Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software ...

pdf/hm-toolbox-matlab-software-for-hodlr-and-hss-matrices-cbh1rbnwkp.pdf

hm-toolbox: MATLAB Software for HODLR and HSS Matrices

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

We present a method for converting tensors into the tensor train format based on actions of the tensor as a vector-valued multilinear function. Existing methods for constructing tensor trains requi...

pdf/tensor-train-construction-from-tensor-actions-with-4whrydmd47.pdf

Tensor train construction from tensor actions, with application to compression of large high order derivative tensors

Gaussian random fields are widely used as building blocks for modeling stochastic processes. This paper is concerned with the efficient representation of d-point correlations for such fields, which in turn enables the representation of more general stochastic processes that can be expressed as a function of one (or several) Gaussian random fields. Our representation consists of two ingredients. In the first step, we replace the random field by a truncated Karhunen--Loeve expansion and analyze the resulting error. The parameters describing the d-point correlation can be arranged in a tensor, but its storage grows exponentially in d. To avoid this, the second step consists of approximating the tensor in a low-rank tensor format, the so-called tensor train decomposition. By exploiting the particular structure of the tensor, an approximation algorithm is derived that does not need to form this tensor explicitly and allows processing correlations of order as high as d=20. The resulting representation is very c...

http://sma.epfl.ch/~anchpcommon/publications/tt-correlation.pdf

Low-Rank Tensor Approximation for High-Order Correlation Functions of Gaussian Random Fields ∗

The propagation of input uncertainty through engineering models allows designers to better understand the range of possible outcomes resulting from design decisions. This could lead to greater trust between modelers and stakeholders in the wind energy industry. In this study, we apply multilevel-multiﬁdelity Monte Carlo sampling to ﬂow over an airfoil, assuming uncertainty in the inﬂow conditions, and characterize the associated computational savings compared to standard Monte Carlo approaches. The “truth” model is provided by an airfoil simulation with a very ﬁne computational time step, and auxiliary lower-level models are provided by simulations with coarser time steps. Reynolds-averaged Navier Stokes and detached eddy simulations are used to obtain two diﬀerent model ﬁdelities. The primary quantity of interest for this analysis is the lift force, which is examined for a range of angles of attack. We launch an initial set of “trial” samples to determine the optimal allocation of model evaluations, and these trial evaluations are used to inform a larger sampling eﬀort. Using the multilevel-multiﬁdelity approach, we achieve roughly an order of magnitude variance reduction in expected lift as compared to the standard Monte Carlo approach with an equivalent computational cost

Multifidelity Uncertainty Quantification with Applications in Wind Turbine Aerodynamics

Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3): A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix. For this purpose, we combine our previous work on the fast computation of spectral projectors in the so called HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. The numerical experiments demonstrate that our algorithm exhibits quasilinear complexity and allows for conveniently dealing with large-scale matrices.

MATHICSE Technical Report : A fast spectral divide-and-conquer method for banded matrices

A CUR approximation of a matrix $A$ is a particular type of low-rank approximation $A \approx C U R$, where $C$ and $R$ consist of columns and rows of $A$, respectively. One way to obtain such an approximation is to apply column subset selection to $A$ and $A^T$. In this work, we describe a numerically robust and much faster variant of the column subset selection algorithm proposed by Deshpande and Rademacher, which guarantees an error close to the best approximation error in the Frobenius norm. For cross approximation, in which $U$ is required to be the inverse of a submatrix of $A$ described by the intersection of $C$ and $R$, we obtain a new algorithm with an error bound that stays within a factor $k + 1$ of the best rank-$k$ approximation error in the Frobenius norm. To the best of our knowledge, this is the first deterministic polynomial-time algorithm for which this factor is bounded by a polynomial in $k$. Our derivation and analysis of the algorithm is based on derandomizing a recent existence result by Zamarashkin and Osinsky. To illustrate the versatility of our new column subset selection algorithm, an extension to low multilinear rank approximations of tensors is provided as well.

MATHICSE Technical Report : Low-rank approximation in the Frobenius norm by column and row subset selection

We study the Darcy problem with log-normal permeability, modeling the fluid flow in a heterogeneous porous medium. A perturbation approach is adopted, expanding the solution in Taylor series around the nominal value of the permeability. The resulting recursive deterministic problem satisfied by the expected value of the stochastic solution, analytically derived and studied in [4], is discretized with a full tensor product finite element technique. To overcome the incurred curse of dimensionality the solution is sought in a low-rank tensor format, the so called Tensor Train format. We develop an algorithm for solving the recursive first moment problem approximately in the Tensor Train format and show its effectiveness with numerical examples.

MATHICSE Technical Report : Tensor train approximation of moment equations for the log-normal Darcy problem

MATHICSE Technical Report : On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices

In this technical report we apply the Continuation Multi Level Monte Carlo (C-MLMC) algorithm to efficiently propagate operating and geometric uncertainties in internal and external aerodynamic simulations modeled by RANS. In particular, we discuss the construction of suitable mesh hierarchies and test the C-MLMC algorithm on the 2D RAE-2822 transonic airfoil and the 3D NASA Rotor 37 affected by operating uncertainties

MATHICSE Technical Report : Continuation Multi-Level Monte-Carlo method for Uncertainty Quantification in Turbulent Compressible Aerodynamics Problems modeled by RANS

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