2,477 research outputs found
Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
We study the approximation of expectations \E(f(X)) for Gaussian random
elements with values in a separable Hilbert space and Lipschitz
continuous functionals . We consider restricted Monte Carlo
algorithms, which may only use random bits instead of random numbers. We
determine the asymptotics (in some cases sharp up to multiplicative constants,
in the other cases sharp up to logarithmic factors) of the corresponding -th
minimal error in terms of the decay of the eigenvalues of the covariance
operator of . It turns out that, within the margins from above, restricted
Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms,
and suitable random bit multilevel algorithms are optimal. The analysis of this
problem leads to a variant of the quantization problem, namely, the optimal
approximation of probability measures on by uniform distributions supported
by a given, finite number of points. We determine the asymptotics (up to
multiplicative constants) of the error of the best approximation for the
one-dimensional standard normal distribution, for Gaussian measures as above,
and for scalar autonomous SDEs
Random Bit Multilevel Algorithms for Stochastic Differential Equations
We study the approximation of expectations \E(f(X)) for solutions of
SDEs and functionals by means of restricted
Monte Carlo algorithms that may only use random bits instead of random numbers.
We consider the worst case setting for functionals from the Lipschitz class
w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm
and establish upper bounds for its error and cost. Furthermore, we derive
matching lower bounds, up to a logarithmic factor, that are valid for all
random bit Monte Carlo algorithms, and we show that, for the given quadrature
problem, random bit Monte Carlo algorithms are at least almost as powerful as
general randomized algorithms
Lattice methods for strongly interacting many-body systems
Lattice field theory methods, usually associated with non-perturbative
studies of quantum chromodynamics, are becoming increasingly common in the
calculation of ground-state and thermal properties of strongly interacting
non-relativistic few- and many-body systems, blurring the interfaces between
condensed matter, atomic and low-energy nuclear physics. While some of these
techniques have been in use in the area of condensed matter physics for a long
time, others, such as hybrid Monte Carlo and improved effective actions, have
only recently found their way across areas. With this topical review, we aim to
provide a modest overview and a status update on a few notable recent
developments. For the sake of brevity we focus on zero-temperature,
non-relativistic problems. After a short introduction, we lay out some general
considerations and proceed to discuss sampling algorithms, observables, and
systematic effects. We show selected results on ground- and excited-state
properties of fermions in the limit of unitarity. The appendix contains details
on group theory on the lattice.Comment: 64 pages, 32 figures; topical review for J. Phys. G; replaced with
published versio
Quantum Monte Carlo in the Interaction Representation --- Application to a Spin-Peierls Model
A quantum Monte Carlo algorithm is constructed starting from the standard
perturbation expansion in the interaction representation. The resulting
configuration space is strongly related to that of the Stochastic Series
Expansion (SSE) method, which is based on a direct power series expansion of
exp(-beta*H). Sampling procedures previously developed for the SSE method can
therefore be used also in the interaction representation formulation. The new
method is first tested on the S=1/2 Heisenberg chain. Then, as an application
to a model of great current interest, a Heisenberg chain including phonon
degrees of freedom is studied. Einstein phonons are coupled to the spins via a
linear modulation of the nearest-neighbor exchange. The simulation algorithm is
implemented in the phonon occupation number basis, without Hilbert space
truncations, and is exact. Results are presented for the magnetic properties of
the system in a wide temperature regime, including the T-->0 limit where the
chain undergoes a spin-Peierls transition. Some aspects of the phonon dynamics
are also discussed. The results suggest that the effects of dynamic phonons in
spin-Peierls compounds such as GeCuO3 and NaV2O5 must be included in order to
obtain a correct quantitative description of their magnetic properties, both
above and below the dimerization temperature.Comment: 23 pages, Revtex, 11 PostScript figure
Intermediate and extreme mass-ratio inspirals — astrophysics, science applications and detection using LISA
Black hole binaries with extreme (gtrsim104:1) or intermediate (~102–104:1) mass ratios are among the most interesting gravitational wave sources that are expected to be detected by the proposed laser interferometer space antenna (LISA). These sources have the potential to tell us much about astrophysics, but are also of unique importance for testing aspects of the general theory of relativity in the strong field regime. Here we discuss these sources from the perspectives of astrophysics, data analysis and applications to testing general relativity, providing both a description of the current state of knowledge and an outline of some of the outstanding questions that still need to be addressed. This review grew out of discussions at a workshop in September 2006 hosted by the Albert Einstein Institute in Golm, Germany
Randomized Complexity of Parametric Integration and the Role of Adaption II. Sobolev Spaces
We study the complexity of randomized computation of integrals depending on a
parameter, with integrands from Sobolev spaces. That is, for
, , , and we are given and we seek to
approximate with error measured in
the -norm. Our results extend previous work of Heinrich and
Sindambiwe (J.\ Complexity, 15 (1999), 317--341) for and Wiegand
(Shaker Verlag, 2006) for . Wiegand's analysis was carried out
under the assumption that is continuously embedded in
(embedding condition). We also study the case that the
embedding condition does not hold. For this purpose a new ingredient is
developed -- a stochastic discretization technique.
The paper is based on Part I, where vector valued mean computation -- the
finite-dimensional counterpart of parametric integration -- was studied.
In Part I a basic problem of Information-Based Complexity on the power of
adaption for linear problems in the randomized setting was solved. Here a
further aspect of this problem is settled.Comment: 32 page
On the Emerging Potential of Quantum Annealing Hardware for Combinatorial Optimization
Over the past decade, the usefulness of quantum annealing hardware for
combinatorial optimization has been the subject of much debate. Thus far,
experimental benchmarking studies have indicated that quantum annealing
hardware does not provide an irrefutable performance gain over state-of-the-art
optimization methods. However, as this hardware continues to evolve, each new
iteration brings improved performance and warrants further benchmarking. To
that end, this work conducts an optimization performance assessment of D-Wave
Systems' most recent Advantage Performance Update computer, which can natively
solve sparse unconstrained quadratic optimization problems with over 5,000
binary decision variables and 40,000 quadratic terms. We demonstrate that
classes of contrived problems exist where this quantum annealer can provide run
time benefits over a collection of established classical solution methods that
represent the current state-of-the-art for benchmarking quantum annealing
hardware. Although this work does not present strong evidence of an irrefutable
performance benefit for this emerging optimization technology, it does exhibit
encouraging progress, signaling the potential impacts on practical optimization
tasks in the future
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