Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

高等学校計算数学学報 = Numerical mathematics

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

http://arturo.imati.cnr.it/brezzi/papers/hitchhikers-preprint.pdf

The Hitchhiker's Guide to the Virtual Element Method

In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirao da Veiga et al., J. Comput. Phys., 228 (2009), pp. 7215--7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one.

/pdf/a-stream-virtual-element-formulation-of-the-stokes-problem-268pr0t0w7.pdf

A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes

In this paper we develop an evolution of the $C^1$ virtual elements of minimal degree for the approximation of the Cahn--Hilliard equation. The proposed method has the advantage of being conforming in $H^2$ and making use of a very simple set of degrees of freedom, namely, 3 degrees of freedom per vertex of the mesh. Moreover, although the present method is new also on triangles, it can make use of general polygonal meshes. As a theoretical and practical support, we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests.

A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes

In this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

The fully nonconforming virtual element method for biharmonic problems

We present a possible framework for the numerical simulation of flow in fractured porous media that couples mimetic finite differences for the porous matrix with a finite volume scheme for the flow in the fractures. The resulting method is theoretically analyzed in the case of a single fracture. Moreover, several numerical experiments show the capability of the method to deal also with complicated networks of fractures. Thanks to the implementation of rather general coupling conditions, it encompasses both “conductive fractures”, i.e. , fractures with high permeability and “sealed fractures”, i.e. , fractures with low permeability which act as a flow barrier.

/pdf/mimetic-finite-difference-approximation-of-flows-in-4apqgtzypr.pdf

Mimetic finite difference approximation of flows in fractured porous media

We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h) , where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.

/pdf/schwarz-domain-decomposition-preconditioners-for-4jyvcm0k5w.pdf

Paola F. Antonietti

Papers

A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes

A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes

The fully nonconforming virtual element method for biharmonic problems

Mimetic finite difference approximation of flows in fractured porous media

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case