analyse the nonlinear Kuramoto-Sivashinsky equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately…
We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural…
This paper develops a unified methodology for probabilistic analysis and optimal control design for jump diffusion processes defined by polynomials. For such systems, the evolution of the moments of the state can be described via a system…
Motivated by applications to mathematical biology, we study the averaging problem for slow-fast systems, {\em in the case in which the fast dynamics is a stochastic process with multiple invariant measures}. We consider both the case in…
In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on $n$-bits in time $\text{poly}(n) 2^{n/2}$. In this work, we investigate whether such general…
We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…
In this short communication, we announce an algorithmic procedure for constructing non-uniqueness counter-examples of classical solutions to initial-boundary-value problems for a wide class of linear evolution partial differential…
This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These…