113,696 research outputs found
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
Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems
In this paper we propose a general framework for the uncertainty
quantification of quantities of interest for high-contrast single-phase flow
problems. It is based on the generalized multiscale finite element method
(GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a
hierarchy of approximations of different resolution, whereas the latter gives
an efficient way to estimate quantities of interest using samples on different
levels. The number of basis functions in the online GMsFEM stage can be varied
to determine the solution resolution and the computational cost, and to
efficiently generate samples at different levels. In particular, it is cheap to
generate samples on coarse grids but with low resolution, and it is expensive
to generate samples on fine grids with high accuracy. By suitably choosing the
number of samples at different levels, one can leverage the expensive
computation in larger fine-grid spaces toward smaller coarse-grid spaces, while
retaining the accuracy of the final Monte Carlo estimate. Further, we describe
a multilevel Markov chain Monte Carlo method, which sequentially screens the
proposal with different levels of approximations and reduces the number of
evaluations required on fine grids, while combining the samples at different
levels to arrive at an accurate estimate. The framework seamlessly integrates
the multiscale features of the GMsFEM with the multilevel feature of the MLMC
methods following the work in \cite{ketelson2013}, and our numerical
experiments illustrate its efficiency and accuracy in comparison with standard
Monte Carlo estimates.Comment: 29 pages, 6 figure
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category
- …