Realisation and approximation of linear infinite-dimensional systems with error bounds
SIAM Journal on Control and Optimization
A Multilinear Singular Value Decomposition
SIAM Journal on Matrix Analysis and Applications
On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns
Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns
Orthogonal Tensor Decompositions
SIAM Journal on Matrix Analysis and Applications
Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics
SIAM Journal on Numerical Analysis
SIAM Journal on Matrix Analysis and Applications
Algorithms for Numerical Analysis in High Dimensions
SIAM Journal on Scientific Computing
Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics)
Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
SIAM Journal on Scientific Computing
Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions
SIAM Journal on Scientific Computing
Singular value decompositions and low rank approximations of tensors
IEEE Transactions on Signal Processing
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The method of Proper Orthogonal Decompositions (POD) is a data-based method that is suitable for the reduction of large-scale distributed systems. In this paper we propose a generalization of the POD method so as to take the ND nature of a distributed model into account. This results in a novel procedure for model reduction of systems with multiple independent variables. Data in multiple independent variables is associated with the mathematical structure of a tensor. We show how orthonormal decompositions of this tensor can be used to derive suitable projection spaces. These projection spaces prove useful for determining reduced order models by performing Galerkin projections on equation residuals. We demonstrate how prior knowledge about the structure of the model reduction problem can be used to improve the quality of approximations. The tensor decomposition techniques are demonstrated on an application in data compression. The proposed model reduction procedure is illustrated on a heat diffusion problem.