158,330 research outputs found
Constrained correlation functions from the Millennium Simulation
Context. In previous work, we developed a quasi-Gaussian approximation for
the likelihood of correlation functions, which, in contrast to the usual
Gaussian approach, incorporates fundamental mathematical constraints on
correlation functions. The analytical computation of these constraints is only
feasible in the case of correlation functions of one-dimensional random fields.
Aims. In this work, we aim to obtain corresponding constraints in the case of
higher-dimensional random fields and test them in a more realistic context.
Methods. We develop numerical methods to compute the constraints on
correlation functions which are also applicable for two- and three-dimensional
fields. In order to test the accuracy of the numerically obtained constraints,
we compare them to the analytical results for the one-dimensional case.
Finally, we compute correlation functions from the halo catalog of the
Millennium Simulation, check whether they obey the constraints, and examine the
performance of the transformation used in the construction of the
quasi-Gaussian likelihood.
Results. We find that our numerical methods of computing the constraints are
robust and that the correlation functions measured from the Millennium
Simulation obey them. Despite the fact that the measured correlation functions
lie well inside the allowed region of parameter space, i.e. far away from the
boundaries of the allowed volume defined by the constraints, we find strong
indications that the quasi-Gaussian likelihood yields a substantially more
accurate description than the Gaussian one.Comment: 11 pages, 13 figures, updated to match version accepted by A&
Integer Polynomial Optimization in Fixed Dimension
We classify, according to their computational complexity, integer
optimization problems whose constraints and objective functions are polynomials
with integer coefficients and the number of variables is fixed. For the
optimization of an integer polynomial over the lattice points of a convex
polytope, we show an algorithm to compute lower and upper bounds for the
optimal value. For polynomials that are non-negative over the polytope, these
sequences of bounds lead to a fully polynomial-time approximation scheme for
the optimization problem.Comment: In this revised version we include a stronger complexity bound on our
algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time
approximation scheme) to maximize a non-negative integer polynomial over the
lattice points of a polytop
Reversibility and Adiabatic Computation: Trading Time and Space for Energy
Future miniaturization and mobilization of computing devices requires energy
parsimonious `adiabatic' computation. This is contingent on logical
reversibility of computation. An example is the idea of quantum computations
which are reversible except for the irreversible observation steps. We propose
to study quantitatively the exchange of computational resources like time and
space for irreversibility in computations. Reversible simulations of
irreversible computations are memory intensive. Such (polynomial time)
simulations are analysed here in terms of `reversible' pebble games. We show
that Bennett's pebbling strategy uses least additional space for the greatest
number of simulated steps. We derive a trade-off for storage space versus
irreversible erasure. Next we consider reversible computation itself. An
alternative proof is provided for the precise expression of the ultimate
irreversibility cost of an otherwise reversible computation without
restrictions on time and space use. A time-irreversibility trade-off hierarchy
in the exponential time region is exhibited. Finally, extreme
time-irreversibility trade-offs for reversible computations in the thoroughly
unrealistic range of computable versus noncomputable time-bounds are given.Comment: 30 pages, Latex. Lemma 2.3 should be replaced by the slightly better
``There is a winning strategy with pebbles and erasures for
pebble games with , for all '' with appropriate
further changes (as pointed out by Wim van Dam). This and further work on
reversible simulations as in Section 2 appears in quant-ph/970300
Computation of generalized matrix functions
We develop numerical algorithms for the efficient evaluation of quantities
associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel,
Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are
based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants
are also investigated. Numerical experiments are performed to illustrate the
effectiveness and efficiency of our techniques in computing generalized matrix
functions arising in the analysis of networks.Comment: 25 paged, 2 figure
Accelerating the CM method
Given a prime q and a negative discriminant D, the CM method constructs an
elliptic curve E/\Fq by obtaining a root of the Hilbert class polynomial H_D(X)
modulo q. We consider an approach based on a decomposition of the ring class
field defined by H_D, which we adapt to a CRT setting. This yields two
algorithms, each of which obtains a root of H_D mod q without necessarily
computing any of its coefficients. Heuristically, our approach uses
asymptotically less time and space than the standard CM method for almost all
D. Under the GRH, and reasonable assumptions about the size of log q relative
to |D|, we achieve a space complexity of O((m+n)log q) bits, where mn=h(D),
which may be as small as O(|D|^(1/4)log q). The practical efficiency of the
algorithms is demonstrated using |D| > 10^16 and q ~ 2^256, and also |D| >
10^15 and q ~ 2^33220. These examples are both an order of magnitude larger
than the best previous results obtained with the CM method.Comment: 36 pages, minor edits, to appear in the LMS Journal of Computation
and Mathematic
Gradient-Bounded Dynamic Programming with Submodular and Concave Extensible Value Functions
We consider dynamic programming problems with finite, discrete-time horizons
and prohibitively high-dimensional, discrete state-spaces for direct
computation of the value function from the Bellman equation. For the case that
the value function of the dynamic program is concave extensible and submodular
in its state-space, we present a new algorithm that computes deterministic
upper and stochastic lower bounds of the value function similar to dual dynamic
programming. We then show that the proposed algorithm terminates after a finite
number of iterations. Finally, we demonstrate the efficacy of our approach on a
high-dimensional numerical example from delivery slot pricing in attended home
delivery.Comment: 6 pages, 2 figures, accepted for IFAC World Congress 202
First steps towards an imprecise Poisson process
The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here consider a rate interval and let it characterise two nested sets of stochastic processes. We call these two sets of stochastic process imprecise Poisson processes, explain why this is justified, and study the corresponding lower and upper (conditional) expectations. Besides a general theoretical framework, we also provide practical methods to compute lower and upper (conditional) expectations of functions that depend on the number of events at a single point in time
- …