Linear and nonlinear reduced models for the numerical approximation of high-dimensional functions

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Partial differential equations (PDEs) are widely used as the mathematical model for problems arising in physics, biology or engineering. In most cases, these problems depend on many parameters, for instance the geometry of the physical domain, the boundary/initial conditions or the coefficients, yielding so-called parametric partial differential equations (PDEs). Nowadays, it is common to include the inherent uncertainty affecting these complex phenomena in the mathematical model. A way to model the uncertainty is to use random variables or random fields. Such PDE with random input has an equivalent parametric deterministic formulation, where the parameter space is endowed with a probability measure. The long-term goal of this program is to design, analyze and implement numerical methods for approximating the solutions to parametric/random PDEs. The main focus will be on model order reduction techniques and adaptive strategies for the so-called forward problem: given a value of the parameter (in some parameter space), find an approximation of the corresponding solution. In order to have methods that are immune to the so-called curse of dimensionality, emphasis will be given to high-dimensional problems, namely when the dimension of the parameter space is large or even infinite. The main objectives are: (I) to compare linear reduced models and apply them to problems of practical interest; (II) to design and analyze nonlinear reduced models with provable performance guarantees; (III) to introduce adaptive strategies for solving random PDEs and compare them to existing methods. A prominent efficient linear reduced model is the reduced basis method. This method hinges on the potential smoothness of the solution with respect to the parameters to build a linear space in which the solution is approximated. The linear space is the span of so-called snapshots, namely the solution of the problem for suitably selected values of the parameters. In some cases, the construction of one linear space for approximating the parameter to solution map is not feasible numerically. It is well-known that nonlinear methods can provide improved efficiency. Recently, several nonlinear reduced methods have been developed and tested numerically. Contrary to linear reduced models, for which the theory is well-understood, little is known in terms of precise performance guarantees for nonlinear strategies. One of the main goals of this program is thus to develop new algorithms for constructing nonlinear reduced models and perform a precise analysis of their performances. This research program will train 3 PhD, 3 MSc and 2 BSc students, who will gain expertise in numerical analysis, and make significant progress in the development of numerical methods for solving parametric/random PDEs. Fast and efficient forward solvers with provable performance guarantees are essential tools in many applications, such as optimal engineering design, weather prediction or medical diagnosis.