Robust and optimal control
A multiscale finite element method for elliptic problems in composite materials and porous media
Journal of Computational Physics
Convergence of a Nonconforming Multiscale Finite Element Method
SIAM Journal on Numerical Analysis
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
Mathematics of Computation
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
Journal of Computational Physics
Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
Accurate multiscale finite element methods for two-phase flow simulations
Journal of Computational Physics
Domain decomposition for multiscale PDEs
Numerische Mathematik
Journal of Computational and Applied Mathematics
Linearized reduced-order models for subsurface flow simulation
Journal of Computational Physics
Multiscale finite element methods for high-contrast problems using local spectral basis functions
Journal of Computational Physics
Brief paper: Balanced truncation model reduction for systems with inhomogeneous initial conditions
Automatica (Journal of IFAC)
Mode decomposition methods for flows in high-contrast porous media. A global approach
Journal of Computational Physics
Hi-index | 31.45 |
In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result.