# Reduced basis method for the rapid and reliable solution of partial differential equations

### Yvon Maday

Université Pierre et Marie Curie, Paris, France

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## Abstract

Numerical approximation of the solution of partial differential equations plays an important role in many areas such as engineering, mechanics, physics, chemistry, biology – for computer-aided design-analysis, computer-aided decision-making or simply better understanding. The ﬁdelity of the simulations with respect to reality is achieved through the combined efforts to derive: (i) better models, (ii) faster numerical algorithm, (iii) more accurate discretization methods and (iv) improved large scale computing resources. In many situations, including optimization and control, the same model, depending on a parameter that is changing, has to be simulated over and over, multiplying by a large factor (up to 100 or 1000) the solution procedure cost of one single simulation. The reduced basis method allows to deﬁne a surrogate solution procedure, that, thanks to the complementary design of ﬁdelity certiﬁcates on outputs, allows to speed up the computations by two to three orders of magnitude while maintaining a sufﬁcient accuracy. We present here the basics of this approach for linear and non linear elliptic and parabolic PDE’s.