This paper is concerned with the numerical solution of high-dimensional Fokker-Planck equations related to multi-dimensional diffusion with polynomial coefficients or Pearson diffusions. ...Classification of multi-dimensional Pearson diffusion follows from the classification of one-dimensional Pearson diffusion. There are six important classes of Pearson diffusion-three of them possess an infinite system of moments (Gaussian, Gamma, Beta) while the other three possess a finite number of moments (inverted Gamma, Student and Fisher-Snedecor). Numerical approximations to the solution of the Fokker-Planck equation are generated using the spectral method. The use of an adaptive reduced basis technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection stages. Numerical results with different boundary conditions are presented to demonstrate the accuracy and efficiency of the numerical scheme.
Schizophrenia is a highly heritable disorder. Genome-wide association studies based largely on common alleles have identified over 100 schizophrenia risk loci, but it is also evident from studies of ...copy number variants (CNVs) and from exome-sequencing studies that rare alleles are also involved. Full characterization of the contribution of rare alleles to the disorder awaits the deployment of sequencing technology in very large sample sizes, meanwhile, as an interim measure, exome arrays allow rare non-synonymous variants to be sampled at a fraction of the cost. In an analysis of exome array data from 13 688 individuals (5585 cases and 8103 controls) from the UK, we found that rare (minor allele frequency < 0.1%) variant association signal was enriched among genes that map to autosomal loci that are genome-wide significant (GWS) in common variant studies of schizophrenia genome-wide association study (PGWAS = 0.01) as well as gene sets known to be enriched for rare variants in sequencing studies (PRARE = 0.026). We also identified the gene-wise equivalent of GWS support for WDR88 (WD repeat-containing protein 88), a gene of unknown function (P = 6.5 × 10(-7)). Rare alleles represented on exome chip arrays contribute to the genetic architecture of schizophrenia, but as is the case for GWAS, very large studies are required to reveal additional susceptibility alleles for the disorder.
The numerical solution of the one-dimensional Fokker–Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer ...dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker–Planck equation, using the
d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.
Leaf area index (LAI) is one of the key parameters for the calculation of the energy budget, photosynthesis, and the interception of precipitation in land-surface models at local to global scales. ...Estimation of LAI from satellite data is a challenging and difficult problem. Studies over the past decades have focused predominantly on the improvement of forward modeling of the radiative transfer problem and on the application of more realistic numerical inversion schemes. Little or no attention has been paid to alternatives for the least squares method as a statistical distance measure or cost function, used to minimize the distance between observations and model predictions. The least-squares method has properties that assume noise with a Gaussian distribution and zero mean, an assumption often violated when LAI is estimated from satellite reflectance data. Here, we test the use of alternative statistical distance measures or cost functions to estimate LAI. We combine a look-up table (LUT)-inversion method based on the FLIGHT radiative transfer model and test how well it estimates LAI from MODIS reflectance data for a large set of alternative cost functions. We consider three classes of statistical distance measures or cost functions: information divergence measures, M-estimates, and minimum contrast methods. We estimate LAI from the Moderate Resolution Imaging Spectrometer (MODIS) surface reflectance product (MOD09GA) for 11 VALERI and BigFoot sites around the globe. These sites consist of a wide range of tree-cover types that include conifer, broadleaf and mixed (conifer, broadleaf, grassland) forest sites. We develop LUTs with FLIGHT for conifer and broadleaf forests and we show that improvements can be obtained for the estimation of LAI by choosing a cost function appropriate for a particular problem. Results show error reductions of 20% compared with the MODIS LAI retrieval (MOD15A2).
•60+ statistical distances were tested to estimate LAI from MODIS reflectances.•LAI inversions were based on LUTs generated by PROSPECT/FLIGHT.•LAI inversions improved when alternative cost functions were used.•LAI estimates improved compared to the MODIS LAI product (MOD15A2).
Correction to: Molecular Psychiatry advance online publication, 12 July 2016; doi:10.1038/mp.2016.97 The ninth author's name was presented incorrectly. It should have been listed as LF Jarskog.
In this paper different types of hyperbolic diffusions and their corresponding transient Fokker–Planck equation are described and numerical solutions are presented. Diffusion models were developed ...that can fit both the marginal distribution and correlation structure and they have found a wide application in finance, turbulence and environmental time series. Hyperbolic diffusions have a complicated structure and variety of parameters and are extremely difficult to study and to model. We propose a numerical technique that solves the one-dimensional hyperbolic Fokker–Planck equation in the time dependent case. Note that this is the first study where transient hyperbolic diffusions are considered. The numerical technique is based on an adaptive reduced basis method using a spectral element discretization. It involves enrichment and projection stages where an optimal basis is found in a dynamic way using the singular value decomposition (SVD). The approach dramatically reduces the number of degrees of freedom required to solve the problem. The numerical simulations of the Fokker–Planck equation are verified with available stationary solutions.
This paper studies the transient solution of the Markovian queue, namely
M
/
E
k
/
1
model. A new approach is introduced for finding exact solution for this queueing system. The transient ...probabilities are expressed recurrently.
The transient solution to M/Ek/1 queue GRIFFITHS, J. D; LEONENKO, G. M; WILLIAMS, J. E
Operations research letters,
05/2006, Letnik:
34, Številka:
3
Journal Article
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