Related papers: On Computation of Approximate Solutions to Large-S…
Large deviations of conservative interacting particle systems, such as the zero range process, about their hydrodynamic limit and their respective rate functions lead to the analysis of the skeleton equation; a degenerate…
Recent progress in deep learning has been driven by increasingly larger models. However, their computational and energy demands have grown proportionally, creating significant barriers to their deployment and to a wider adoption of deep…
Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to…
In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that…
This paper deals with the backstepping design of observer-based compensators for parabolic ODE-PDE-ODE systems. The latter consist of n coupled parabolic PDEs with distinct diffusion coefficients and spatially-varying coefficients, that are…
Subgradient algorithms for training support vector machines have been quite successful for solving large-scale and online learning problems. However, they have been restricted to linear kernels and strongly convex formulations. This paper…
This paper addresses the stabilization of a chain of three coupled hyperbolic partial differential equations actuated by two control inputs applied at arbitrary nodes of the network. With the exception of configurations where one input is…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
Solvers for partial differential equations (PDE) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that needs to be stored and transmitted on all levels of the memory hierarchy.…
In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic…
Principal component regression (PCR) is a useful method for regularizing linear regression. Although conceptually simple, straightforward implementations of PCR have high computational costs and so are inappropriate when learning with large…
We provide a `user guide' to the literature of the past twenty years concerning the modeling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes…
Neural Stochastic Differential Equations (NSDEs) model the drift and diffusion functions of a stochastic process as neural networks. While NSDEs are known to make accurate predictions, their uncertainty quantification properties have been…
We construct deterministic particle solutions for linear and fast diffusion equations using a nonlocal approximation. We exploit the $2$-Wasserstein gradient flow structure of the equations in order to obtain the nonlocal approximating PDEs…
We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
The paper is concerned with the strict-feedback form of hyperbolic distributed-parameter systems. Such a system structure is well known to be the basis for the recursive backstepping control design for nonlinear ODEs and is also reflected…
In this paper, we design Dirichlet-Neumann boundary feedback controllers for the Korteweg-de Vries (KdV) equation that act at the right endpoint of the domain. The length of the domain is allowed to be critical. Constructing backstepping…
The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years,…