6,880 research outputs found
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with
significantly extended numerical experiments. Some unused material is remove
Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems
Coupled problems with various combinations of multiple physics, scales, and
domains are found in numerous areas of science and engineering. A key challenge
in the formulation and implementation of corresponding coupled numerical models
is to facilitate the communication of information across physics, scale, and
domain interfaces, as well as between the iterations of solvers used for
response computations. In a probabilistic context, any information that is to
be communicated between subproblems or iterations should be characterized by an
appropriate probabilistic representation. Although the number of sources of
uncertainty can be expected to be large in most coupled problems, our
contention is that exchanged probabilistic information often resides in a
considerably lower dimensional space than the sources themselves. In this work,
we thus use a dimension-reduction technique for obtaining the representation of
the exchanged information. The main subject of this work is the investigation
of a measure-transformation technique that allows implementations to exploit
this dimension reduction to achieve computational gains. The effectiveness of
the proposed dimension-reduction and measure-transformation methodology is
demonstrated through a multiphysics problem relevant to nuclear engineering
Stochastic Testing Simulator for Integrated Circuits and MEMS: Hierarchical and Sparse Techniques
Process variations are a major concern in today's chip design since they can
significantly degrade chip performance. To predict such degradation, existing
circuit and MEMS simulators rely on Monte Carlo algorithms, which are typically
too slow. Therefore, novel fast stochastic simulators are highly desired. This
paper first reviews our recently developed stochastic testing simulator that
can achieve speedup factors of hundreds to thousands over Monte Carlo. Then, we
develop a fast hierarchical stochastic spectral simulator to simulate a complex
circuit or system consisting of several blocks. We further present a fast
simulation approach based on anchored ANOVA (analysis of variance) for some
design problems with many process variations. This approach can reduce the
simulation cost and can identify which variation sources have strong impacts on
the circuit's performance. The simulation results of some circuit and MEMS
examples are reported to show the effectiveness of our simulatorComment: Accepted to IEEE Custom Integrated Circuits Conference in June 2014.
arXiv admin note: text overlap with arXiv:1407.302
Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition
Hierarchical uncertainty quantification can reduce the computational cost of
stochastic circuit simulation by employing spectral methods at different
levels. This paper presents an efficient framework to simulate hierarchically
some challenging stochastic circuits/systems that include high-dimensional
subsystems. Due to the high parameter dimensionality, it is challenging to both
extract surrogate models at the low level of the design hierarchy and to handle
them in the high-level simulation. In this paper, we develop an efficient
ANOVA-based stochastic circuit/MEMS simulator to extract efficiently the
surrogate models at the low level. In order to avoid the curse of
dimensionality, we employ tensor-train decomposition at the high level to
construct the basis functions and Gauss quadrature points. As a demonstration,
we verify our algorithm on a stochastic oscillator with four MEMS capacitors
and 184 random parameters. This challenging example is simulated efficiently by
our simulator at the cost of only 10 minutes in MATLAB on a regular personal
computer.Comment: 14 pages (IEEE double column), 11 figure, accepted by IEEE Trans CAD
of Integrated Circuits and System
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