27,034 research outputs found
Sampling and reconstruction of operators
We study the recovery of operators with bandlimited Kohn-Nirenberg symbol
from the action of such operators on a weighted impulse train, a procedure we
refer to as operator sampling. Kailath, and later Kozek and the authors have
shown that operator sampling is possible if the symbol of the operator is
bandlimited to a set with area less than one. In this paper we develop explicit
reconstruction formulas for operator sampling that generalize reconstruction
formulas for bandlimited functions. We give necessary and sufficient conditions
on the sampling rate that depend on size and geometry of the bandlimiting set.
Moreover, we show that under mild geometric conditions, classes of operators
bandlimited to an unknown set of area less than one-half permit sampling and
reconstruction. A similar result considering unknown sets of area less than one
was independently achieved by Heckel and Boelcskei.
Operators with bandlimited symbols have been used to model doubly dispersive
communication channels with slowly-time-varying impulse response. The results
in this paper are rooted in work by Bello and Kailath in the 1960s.Comment: Submitted to IEEE Transactions on Information Theor
Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information
We consider variants of trust-region and cubic regularization methods for
non-convex optimization, in which the Hessian matrix is approximated. Under
mild conditions on the inexact Hessian, and using approximate solution of the
corresponding sub-problems, we provide iteration complexity to achieve -approximate second-order optimality which have shown to be tight.
Our Hessian approximation conditions constitute a major relaxation over the
existing ones in the literature. Consequently, we are able to show that such
mild conditions allow for the construction of the approximate Hessian through
various random sampling methods. In this light, we consider the canonical
problem of finite-sum minimization, provide appropriate uniform and non-uniform
sub-sampling strategies to construct such Hessian approximations, and obtain
optimal iteration complexity for the corresponding sub-sampled trust-region and
cubic regularization methods.Comment: 32 page
Finite-Temperature Auxiliary-Field Quantum Monte Carlo for Bose-Fermi Mixtures
We present a quantum Monte Carlo (QMC) technique for calculating the exact
finite-temperature properties of Bose-Fermi mixtures. The Bose-Fermi
Auxiliary-Field Quantum Monte Carlo (BF-AFQMC) algorithm combines two methods,
a finite-temperature AFQMC algorithm for bosons and a variant of the standard
AFQMC algorithm for fermions, into one algorithm for mixtures. We demonstrate
the accuracy of our method by comparing its results for the Bose-Hubbard and
Bose-Fermi-Hubbard models against those produced using exact diagonalization
for small systems. Comparisons are also made with mean-field theory and the
worm algorithm for larger systems. As is the case with most fermion
Hamiltonians, a sign or phase problem is present in BF-AFQMC. We discuss the
nature of these problems in this framework and describe how they can be
controlled with well-studied approximations to expand BF-AFQMC's reach. The new
algorithm can serve as an essential tool for answering many unresolved
questions about many-body physics in mixed Bose-Fermi systems.Comment: 19 pages, 6 figure
Computing with functions in spherical and polar geometries I. The sphere
A collection of algorithms is described for numerically computing with smooth
functions defined on the unit sphere. Functions are approximated to essentially
machine precision by using a structure-preserving iterative variant of Gaussian
elimination together with the double Fourier sphere method. We show that this
procedure allows for stable differentiation, reduces the oversampling of
functions near the poles, and converges for certain analytic functions.
Operations such as function evaluation, differentiation, and integration are
particularly efficient and can be computed by essentially one-dimensional
algorithms. A highlight is an optimal complexity direct solver for Poisson's
equation on the sphere using a spectral method. Without parallelization, we
solve Poisson's equation with million degrees of freedom in one minute on
a standard laptop. Numerical results are presented throughout. In a companion
paper (part II) we extend the ideas presented here to computing with functions
on the disk.Comment: 23 page
Cornerstones of Sampling of Operator Theory
This paper reviews some results on the identifiability of classes of
operators whose Kohn-Nirenberg symbols are band-limited (called band-limited
operators), which we refer to as sampling of operators. We trace the motivation
and history of the subject back to the original work of the third-named author
in the late 1950s and early 1960s, and to the innovations in spread-spectrum
communications that preceded that work. We give a brief overview of the NOMAC
(Noise Modulation and Correlation) and Rake receivers, which were early
implementations of spread-spectrum multi-path wireless communication systems.
We examine in detail the original proof of the third-named author
characterizing identifiability of channels in terms of the maximum time and
Doppler spread of the channel, and do the same for the subsequent
generalization of that work by Bello.
The mathematical limitations inherent in the proofs of Bello and the third
author are removed by using mathematical tools unavailable at the time. We
survey more recent advances in sampling of operators and discuss the
implications of the use of periodically-weighted delta-trains as identifiers
for operator classes that satisfy Bello's criterion for identifiability,
leading to new insights into the theory of finite-dimensional Gabor systems. We
present novel results on operator sampling in higher dimensions, and review
implications and generalizations of the results to stochastic operators, MIMO
systems, and operators with unknown spreading domains
Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems
This work focuses on providing accurate low-cost approximations of stochastic Âżnite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments.Postprint (author's final draft
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