Publication

Linearized numerical homogenization method for nonlinear monotone parabolic multiscale problems

Assyr Abdulle, Gilles Vilmart, Martin Ernst Huber
2015
Journal Articles

Abstract

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 microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the methodfor various nonlinear problems.

Official source

https://infoscience.epfl.ch/entities/publication/5c8faa73-2f4e-4916-ad53-03cdc5d02f71

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Assyr Abdulle, Gilles Vilmart, Martin Ernst Huber
2015
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/5c8faa73-2f4e-4916-ad53-03cdc5d02f71

About this result

Related concepts (20)

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

Mean squared error

In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate.

Related publications (28)

Quantifying the Unknown: Data-Driven Approaches and Applications in Energy Systems

Paul Scharnhorst

In light of the challenges posed by climate change and the goals of the Paris Agreement, electricity generation is shifting to a more renewable and decentralized pattern, while the operation of systems like buildings is increasingly electrified. This calls ...

EPFL2024

SIMPLE HIERARCHICAL PLANNING WITH DIFFUSION

Chang Chen, Caglar Gulcehre

Diffusion-based generative methods have proven effective in modeling trajectories with offline datasets. However, they often face computational challenges and can falter in generalization, especially in capturing temporal abstractions for long- horizon tas ...

2024

Iterative removal of sources to model the turbulent electromotive force

Jennifer Waltraud Schober, Abhijit Bhausaheb Bendre

We describe a novel method to compute the components of dynamo tensors from direct magnetohydrodynamic (MHD) simulations. Our method relies upon an extension and generalization of the standard H & ouml;gbom CLEAN algorithm widely used in radio astronomy to ...

Oxford Univ Press2024