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Publication# Model reduction of coupled systems based on non-intrusive approximations of the boundary response maps

Abstract

We propose a local, non -intrusive model order reduction technique to accurately approximate the solution of coupled multi -component parametrized systems governed by partial differential equations. Our approach is based on the approximation of the boundary response maps arising from a non -overlapping domain decomposition method. To construct the surrogate, we combine dimensionality reduction techniques with interpolation or regression approaches, such as kernel interpolation methods and artificial neural networks. Two alternative training strategies, making use of the full coupled problem or an artificial parametrization of the boundary conditions, are proposed and discussed. We show the potential of our approach in a series of test cases, ranging from linear diffusion -like models to nonlinear multi -physics problems. High levels of accuracy and computational efficiency are achieved in all cases.

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Numerical methods for partial differential equations

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.

Boundary value problem

In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems.

Dirichlet boundary condition

In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.

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