1,457 research outputs found
Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors
Sparsity has become a key concept for solving of high-dimensional inverse
problems using variational regularization techniques. Recently, using similar
sparsity-constraints in the Bayesian framework for inverse problems by encoding
them in the prior distribution has attracted attention. Important questions
about the relation between regularization theory and Bayesian inference still
need to be addressed when using sparsity promoting inversion. A practical
obstacle for these examinations is the lack of fast posterior sampling
algorithms for sparse, high-dimensional Bayesian inversion: Accessing the full
range of Bayesian inference methods requires being able to draw samples from
the posterior probability distribution in a fast and efficient way. This is
usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this
article, we develop and examine a new implementation of a single component
Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that
the efficiency of our Gibbs sampler increases when the level of sparsity or the
dimension of the unknowns is increased. This property is contrary to the
properties of the most commonly applied Metropolis-Hastings (MH) sampling
schemes: We demonstrate that the efficiency of MH schemes for L1-type priors
dramatically decreases when the level of sparsity or the dimension of the
unknowns is increased. Practically, Bayesian inversion for L1-type priors using
MH samplers is not feasible at all. As this is commonly believed to be an
intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also
challenges common beliefs about the applicability of sample based Bayesian
inference.Comment: 33 pages, 14 figure
Size dependent electronic properties of silicon quantum dots - an analysis with hybrid, screened hybrid and local density functional theory
We use an efficient projection scheme for the Fock operator to analyze the
size dependence of silicon quantum dots (QDs) electronic properties. We compare
the behavior of hybrid, screened hybrid and local density functionals as a
function of the dot size up to 800 silicon atoms and volume of up to
20nm. This allows comparing the calculations of hybrid and screened
hybrid functionals to experimental results over a wide range of QD sizes. We
demonstrate the size dependent behavior of the band gap, density of states,
ionization potential and HOMO level shift after ionization. Those results are
compared to experiment and to other theoretical approaches, such as
tight-binding, empirical pseudopotentials, TDDFT and GW
Analysing Ewald\u27s Method for the Evaluation of Green\u27s Functions for Periodic Media
Expressions for periodic Green\u27s functions for the Helmholtz equation in two and three dimensions are derived via Ewald\u27s method. The decay rate of the series occurring in these expressions is analysed and rigorous estimates for the remainder are derived when the series are truncated and replaced by finite sums. The effect of choosing a control parameter occurring in Ewald\u27s expressions is discussed and some recommendations for its choice are given based on the aforementioned estimates. We present various numerical examples for the resulting method for evaluating the Green\u27s functions. The results can also be carried over to evaluating the partial derivatives
- …