Fourier Analysis of Correlated Monte Carlo Importance Sampling

International audienceFourier analysis is gaining popularity in image synthesis, as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyze convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling-based integrators to unify non-uniform sample distributions (importance sampling) as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier-based tools in performing variance analysis for MC integration. This non-trivial exercise also provides exciting insight into the effects of importance sampling on the convergence rate of estimators because of the introduction or removal of discontinuities. Specifically, we demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy that converges slowest. We propose two simple and practical approaches to limit the impact of discontinuities on the convergence rate of estimators: The first one involves mirroring the integrand to cancel out the effect of boundary discontinuities. This is followed by two novel mirror sampling techniques for MC estimation in this mirrored domain. The second approach improves direct illumination light sampling by smoothing out discontinuities within the domain at the cost of introducing a small amount of bias. Our approaches are simple, practical and can be easily incorporated in production renderers

Application of Monte Carlo Algorithms to the Bayesian Analysis of the Cosmic Microwave Background

Power spectrum estimation and evaluation of associated errors in the presence of incomplete sky coverage; non-homogeneous, correlated instrumental noise; and foreground emission is a problem of central importance for the extraction of cosmological information from the cosmic microwave background. We develop a Monte Carlo approach for the maximum likelihood estimation of the power spectrum. The method is based on an identity for the Bayesian posterior as a marginalization over unknowns. Maximization of the posterior involves the computation of expectation values as a sample average from maps of the cosmic microwave background and foregrounds given some current estimate of the power spectrum or cosmological model, and some assumed statistical characterization of the foregrounds. Maps of the CMB are sampled by a linear transform of a Gaussian white noise process, implemented numerically with conjugate gradient descent. For time series data with N_{t} samples, and N pixels on the sphere, the method has a computational expense $KO[N^{2} +- N_{t} +AFw-log N_{t}], where K is a prefactor determined by the convergence rate of conjugate gradient descent. Preconditioners for conjugate gradient descent are given for scans close to great circle paths, and the method allows partial sky coverage for these cases by numerically marginalizing over the unobserved, or removed, region.Comment: submitted to Ap

Methods for Bayesian power spectrum inference with galaxy surveys

We derive and implement a full Bayesian large scale structure inference method aiming at precision recovery of the cosmological power spectrum from galaxy redshift surveys. Our approach improves over previous Bayesian methods by performing a joint inference of the three dimensional density field, the cosmological power spectrum, luminosity dependent galaxy biases and corresponding normalizations. We account for all joint and correlated uncertainties between all inferred quantities. Classes of galaxies with different biases are treated as separate sub samples. The method therefore also allows the combined analysis of more than one galaxy survey. In particular, it solves the problem of inferring the power spectrum from galaxy surveys with non-trivial survey geometries by exploring the joint posterior distribution with efficient implementations of multiple block Markov chain and Hybrid Monte Carlo methods. Our Markov sampler achieves high statistical efficiency in low signal to noise regimes by using a deterministic reversible jump algorithm. We test our method on an artificial mock galaxy survey, emulating characteristic features of the Sloan Digital Sky Survey data release 7, such as its survey geometry and luminosity dependent biases. These tests demonstrate the numerical feasibility of our large scale Bayesian inference frame work when the parameter space has millions of dimensions. The method reveals and correctly treats the anti-correlation between bias amplitudes and power spectrum, which are not taken into account in current approaches to power spectrum estimation, a 20 percent effect across large ranges in k-space. In addition, the method results in constrained realizations of density fields obtained without assuming the power spectrum or bias parameters in advance

Condensate fraction in liquid 4He at zero temperature

We present results of the one-body density matrix (OBDM) and the condensate fraction n_0 of liquid 4He calculated at zero temperature by means of the Path Integral Ground State Monte Carlo method. This technique allows to generate a highly accurate approximation for the ground state wave function Psi_0 in a totally model-independent way, that depends only on the Hamiltonian of the system and on the symmetry properties of Psi_0. With this unbiased estimation of the OBDM, we obtain precise results for the condensate fraction n_0 and the kinetic energy K of the system. The dependence of n_0 with the pressure shows an excellent agreement of our results with recent experimental measurements. Above the melting pressure, overpressurized liquid 4He shows a small condensate fraction that has dropped to 0.8% at the highest pressure of p = 87 bar.Comment: 12 pages. 4 figures. Accepted for publication on "Journal of Low Temperature Physics

Gravitational waves from coalescing binaries: detection strategies and Monte Carlo estimation of parameters

The paper deals with issues pertaining the detection of gravitational waves from coalescing binaries. We introduce the application of differential geometry to the problem of optimal detection of the `chirp signal'. We have also carried out extensive Monte Carlo simulations to understand the errors in the estimation of parameters of the binary system. We find that the errors are much more than those predicted by the covariance matrix even at a high SNR of 10-15. We also introduce the idea of using the instant of coalescence rather than the time of arrival to determine the direction to the source.Comment: 28 pages, REVTEX, 12 figures (bundled via uufiles command along with this paper) submitted to Phys. Rev.

Monte Carlo Analysis of a New Interatomic Potential for He

By means of a Quadratic Diffusion Monte Carlo method we have performed a comparative analysis between the Aziz potential and a revised version of it. The results demonstrate that the new potential produces a better description of the equation of state for liquid $^4$He. In spite of the improvement in the description of derivative magnitudes of the energy, as the pressure or the compressibility, the energy per particle which comes from this new potential is lower than the experimental one. The inclusion of three-body interactions, which give a repulsive contribution to the potential energy, makes it feasible that the calculated energy comes close to the experimental result.Comment: 36 pages, LaTex, 11 PostScript figures include