37,391 research outputs found
Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm
We present an iterative algorithm, called the symmetric tensor eigen-rank-one
iterative decomposition (STEROID), for decomposing a symmetric tensor into a
real linear combination of symmetric rank-1 unit-norm outer factors using only
eigendecompositions and least-squares fitting. Originally designed for a
symmetric tensor with an order being a power of two, STEROID is shown to be
applicable to any order through an innovative tensor embedding technique.
Numerical examples demonstrate the high efficiency and accuracy of the proposed
scheme even for large scale problems. Furthermore, we show how STEROID readily
solves a problem in nonlinear block-structured system identification and
nonlinear state-space identification
Multidimensional approximation of nonlinear dynamical systems
A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems
Tensor Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
Dynamic mode decomposition with control
We develop a new method which extends Dynamic Mode Decomposition (DMD) to
incorporate the effect of control to extract low-order models from
high-dimensional, complex systems. DMD finds spatial-temporal coherent modes,
connects local-linear analysis to nonlinear operator theory, and provides an
equation-free architecture which is compatible with compressive sensing. In
actuated systems, DMD is incapable of producing an input-output model;
moreover, the dynamics and the modes will be corrupted by external forcing. Our
new method, Dynamic Mode Decomposition with control (DMDc), capitalizes on all
of the advantages of DMD and provides the additional innovation of being able
to disambiguate between the underlying dynamics and the effects of actuation,
resulting in accurate input-output models. The method is data-driven in that it
does not require knowledge of the underlying governing equations, only
snapshots of state and actuation data from historical, experimental, or
black-box simulations. We demonstrate the method on high-dimensional dynamical
systems, including a model with relevance to the analysis of infectious disease
data with mass vaccination (actuation).Comment: 10 pages, 4 figure
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