6,033 research outputs found
PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator
PyKoopman is a Python package for the data-driven approximation of the
Koopman operator associated with a dynamical system. The Koopman operator is a
principled linear embedding of nonlinear dynamics and facilitates the
prediction, estimation, and control of strongly nonlinear dynamics using linear
systems theory. In particular, PyKoopman provides tools for data-driven system
identification for unforced and actuated systems that build on the
equation-free dynamic mode decomposition (DMD) and its variants. In this work,
we provide a brief description of the mathematical underpinnings of the Koopman
operator, an overview and demonstration of the features implemented in
PyKoopman (with code examples), practical advice for users, and a list of
potential extensions to PyKoopman. Software is available at
http://github.com/dynamicslab/pykoopmanComment: 16 page
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
Randomized Dynamic Mode Decomposition
This paper presents a randomized algorithm for computing the near-optimal
low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging
techniques to compute low-rank matrix approximations at a fraction of the cost
of deterministic algorithms, easing the computational challenges arising in the
area of `big data'. The idea is to derive a small matrix from the
high-dimensional data, which is then used to efficiently compute the dynamic
modes and eigenvalues. The algorithm is presented in a modular probabilistic
framework, and the approximation quality can be controlled via oversampling and
power iterations. The effectiveness of the resulting randomized DMD algorithm
is demonstrated on several benchmark examples of increasing complexity,
providing an accurate and efficient approach to extract spatiotemporal coherent
structures from big data in a framework that scales with the intrinsic rank of
the data, rather than the ambient measurement dimension. For this work we
assume that the dynamics of the problem under consideration is evolving on a
low-dimensional subspace that is well characterized by a fast decaying singular
value spectrum
- …