124,790 research outputs found
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
An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator
In many applications, and in systems/synthetic biology, in particular, it is
desirable to compute control policies that force the trajectory of a bistable
system from one equilibrium (the initial point) to another equilibrium (the
target point), or in other words to solve the switching problem. It was
recently shown that, for monotone bistable systems, this problem admits
easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step
functions of fixed length and fixed magnitude). In this paper, we develop this
idea further and formulate a problem of convergence to an equilibrium from an
arbitrary initial point. We show that this problem can be solved using a static
optimization problem in the case of monotone systems. Changing the initial
point to an arbitrary state allows to build closed-loop, event-based or
open-loop policies for the switching/convergence problems. In our derivations
we exploit the Koopman operator, which offers a linear infinite-dimensional
representation of an autonomous nonlinear system. One of the main advantages of
using the Koopman operator is the powerful computational tools developed for
this framework. Besides the presence of numerical solutions, the
switching/convergence problem can also serve as a building block for solving
more complicated control problems and can potentially be applied to
non-monotone systems. We illustrate this argument on the problem of
synchronizing cardiac cells by defibrillation. Potentially, our approach can be
extended to problems with different parametrizations of control signals since
the only fundamental limitation is the finite time application of the control
signal.Comment: corrected typo
A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton-Jacobi Integrodifferential Systems on Networks
We study optimal control problems in infinite horizon when the dynamics
belong to a specific class of piecewise deterministic Markov processes
constrained to star-shaped networks (inspired by traffic models). We adapt the
results in [H. M. Soner. Optimal control with state-space constraint. II. SIAM
J. Control Optim., 24(6):1110.1122, 1986] to prove the regularity of the value
function and the dynamic programming principle. Extending the networks and
Krylov's ''shaking the coefficients'' method, we prove that the value function
can be seen as the solution to a linearized optimization problem set on a
convenient set of probability measures. The approach relies entirely on
viscosity arguments. As a by-product, the dual formulation guarantees that the
value function is the pointwise supremum over regular subsolutions of the
associated Hamilton-Jacobi integrodifferential system. This ensures that the
value function satisfies Perron's preconization for the (unique) candidate to
viscosity solution. Finally, we prove that the same kind of linearization can
be obtained by combining linearization for classical (unconstrained) problems
and cost penalization. The latter method works for very general near-viable
systems (possibly without further controllability) and discontinuous costs.Comment: accepted to Applied Mathematics and Optimization (01/10/2015
Sudden switch of generalized Lieb-Robinson velocity in a transverse field Ising spin chain
The Lieb-Robinson theorem states that the speed at which the correlations
between two distant nodes in a spin network can be built through local
interactions has an upper bound, which is called the Lieb-Robinson velocity.
Our central aim is to demonstrate how to observe the Lieb-Robinson velocity in
an Ising spin chain with a strong transverse field. We adopt and compare four
correlation measures for characterizing different types of correlations, which
include correlation function, mutual information, quantum discord, and
entanglement of formation. We prove that one of correlation functions shows a
special behavior depending on the parity of the spin number. All the
information-theoretical correlation measures demonstrate the existence of the
Lieb-Robinson velocity. In particular, we find that there is a sudden switch of
the Lieb-Robinson speed with the increasing of the number of spin
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