291,962 research outputs found
On the performance of algorithms for the minimization of -penalized functionals
The problem of assessing the performance of algorithms used for the
minimization of an -penalized least-squares functional, for a range of
penalty parameters, is investigated. A criterion that uses the idea of
`approximation isochrones' is introduced. Five different iterative minimization
algorithms are tested and compared, as well as two warm-start strategies. Both
well-conditioned and ill-conditioned problems are used in the comparison, and
the contrast between these two categories is highlighted.Comment: 18 pages, 10 figures; v3: expanded version with an additional
synthetic test problem
Approximation Complexity of Complex-Weighted Degree-Two Counting Constraint Satisfaction Problems
Constraint satisfaction problems have been studied in numerous fields with
practical and theoretical interests. In recent years, major breakthroughs have
been made in a study of counting constraint satisfaction problems (or #CSPs).
In particular, a computational complexity classification of bounded-degree
#CSPs has been discovered for all degrees except for two, where the "degree" of
an input instance is the maximal number of times that each input variable
appears in a given set of constraints. Despite the efforts of recent studies,
however, a complexity classification of degree-2 #CSPs has eluded from our
understandings. This paper challenges this open problem and gives its partial
solution by applying two novel proof techniques--T_{2}-constructibility and
parametrized symmetrization--which are specifically designed to handle
"arbitrary" constraints under randomized approximation-preserving reductions.
We partition entire constraints into four sets and we classify the
approximation complexity of all degree-2 #CSPs whose constraints are drawn from
two of the four sets into two categories: problems computable in
polynomial-time or problems that are at least as hard as #SAT. Our proof
exploits a close relationship between complex-weighted degree-2 #CSPs and
Holant problems, which are a natural generalization of complex-weighted #CSPs.Comment: A4, 10pt, 23 pages. This is a complete version of the paper that
appeared in the Proceedings of the 17th Annual International Computing and
Combinatorics Conference (COCOON 2011), Lecture Notes in Computer Science,
vol.6842, pp.122-133, Dallas, Texas, USA, August 14-16, 201
A Dichotomy Theorem for the Approximate Counting of Complex-Weighted Bounded-Degree Boolean CSPs
We determine the computational complexity of approximately counting the total
weight of variable assignments for every complex-weighted Boolean constraint
satisfaction problem (or CSP) with any number of additional unary (i.e., arity
1) constraints, particularly, when degrees of input instances are bounded from
above by a fixed constant. All degree-1 counting CSPs are obviously solvable in
polynomial time. When the instance's degree is more than two, we present a
dichotomy theorem that classifies all counting CSPs admitting free unary
constraints into exactly two categories. This classification theorem extends,
to complex-weighted problems, an earlier result on the approximation complexity
of unweighted counting Boolean CSPs of bounded degree. The framework of the
proof of our theorem is based on a theory of signature developed from Valiant's
holographic algorithms that can efficiently solve seemingly intractable
counting CSPs. Despite the use of arbitrary complex weight, our proof of the
classification theorem is rather elementary and intuitive due to an extensive
use of a novel notion of limited T-constructibility. For the remaining degree-2
problems, in contrast, they are as hard to approximate as Holant problems,
which are a generalization of counting CSPs.Comment: A4, 10pt, 20 pages. This revised version improves its preliminary
version published under a slightly different title in the Proceedings of the
4th International Conference on Combinatorial Optimization and Applications
(COCOA 2010), Lecture Notes in Computer Science, Springer, Vol.6508 (Part I),
pp.285--299, Kailua-Kona, Hawaii, USA, December 18--20, 201
FPTAS for Counting Monotone CNF
A monotone CNF formula is a Boolean formula in conjunctive normal form where
each variable appears positively. We design a deterministic fully
polynomial-time approximation scheme (FPTAS) for counting the number of
satisfying assignments for a given monotone CNF formula when each variable
appears in at most clauses. Equivalently, this is also an FPTAS for
counting set covers where each set contains at most elements. If we allow
variables to appear in a maximum of clauses (or sets to contain
elements), it is NP-hard to approximate it. Thus, this gives a complete
understanding of the approximability of counting for monotone CNF formulas. It
is also an important step towards a complete characterization of the
approximability for all bounded degree Boolean #CSP problems. In addition, we
study the hypergraph matching problem, which arises naturally towards a
complete classification of bounded degree Boolean #CSP problems, and show an
FPTAS for counting 3D matchings of hypergraphs with maximum degree .
Our main technique is correlation decay, a powerful tool to design
deterministic FPTAS for counting problems defined by local constraints among a
number of variables. All previous uses of this design technique fall into two
categories: each constraint involves at most two variables, such as independent
set, coloring, and spin systems in general; or each variable appears in at most
two constraints, such as matching, edge cover, and holant problem in general.
The CNF problems studied here have more complicated structures than these
problems and require new design and proof techniques. As it turns out, the
technique we developed for the CNF problem also works for the hypergraph
matching problem. We believe that it may also find applications in other CSP or
more general counting problems.Comment: 24 pages, 2 figures. version 1=>2: minor edits, highlighted the
picture of set cover/packing, and an implication of our previous result in 3D
matchin
Approximation of high-dimensional parametric PDEs
Parametrized families of PDEs arise in various contexts such as inverse
problems, control and optimization, risk assessment, and uncertainty
quantification. In most of these applications, the number of parameters is
large or perhaps even infinite. Thus, the development of numerical methods for
these parametric problems is faced with the possible curse of dimensionality.
This article is directed at (i) identifying and understanding which properties
of parametric equations allow one to avoid this curse and (ii) developing and
analyzing effective numerical methodd which fully exploit these properties and,
in turn, are immune to the growth in dimensionality. The first part of this
article studies the smoothness and approximability of the solution map, that
is, the map where is the parameter value and is the
corresponding solution to the PDE. It is shown that for many relevant
parametric PDEs, the parametric smoothness of this map is typically holomorphic
and also highly anisotropic in that the relevant parameters are of widely
varying importance in describing the solution. These two properties are then
exploited to establish convergence rates of -term approximations to the
solution map for which each term is separable in the parametric and physical
variables. These results reveal that, at least on a theoretical level, the
solution map can be well approximated by discretizations of moderate
complexity, thereby showing how the curse of dimensionality is broken. This
theoretical analysis is carried out through concepts of approximation theory
such as best -term approximation, sparsity, and -widths. These notions
determine a priori the best possible performance of numerical methods and thus
serve as a benchmark for concrete algorithms. The second part of this article
turns to the development of numerical algorithms based on the theoretically
established sparse separable approximations. The numerical methods studied fall
into two general categories. The first uses polynomial expansions in terms of
the parameters to approximate the solution map. The second one searches for
suitable low dimensional spaces for simultaneously approximating all members of
the parametric family. The numerical implementation of these approaches is
carried out through adaptive and greedy algorithms. An a priori analysis of the
performance of these algorithms establishes how well they meet the theoretical
benchmarks
Krylov Methods for Large Sparse Systems: A Comprehensive Overview
In this paper are analyzed behavior and properties for different Krylov methods applied in different categories of problems. These categories often include PDEs, econometrics and network models, which are represented by large sparse systems. For our empirical analysis are taken into consideration size, the density of non-zero elements, symmetry/un-symmetry, eigenvalue distribution, also well/ill-conditioned and random systems. Convergence, approximation error and residuals are compared for the full version of methods, some restarted methods and preconditioned methods. Two preconditioners are considered respectively, ILU(0) and IC(0) by using at least five preconditioning techniques. In each case, empirical results show which technique is best to use based on properties of the system and are backed up by general theoretical information already found on Krylov space methods
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