4,664 research outputs found
Adaptive Mesh Refinement for Characteristic Grids
I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when
numerically solving partial differential equations with wave-like solutions,
using characteristic (double-null) grids. Such AMR algorithms are naturally
recursive, and the best-known past Berger-Oliger characteristic AMR algorithm,
that of Pretorius & Lehner (J. Comp. Phys. 198 (2004), 10), recurses on
individual "diamond" characteristic grid cells. This leads to the use of
fine-grained memory management, with individual grid cells kept in
2-dimensional linked lists at each refinement level. This complicates the
implementation and adds overhead in both space and time.
Here I describe a Berger-Oliger characteristic AMR algorithm which instead
recurses on null \emph{slices}. This algorithm is very similar to the usual
Cauchy Berger-Oliger algorithm, and uses relatively coarse-grained memory
management, allowing entire null slices to be stored in contiguous arrays in
memory. The algorithm is very efficient in both space and time.
I describe discretizations yielding both 2nd and 4th order global accuracy.
My code implementing the algorithm described here is included in the electronic
supplementary materials accompanying this paper, and is freely available to
other researchers under the terms of the GNU general public license.Comment: 37 pages, 15 figures (40 eps figure files, 8 of them color; all are
viewable ok in black-and-white), 1 mpeg movie, uses Springer-Verlag svjour3
document class, includes C++ source code. Changes from v1: revised in
response to referee comments: many references added, new figure added to
better explain the algorithm, other small changes, C++ code updated to latest
versio
An exercise in transformational programming: Backtracking and Branch-and-Bound
We present a formal derivation of program schemes that are usually called Backtracking programs and Branch-and-Bound programs. The derivation consists of a series of transformation steps, specifically algebraic manipulations, on the initial specification until the desired programs are obtained. The well-known notions of linear recursion and tail recursion are extended, for structures, to elementwise linear recursion and elementwise tail recursion; and a transformation between them is derived too
On-the-fly reduction of open loops
Building on the open-loop algorithm we introduce a new method for the
automated construction of one-loop amplitudes and their reduction to scalar
integrals. The key idea is that the factorisation of one-loop integrands in a
product of loop segments makes it possible to perform various operations
on-the-fly while constructing the integrand. Reducing the integrand on-the-fly,
after each segment multiplication, the construction of loop diagrams and their
reduction are unified in a single numerical recursion. In this way we entirely
avoid objects with high tensor rank, thereby reducing the complexity of the
calculations in a drastic way. Thanks to the on-the-fly approach, which is
applied also to helicity summation and for the merging of different diagrams,
the speed of the original open-loop algorithm can be further augmented in a
very significant way. Moreover, addressing spurious singularities of the
employed reduction identities by means of simple expansions in rank-two Gram
determinants, we achieve a remarkably high level of numerical stability. These
features of the new algorithm, which will be made publicly available in a
forthcoming release of the OpenLoops program, are particularly attractive for
NLO multi-leg and NNLO real-virtual calculations.Comment: v2 as accepted by EPJ C: extended discussion of the triangle
reduction and its numerical stability in section 5.4.2; speed benchmarks for
2->5 processes included in section 6.2.1; ref. adde
On space efficiency of algorithms working on structural decompositions of graphs
Dynamic programming on path and tree decompositions of graphs is a technique
that is ubiquitous in the field of parameterized and exponential-time
algorithms. However, one of its drawbacks is that the space usage is
exponential in the decomposition's width. Following the work of Allender et al.
[Theory of Computing, '14], we investigate whether this space complexity
explosion is unavoidable. Using the idea of reparameterization of Cai and
Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely
related to a conjecture that the Longest Common Subsequence problem
parameterized by the number of input strings does not admit an algorithm that
simultaneously uses XP time and FPT space. Moreover, we complete the complexity
landscape sketched for pathwidth and treewidth by Allender et al. by
considering the parameter tree-depth. We prove that computations on tree-depth
decompositions correspond to a model of non-deterministic machines that work in
polynomial time and logarithmic space, with access to an auxiliary stack of
maximum height equal to the decomposition's depth. Together with the results of
Allender et al., this describes a hierarchy of complexity classes for
polynomial-time non-deterministic machines with different restrictions on the
access to working space, which mirrors the classic relations between treewidth,
pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new
version is augmented with a space-efficient algorithm for Dominating Set
using the Chinese remainder theore
Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients
Building on an insight due to Avramidi, we provide a system of transport
equations for determining key fundamental bi-tensors, including derivatives of
the world-function, \sigma(x,x'), the square root of the Van Vleck determinant,
\Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard form
of the Green function. These bi-tensors are central to a broad range of
problems from radiation reaction to quantum field theory in curved spacetime
and quantum gravity. Their transport equations may be used either in a
semi-recursive approach to determining their covariant Taylor series
expansions, or as the basis of numerical calculations. To illustrate the power
of the semi-recursive approach, we present an implementation in
\textsl{Mathematica} which computes very high order covariant series expansions
of these objects. Using this code, a moderate laptop can, for example,
calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20}
in a matter of minutes. Results may be output in either a compact notation or
in xTensor form. In a second application of the approach, we present a scheme
for numerically integrating the transport equations as a system of coupled
ordinary differential equations. As an example application of the scheme, we
integrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschild
spacetimes.Comment: 32 pages, 5 figures. Final published version with correction to Eq.
(3.24
Relating goal scheduling, precedence, and memory management in and-parallel execution of logic programs
The interactions among three important issues involved in the implementation of logic programs in parallel (goal scheduling, precedence, and memory management) are discussed. A simplified, parallel memory management model and an efficient, load-balancing goal scheduling strategy are presented. It is shown how, for systems which support "don't know" non-determinism, special care has to be taken during goal scheduling if the space recovery characteristics
of sequential systems are to be preserved. A solution based on selecting only "newer" goals for execution is described, and an algorithm is proposed for efficiently maintaining and determining precedence relationships and variable ages across parallel goals. It is argued that the proposed schemes and algorithms make it possible to extend the storage performance of sequential systems to parallel execution without the considerable overhead previously associated with it. The results are applicable to a wide class of parallel and coroutining systems, and they represent an efficient alternative to "all heap" or "spaghetti stack" allocation models
Optical absorption and single-particle excitations in the 2D Holstein t-J model
To discuss the interplay of electronic and lattice degrees of freedom in
systems with strong Coulomb correlations we have performed an extensive
numerical study of the two-dimensional Holstein t-J model. The model describes
the interaction of holes, doped in a quantum antiferromagnet, with a
dispersionsless optical phonon mode. We apply finite-lattice Lanczos
diagonalization, combined with a well-controlled phonon Hilbert space
truncation, to the Hamiltonian. The focus is on the dynamical properties. In
particular we have evaluated the single-particle spectral function and the
optical conductivity for characteristic hole-phonon couplings, spin exchange
interactions and phonon frequencies. The results are used to analyze the
formation of hole polarons in great detail. Links with experiments on layered
perovskites are made. Supplementary we compare the Chebyshev recursion and
maximum entropy algorithms, used for calculating spectral functions, with
standard Lanczos methods.Comment: 32 pages, 12 figures, submitted to Phys. Rev.
Greedy adaptive walks on a correlated fitness landscape
We study adaptation of a haploid asexual population on a fitness landscape
defined over binary genotype sequences of length . We consider greedy
adaptive walks in which the population moves to the fittest among all single
mutant neighbors of the current genotype until a local fitness maximum is
reached. The landscape is of the rough mount Fuji type, which means that the
fitness value assigned to a sequence is the sum of a random and a deterministic
component. The random components are independent and identically distributed
random variables, and the deterministic component varies linearly with the
distance to a reference sequence. The deterministic fitness gradient is a
parameter that interpolates between the limits of an uncorrelated random
landscape () and an effectively additive landscape ().
When the random fitness component is chosen from the Gumbel distribution,
explicit expressions for the distribution of the number of steps taken by the
greedy walk are obtained, and it is shown that the walk length varies
non-monotonically with the strength of the fitness gradient when the starting
point is sufficiently close to the reference sequence. Asymptotic results for
general distributions of the random fitness component are obtained using
extreme value theory, and it is found that the walk length attains a
non-trivial limit for , different from its values for and
, if is scaled with in an appropriate combination.Comment: minor change
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