23,604 research outputs found
Using the distribution of cells by dimension in a cylindrical algebraic decomposition
We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
This insight is then combined with an algorithm that produces only
full-dimensional cells to give an accurate method of predicting the number of
cells in a complete CAD. Since constructing only full-dimensional cells is
relatively inexpensive (involving no costly algebraic number calculations) this
leads to heuristics for helping with various questions of problem formulation
for CAD, such as choosing an optimal variable ordering. Our experiments
demonstrate that this approach can be highly effective.Comment: 8 page
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
Far-Field Compression for Fast Kernel Summation Methods in High Dimensions
We consider fast kernel summations in high dimensions: given a large set of
points in dimensions (with ) and a pair-potential function (the
{\em kernel} function), we compute a weighted sum of all pairwise kernel
interactions for each point in the set. Direct summation is equivalent to a
(dense) matrix-vector multiplication and scales quadratically with the number
of points. Fast kernel summation algorithms reduce this cost to log-linear or
linear complexity.
Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by
constructing approximate representations of interactions of points that are far
from each other. In algebraic terms, these representations correspond to
low-rank approximations of blocks of the overall interaction matrix. Existing
approaches require an excessive number of kernel evaluations with increasing
and number of points in the dataset.
To address this issue, we use a randomized algebraic approach in which we
first sample the rows of a block and then construct its approximate, low-rank
interpolative decomposition. We examine the feasibility of this approach
theoretically and experimentally. We provide a new theoretical result showing a
tighter bound on the reconstruction error from uniformly sampling rows than the
existing state-of-the-art. We demonstrate that our sampling approach is
competitive with existing (but prohibitively expensive) methods from the
literature. We also construct kernel matrices for the Laplacian, Gaussian, and
polynomial kernels -- all commonly used in physics and data analysis. We
explore the numerical properties of blocks of these matrices, and show that
they are amenable to our approach. Depending on the data set, our randomized
algorithm can successfully compute low rank approximations in high dimensions.
We report results for data sets with ambient dimensions from four to 1,000.Comment: 43 pages, 21 figure
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
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
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