Numerical simulation based on Partial Differential Equation models is by now an established methodology of investigation in Science, Engineering and Society. In many fields numerical methods have reached such a level of maturity to provide reliable, efficient and easy-to-use tools of analysis. This growing success pushes numerical simulation towards new frontiers, where the complexity of the problems to be tackled calls for the development of innovative concepts and methods. The School aims precisely at presenting some of the challenges taken up in recent years in this field, illustrating new perspectives leading to satisfactory solutions.

Nonlinear approximation deals with the approximation of a function in a sequence space by picking up its most relevant components with respect to a given basis or frame, thus minimizing the quantity of information needed to encode the function with sufficient accuracy; the concepts of best n-term approximation and of greedy algorithm are central here. A further level of nonlinearity arises if the representation basis itself is left free to be selected from a dictionary of bases.

A formidable task consists in performing a nonlinear approximation of the solution of a PDE problem, since such a function is only indirectly accessible through the inspection of the operator and the data. This leads to adaptive discretization methods (by wavelet, spectral, h- or hp-type finite element methods), which for various classes of problems are guaranteed to reach any target accuracy with nearly optimal computational complexity.

Complexity in approximating a function may depend on its sparsity in the chosen representation system, as measured in a proper scale of functional classes. Sparsity is also crucial in dealing with high-dimensional problems, such as those arising from PDEs with stochastic coefficients or in Quantum Chemistry. Sparse grids or more generally tensor products mitigate the curse of dimensionality; the use of solution dependent tensor bases in suitable stable formats may rigorously guarantee near-minimal rank approximations for any given target error, thereby even breaking the curse in certain cases.

Complexity reduction may be also achieved by resorting to reduced models, that surrogate the exact ones with a lower computational burden. They are essential in handling problems, such as parametric optimization or uncertainty quantification, in which an expensive numerical PDE solver should be invoked many times with different input data. Concepts like greedy space search, n-width performance, reduced basis methods are relevant in this field, and bear intimate connections with the other topics treated in the School.