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Rank-adaptive tensor methods for high-dimensional nonlinear PDEs
Alec Dektor, Abram Rodgers, Daniele Venturi
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansio…
Dynamic tensor approximation of high-dimensional nonlinear PDEs
Alec Dektor, Daniele Venturi
We present a new method based on functional tensor decomposition and dynamic tensor approximation to compute the solution of a high-dimensional time-dependent nonlinear partial dif…
Spectral methods for nonlinear functionals and functional differential equations
Daniele Venturi, Alec Dektor
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpos…
A new scalable algorithm for computational optimal control under uncertainty
Panos Lambrianides, Qi Gong, Daniele Venturi
We address the design and synthesis of optimal control strategies for high-dimensional stochastic dynamical systems. Such systems may be deterministic nonlinear systems evolving fr…
Stability analysis of hierarchical tensor methods for time-dependent PDEs
Abram Rodgers, Daniele Venturi
In this paper we address the question of whether it is possible to integrate time-dependent high-dimensional PDEs with hierarchical tensor methods and explicit time stepping scheme…
Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs
Alec Dektor, Daniele Venturi
We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equati…