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A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
As the dimension of a system increases, traditional methods for control and differential games rapidly become intractable, making the design of safe autonomous agents challenging in complex or team settings. Deep-learning approaches avoid…
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint,…
We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into…
We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand,…
In this paper we introduce a novel approach for an important problem of break detection. Specifically, we are interested in detection of an abrupt change in the covariance structure of a high-dimensional random process -- a problem, which…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-\Delta$. The problem is variational in nature and we apply a nonlinear reduction method which allows…
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few…
This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the…
This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to…
High-dimensional self-exciting point processes have been widely used in many application areas to model discrete event data in which past and current events affect the likelihood of future events. In this paper, we are concerned with…
Bayesian methods have been widely used in the last two decades to infer statistical properties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…
The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple…
The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In…
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are…
Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution…
We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and…
We discuss some aspects of numerical continuation and bifurcation for partial differential equations, specifically pattern formation and coherent structures. For the sake of clarity we focus on wavetrains, stability and associated invasion…