15. Numerical Analysis and
Numerical Analysis and Scientific Computing
IL15.1
On a class of high order schemes for hyperbolic problems Rémi Abgrall
University of Zurich, Switzerland [email protected]
2010Mathematics Subject Classification. 65, 76
Keywords. Numerical approximation of hyperbolic problems, Non oscillatory schemes, Unstructured meshes, High order methods
This paper provides a review about a family of non oscillatory and parameter free finite ele- ment type methods for advection-diffusion problems. Due to space limitation, only the scalar hyperbolic problem is considered. We also show that this class of schemes can be interpreted as finite volume schemes with multidimensional fluxes.
IL15.4
Spline differential forms Annalisa Buffa
Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, C.N.R., Italy annalisa.buff[email protected]
2010Mathematics Subject Classification. 65N30, 65D07
Keywords. Numerical analysis, spline theory, discretization of partial differential equations
We introduce spline discretization of differential forms and study their properties. We analyse their geometric and topological structure, as related to the connectivity of the underlying mesh, we present degrees of freedom and we construct commuting projection operators, with optimal stability and approximation properties.
IL15.3
Multiscale model reduction with generalized multiscale finite element methods
Yalchin Efendiev
Texas A&M University, United States of America [email protected]
2010Mathematics Subject Classification. 65N99, 65N30
Keywords. Multiscale, finite element, model reduction, homogenization, porous media
Many application problems have multiscale nature. Due to disparity of scales, the simula- tions of these problems are prohibitively expensive. Some types of upscaling or model reduc- tion techniques are needed to solve many multiscale problems. In this talk, we discuss a few known techniques that are used for problems with scale separation and focus on Generalized Multiscale Finite Element Method (GMsFEM) that has been recently proposed for solving problems with non-separable scales and high contrast. The main objective of the method is to provide local reduced-order approximations for linear and nonlinear PDEs via multiscale
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International Congress of Mathematicians, Seoul, 2014
spaces on a coarse computational grid. In the talk, we briefly discuss some main concepts of constructing multiscale spaces and applications of GMsFEMs.
IL15.2
Discontinuous Galerkin method for time-dependent convection dominated partial differential equations
Chi-Wang Shu
Brown University, United States of America [email protected]
2010Mathematics Subject Classification. 65M60, 65M20, 65M12, 65M15
Keywords. Discontinuous Galerkin method, time-dependent convection dominated partial differential equations, hyperbolic equations, convection-diffusion equations, stability
In this lecture we give an introduction to discontinuous Galerkin (DG) methods for solv- ing time-dependent convection dominated partial differential equations (PDEs). DG meth- ods form a class of finite element methods. Differently from classical finite element meth- ods, which are built upon spaces containing continuous, piecewise polynomial functions, DG methods are built upon function spaces containing piecewise polynomials (or other simple functions) which are allowed to be completely discontinuous across element interfaces. Us- ing finite element terminologies, DG methods are the most extreme case of nonconforming finite element methods. DG methods are most natural and most successful for solving hy- perbolic conservation laws which have generic discontinuous solutions. Moreover, in recent years stable and convergent DG methods have also been designed for convection dominated PDEs containing higher order spatial derivatives, such as convection diffusion equations and KdV equations. We will emphasize the guiding principles for the design and analysis, and re- cent development and applications of the DG methods for solving time-dependent convection dominated PDEs.
IL15.5
Singular stochastic computational models, stochastic analysis, PDE analysis, and numerics
Denis Talay
Inria Sophia Antipolis, France [email protected]
2010Mathematics Subject Classification. 60H30, 60H35, 65C05, 65C30, 60C35
Keywords. Stochastic numerics, Applications of stochastic analysis to partial differential equations and numerical analysis
Stochastic computational models are aimed to simulate complex physical or biological phe- nomena and to approximate (deterministic) macroscopic physical quantities by means of probabilistic numerical methods. By nature, they often involve singularities and are subject to the curse of dimensionality. Their efficient and accurate simulation is still an open question in many aspects.
The aim of this lecture is to review some recent developments concerning the numerical
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approximation of singular stochastic dynamics, and to show novel issues in stochastic analysis and PDE analysis they lead to.
IL15.6
A review on subspace methods for nonlinear optimization Ya-xiang Yuan
Chinese Academy of Sciences, China [email protected]
2010Mathematics Subject Classification. 65K05, 90C30
Keywords. Numerical methods, nonlinear optimization, subspace techniques, subproblems
In this paper, we review various subspace techniques that have been used in constructing numerical methods for solving nonlinear optimization problems. As large scale optimization problems are attracting more and more attention in recent years, subspace methods are getting more and more important since they do not require solving large scale subproblems in each iteration. The essential parts of a subspace method are how to construct subproblems defined in lower dimensional subspaces and how to choose the subspaces in which the subproblems are defined. Various subspace methods for unconstrained optimization, constrained optimiza- tion, nonlinear equations and nonlinear least squares, and matrix optimization problems are given respectively, and different proposals are made on how to choose the subspaces.
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