Spectral tensor-train decomposition

The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the "cores") comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting \textit{spectral tensor-train decomposition} combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modifed set of Genz functions with dimension up to $100$, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online.Comment: 33 pages, 19 figure

Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks

Nonlinear model predictive control (NMPC) often requires real-time solution to optimization problems. However, in cases where the mathematical model is of high dimension in the solution space, e.g. for solution of partial differential equations (PDEs), black-box optimizers are rarely sufficient to get the required online computational speed. In such cases one must resort to customized solvers. This paper present a new solver for nonlinear time-dependent PDE-constrained optimization problems. It is composed of a sequential quadratic programming (SQP) scheme to solve the PDE-constrained problem in an offline phase, a proper orthogonal decomposition (POD) approach to identify a lower dimensional solution space, and a neural network (NN) for fast online evaluations. The proposed method is showcased on a regularized least-square optimal control problem for the viscous Burgers' equation. It is concluded that significant online speed-up is achieved, compared to conventional methods using SQP and finite elements, at a cost of a prolonged offline phase and reduced accuracy.Comment: Accepted for publishing at the 58th IEEE Conference on Decision and Control, Nice, France, 11-13 December, https://cdc2019.ieeecss.org

A Spectral Element Method for Nonlinear and Dispersive Water Waves

The use of flexible mesh discretisation methods are im-portant for simulation of nonlinear wave-structure inter-actions in offshore and marine settings such as harbour and coastal areas. For real applications, development of efficient models for wave propagation based on un-structured discretisation methods is of key interest. We present a high-order general-purpose three-dimensional numerical model solving fully nonlinear and dispersive potential flow equations with a free surface. Figure 1: Snapshot of scaled free surface showing diffraction and refraction patterns in the free surface. Governing equations Let both Ω ⊂ Rd (d = 2, 3) and Ω ′ ⊂ Rd−1 be bounded, connected domains with piecewise smooth boundaries Γ and Γ′, respectively. Let T: t ≥ 0 be the time domain. Introduce the free surface boundary ΓFS ⊂ Γ and the bottom boundary Γb ⊂ Γ. The mathematical problem is to find a scalar velocity potential function φ(x, z, t): Ω × T → R and to determine the evolution of the free surface elevation η(x, t) : Ω ′ × T → R. The Eulerian description of the unsteady kinematic and a dynamic free surface boundary conditions is ex-pressed in the Zakharov form. In Ω ′ × T, find η, φ̃ ∂tη = −∇η · ∇φ̃+ w̃(1 +∇η · ∇η) ∂tφ ̃ = −gη −