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Assyr Abdulle

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Related publications (63)

Page 1 of 4

A posteriori error estimate in quantities of interest for the finite element heterogeneous multiscale method

Assyr Abdulle, Achim Nonnenmacher

We present an "a posteriori" error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro-to-micro formulation, where the macrosco ...

Wiley-Blackwell2013

Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: A fully discrete spacetime analysis

Assyr Abdulle, Gilles Vilmart

Numerical methods for parabolic homogenization problems combining finite element methods (FEMs) in space with Runge-Kutta methods in time are proposed. The space discretization is based on the coupling of macro and micro finite element methods following th ...

2012

Numerical methods for multiscale problems

Assyr Abdulle

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation: symplectic and multisymplectic integrators for Hamiltonian systems, symmetric integrators for reversible syst ...

EMS2006

Ensemble Kalman Filter For Multiscale Inverse Problems

Assyr Abdulle, Andrea Zanoni, Giacomo Garegnani

We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows us to reco ...

SIAM PUBLICATIONS2020

Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales

Assyr Abdulle

An analysis of a multiscale symmetric interior penalty discontinuous Galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed. This new method, first described in [A. Abdulle, C.R. Acad. Sci. Par ...

2012

A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations

Assyr Abdulle, Giacomo Rosilho De Souza

A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gra ...

EDP SCIENCES S A2019

High weak order methods for stochastic equations based on modified equations

Assyr Abdulle, Gilles Vilmart, David Cohen

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrat ...

Siam Publications2012

Explicit Stabilized Multirate Method For Stiff Differential Equations

Assyr Abdulle, Giacomo Rosilho De Souza

Stabilized Runge???Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge???Kutta me ...

AMER MATHEMATICAL SOC2022

Numerical methods for multilattices

Assyr Abdulle, Alexander Shapeev

Among the efficient numerical methods based on atomistic models, the quasi-continuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to mult ...

Siam Publications2012

Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

Assyr Abdulle, Giacomo Garegnani

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, ...

2020

A parabolic local problem with exponential decay of the resonance error for numerical homogenization

Assyr Abdulle, Doghonay Arjmand, Edoardo Paganoni

This paper aims at an accurate and efficient computation of effective quantities, e.g. the homogenized coefficients for approximating the solutions to partial differential equations with oscillatory coefficients. Typical multiscale methods are based on a m ...

WORLD SCIENTIFIC PUBL CO PTE LTD2021

Linearized numerical homogenization method for nonlinear monotone parabolic multiscale problems

Assyr Abdulle, Gilles Vilmart, Martin Ernst Huber

We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the mic ...

Siam Publications2015

Numerical methods for stochastic partial differential equations with multiple scales

Assyr Abdulle

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, An ...

2012

Effective velocity for transport in heterogeneous compressible flows with mean drift

Assyr Abdulle

Solving transport equations in heterogeneous flows might give rise to scale dependent transport behavior with effective large scale transport parameters differing from those found on smaller scales. For incompressible velocity fields, homogenization method ...

2008

Adaptive reduced basis finite element heterogeneous multiscale method

Assyr Abdulle, Yun Bai

An adaptive reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is proposed for elliptic problems with multiple scales. The multiscale method is based on the RB-FE-HMM introduced in [A. Abdulle, Y. Bai, Reduced basis finite element het ...

Elsevier2013

Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales

Assyr Abdulle, Martin Ernst Huber

A discontinuous Galerkin finite element heterogeneous multiscale method is proposed for advectiondiffusion problems with highly oscillatory coefficients. The method is based on a coupling of a discontinuous Galerkin discretization for an effective advectio ...

Springer2014

Drift Estimation of Multiscale Diffusions Based on Filtered Data

Assyr Abdulle, Andrea Zanoni, Giacomo Garegnani

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coeffic ...

2023

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

Assyr Abdulle, Gilles Vilmart

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L^2 and the H^1 ...

Springer2012

A probabilistic finite element method based on random meshes: A posteriori error estimators and Bayesian inverse problems

Assyr Abdulle, Giacomo Garegnani

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a pr ...

ELSEVIER SCIENCE SA2021

Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Assyr Abdulle, Timothée Noé Pouchon

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain Omega subset of R-d is proposed and analyzed. For a wave equation with highly oscillatory ...

World Scientific Publ Co Pte Ltd2016