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Random projection RBF nets for multidimensional density estimation

Ewa Skubalska-Rafajłowicz (2008)

International Journal of Applied Mathematics and Computer Science

The dimensionality and the amount of data that need to be processed when intensive data streams are observed grow rapidly together with the development of sensors arrays, CCD and CMOS cameras and other devices. The aim of this paper is to propose an approach to dimensionality reduction as a first stage of training RBF nets. As a vehicle for presenting the ideas, the problem of estimating multivariate probability densities is chosen. The linear projection method is briefly surveyed. Using random...

Recovery of an unknown flux in parabolic problems with nonstandard boundary conditions: Error estimates

Marián Slodička (2003)

Applications of Mathematics

In this paper, we consider a 2nd order semilinear parabolic initial boundary value problem (IBVP) on a bounded domain Ω ⊂ ℝ N , with nonstandard boundary conditions (BCs). More precisely, at some part of the boundary we impose a Neumann BC containing an unknown additive space-constant α ( t ) , accompanied with a nonlocal (integral) Dirichlet side condition. We design a numerical scheme for the approximation of a weak solution to the IBVP and derive error estimates for the approximation of the solution u and...

Reduced basis method for finite volume approximations of parametrized linear evolution equations

Bernard Haasdonk, Mario Ohlberger (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations....

Reduced basis solver for stochastic Galerkin formulation of Darcy flow with uncertain material parameters

Béreš, Michal (2023)

Programs and Algorithms of Numerical Mathematics

In this contribution, we present a solution to the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material coefficients in the separable form. The SG system of equations is kept in the compressed tensor form and its solution is a very challenging task. Here, we present the reduced basis (RB) method as a solver which looks for a low-rank representation of the solution. The construction of the RB consists of iterative expanding of the basis using Monte...

Reduced order controllers for Burgers' equation with a nonlinear observer

Jeanne Atwell, Jeffrey Borggaard, Belinda King (2001)

International Journal of Applied Mathematics and Computer Science

A method for reducing controllers for systems described by partial differential equations (PDEs) is applied to Burgers' equation with periodic boundary conditions. This approach differs from the typical approach of reducing the model and then designing the controller, and has developed over the past several years into its current form. In earlier work it was shown that functional gains for the feedback control law served well as a dataset for reduced order basis generation via the proper orthogonal...

Relating phase field and sharp interface approaches to structural topology optimization

Luise Blank, Harald Garcke, M. Hassan Farshbaf-Shaker, Vanessa Styles (2014)

ESAIM: Control, Optimisation and Calculus of Variations

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...

Relaxation models of phase transition flows

Philippe Helluy, Nicolas Seguin (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimization problem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.

Relaxation schemes for the multicomponent Euler system

Stéphane Dellacherie (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret...

Relaxation schemes for the multicomponent Euler system

Stéphane Dellacherie (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We show that it is possible to construct a class of entropic schemes for the multicomponent Euler system describing a gas or fluid homogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. A first order Chapman–Enskog expansion shows that the relaxed system formally converges when the relaxation frequencies go to the infinity toward a multicomponent Navier–Stokes system with the classical Fick and Newton laws, with a thermal diffusion which can be assimilated to a Soret...

Reliable numerical modelling of malaria propagation

István Faragó, Miklós Emil Mincsovics, Rahele Mosleh (2018)

Applications of Mathematics

We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some given qualitative property, namely, to be between zero and one. The conditions which guarantee this requirement are related to the time-discretization...

Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

Shanghui Jia, Deli Li, Tang Liu, Shu Hua Zhang (2008)

Applications of Mathematics

Asymptotic error expansions in the sense of L ∞ -norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing...

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