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The papers in this volume start with a description of the construction of reduced models through a review of Proper Orthogonal Decomposition (POD) and reduced basis models, including their mathematical foundations and some challenging applications, then followed by a description of a new generation of simulation strategies based on the use of separated representations (space-parameters, space-time, space-time-parameters, space-space,...), which have led to what is known as Proper Generalized Decomposition (PGD) techniques. The models can be enriched by treating parameters as additional coordinates, leading to fast and inexpensive online calculations based on richer offline parametric solutions. Separated representations are analyzed in detail in the course, from their mathematical foundations to their most spectacular applications. It is also shown how such an approximation could evolve into a new paradigm in computational science, enabling one to circumvent various computational issues in a vast array of applications in engineering science.

Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications

Model order reduction based on proper orthogonal decomposition: Model reduction: extracting relevant information.- Interpolation of reduced basis: a geometrical approach.- POD for non-linear models.- Conclusions.- PGD for solving multidimensional and parametric models: Introduction.- Separated representations.- Advanced topics.- Models defined in plate and shell geometries.- Computational vademecums.- PGD in linear and nonlinear Computational Solid Mechanics: Introduction.- PGD -Verification for linear problems (elliptic and parabolic).- PGD for time dependent nonlinear problems (monoscale and multiscale problems).- Reduced basis approximation and error estimation for parameterized elliptic partial differential equations and applications: Introduction and motivation.- Parameterized problems.- High order and reduced order models with reduced basis method: greedy algorithm and a posteriori error estimation.- Applications.- Conclusion.