67,654 research outputs found
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Black hole determinants and quasinormal modes
We derive an expression for functional determinants in thermal spacetimes as
a product over the corresponding quasinormal modes. As simple applications we
give efficient computations of scalar determinants in thermal AdS, BTZ black
hole and de Sitter spacetimes. We emphasize the conceptual utility of our
formula for discussing `1/N' corrections to strongly coupled field theories via
the holographic correspondence.Comment: 28 pages. v2: slightly improved exposition, references adde
Fermions and Loops on Graphs. I. Loop Calculus for Determinant
This paper is the first in the series devoted to evaluation of the partition
function in statistical models on graphs with loops in terms of the
Berezin/fermion integrals. The paper focuses on a representation of the
determinant of a square matrix in terms of a finite series, where each term
corresponds to a loop on the graph. The representation is based on a fermion
version of the Loop Calculus, previously introduced by the authors for
graphical models with finite alphabets. Our construction contains two levels.
First, we represent the determinant in terms of an integral over anti-commuting
Grassman variables, with some reparametrization/gauge freedom hidden in the
formulation. Second, we show that a special choice of the gauge, called BP
(Bethe-Peierls or Belief Propagation) gauge, yields the desired loop
representation. The set of gauge-fixing BP conditions is equivalent to the
Gaussian BP equations, discussed in the past as efficient (linear scaling)
heuristics for estimating the covariance of a sparse positive matrix.Comment: 11 pages, 1 figure; misprints correcte
Dimers and the Critical Ising Model on Lattices of genus>1
We study the partition function of both Close-Packed Dimers and the Critical
Ising Model on a square lattice embedded on a genus two surface. Using
numerical and analytical methods we show that the determinants of the Kasteleyn
adjacency matrices have a dependence on the boundary conditions that, for large
lattice size, can be expressed in terms of genus two theta functions. The
period matrix characterizing the continuum limit of the lattice is computed
using a discrete holomorphic structure. These results relate in a direct way
the lattice combinatorics with conformal field theory, providing new insight to
the lattice regularization of conformal field theories on higher genus Riemann
Surfaces.Comment: 44 pages, eps figures included; typos corrected, figure and comments
added to section
Estimation of properties of low-lying excited states of Hubbard models : a multi-configurational symmetrized projector quantum Monte Carlo approach
We present in detail the recently developed multi-configurational symmetrized
projector quantum Monte Carlo (MSPQMC) method for excited states of the Hubbard
model. We describe the implementation of the Monte Carlo method for a
multi-configurational trial wavefunction. We give a detailed discussion of
issues related to the symmetry of the projection procedure which validates our
Monte Carlo procedure for excited states and leads naturally to the idea of
symmetrized sampling for correlation functions, developed earlier in the
context of ground state simulations. It also leads to three possible averaging
schemes. We have analyzed the errors incurred in these various averaging
procedures and discuss and detail the preferred averaging procedure for
correlations that do not have the full symmetry of the Hamiltonian. We study
the energies and correlation functions of the low-lying excited states of the
half-filled Hubbard model in 1-D. We have used this technique to study the
pair-binding energies of two holes in and systems, which compare
well the Bethe ansatz data of Fye, Martins and Scalettar. We have also studied
small clusters amenable to exact diagonalization studies in 2-D and have
reproduced their energies and correlation functions by the MSPQMC method. We
identify two ways in which a multiconfigurational trial wavefunction can lead
to a negative sign problem. We observe that this effect is not severe in 1-D
and tends to vanish with increasing system size. We also note that this does
not enhance the severity of the sign problem in two dimensions.Comment: 29 pages, 2 figures available on request, submitted to Phys. Rev.
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