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

New data on eudialyte decomposition minerals from kakortokites and associated pegmatites of the Ilimaussaq complex, South Greenland

A suite of samples with eudialyte and eudialyte decomposition minerals from the kakortokite and associated pegmatites of the Ilimaussaq complex in South Greenland has been investigated by electron microprobe analysis. Extensive decomposition of eudialyte has resulted in the formation of catapleiite as host for a number of rare and hitherto unknown REE minerals besides known minerals such as monazite and kainosite. Mineral A1 is present in very small amounts in nearly all eudialyte decomposition aggregates and comprises two varieties: Ca-rich A1 with composition HCa3REE6(SiO4)6(Fsquare) and presumed apatite structure, and Ca-poor A1 with composition (Fe,Mn,Ca)1.5REE6Si6FO22 and unknown structure. Mineral A2 with composition (Ca,Fe)1.2 REE4Si6O19-y(OH)2y.nH2O is indistinguishable from A1 in EMP-backscattered light and has only been found at a limited number of localities. Mineral A2 also occurs as a primary mineral at one locality. Additional rare and new REE-minerals are mineral A3 with composition Na0.2Ca0.6Fe0.2Mn0.5Al0.5REE2.8Si6F0.5O)18-y(OH)2y . nH2O; mineral Uk2 with composition REE2.00F1.50O2.25-y(OH)2y . nH2O; mineral Uk3 with composition CaREE4O7-yOH)2y . nH2O; and mineral Y1 with composition Na2Ca4Y2.7REE1.3F18 (OH)4. The Ce:(Y+La+Pr+Nd+Sm+Gd) molar ratio for A1, A2, A3, Uk2, Uk3 and monazite is close to 1:1. Characteristic for A1, A2 and monazite are substantial solid solutions between La and (Pr+Nd+Sm+Gd) with slowly increasing content of Ce as the content of La increases. A similar pattern does not exist for the REE in fresh eudialyte. Kainosite, identified in one decomposition aggregate, has not previously been found in the Il maussaq complex.</p

Efficient p-multigrid spectral element model for water waves and marine offshore structures

In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave-body interaction with high computational efficiency within a single modellingapproach. We design and propose an efficientO(n)-scalable computational procedure based on geometric p-multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high-order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p-multigrid spectralelement model can take advantage of the high-order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h-multigrid approaches. We provide numerical benchmarks for the algorithmic and numerical efficiency of the iterative geometric p-multigrid solver. Results of numerical experiments are presented for wave propagation and for wave-body interaction in an advanced case for focusing design waves interacting with a FPSO. Our study shows, that the use of iterative geometric p-multigrid methods for theLaplace problem can significantly improve run-time efficiency of FNPF simulators.Comment: Submitted to an international journal for peer revie