Iterative Methods for Model Reduction by Domain Decomposition

It is proposed a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including Laplace equation with non-linear boundary conditions and compressible Euler equations.

[1]  L. Zannetti,et al.  About the numerical modelling of multidimensional unsteady compressible flow , 1989 .

[2]  P. Christofides,et al.  Finite-dimensional approximation and control of non-linear parabolic PDE systems , 2000 .

[3]  P. Christofides,et al.  Dynamic optimization of dissipative PDE systems using nonlinear order reduction , 2002 .

[4]  David J. Lucia,et al.  Reduced order modeling of a two-dimensional flow with moving shocks , 2003 .

[5]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[6]  Marco Amabili,et al.  Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method , 2006 .

[7]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[8]  V. A. Krasil’nikov,et al.  Atmospheric turbulence and radio-wave propagation , 1962 .

[9]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[10]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[11]  Ioannis G. Kevrekidis,et al.  Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis , 1996 .

[12]  Youcef Saad,et al.  A Basic Tool Kit for Sparse Matrix Computations , 1990 .

[13]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[14]  Juan J. Alonso,et al.  Dynamic Domain Decomposition and Error Correction for Reduced Order Models , 2003 .

[15]  K. Willcox,et al.  Aerodynamic Data Reconstruction and Inverse Design Using Proper Orthogonal Decomposition , 2004 .