Greedy Kernel Surrogates for Nonlinear PDEs in High Dimensions

• Applicant: Professor Dr. Bernard Haasdonk
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 540080351

Partial differential equations (PDEs) on high-dimensional domains appear in many contexts, such as optimal control or parametric problems. In the field of surrogate modelling, approximants are sought, that can be rapidly evaluated for new parameter settings, new point evaluations, etc. in order to apply them in multi-query or real-time contexts. Such scenarios pose problems for traditional mesh-based numerical methods, such as finite element methods. These methods usually suffer from the curse of dimensionality, i.e. with increasing dimension of the input space, the expansion size in the number of basis functions and thus the evaluation effort increases exponentially. In the current project, we want to develop and investigate the use of meshfree kernel methods for these tasks. These methods are defined by selecting the appropriate kernel function, center points, and approximation scheme. So-called greedy schemes produce extremely compact approximation expansions, since they build an expansion incrementally by "optimally" expanding a given model until a suitable accuracy is achieved. In pure function approximation, such schemes provably break the curse of dimension, since the convergence rates have a dimension-independent decay factor. Their small expansion size makes the corresponding models ideal for use as surrogates. We aim at the formulation of PDE greedy approximation schemes for nonlinear problems. This will comprise suitable nonlinear iteration loop around linearized subproblems for which different options will be investigated. We will construct problem-dependent kernel functions as correctly included prior knowledge may further reduce the expansion size and therefore improve convergence rates and evaluation cost. This kernel construction will first aim to correctly model the boundaries of the domain by appropriate weighting or transformation of a base kernel. Secondly, we will adopt kernels to the governing physics of the PDE by defining suitable multilayer kernels. The greedy approximation methods are subjected to convergence analysis in terms of nonlinear fixed point iteration and asymptotic approximation rates. We will implement, apply, validate, and numerically compare the methods with other approximation schemes. This will be realized by suitable benchmark problems from the literature. Overall we anticipate to derive a powerful class of approximation schemes for general PDEs, high dimensions, arbitrary geometries and boundary conditions that are additionally easy to use and are founded on good convergence statements.

• DFG Programme: Research Grants
• International Connection: Italy
• Cooperation Partner: Professor Dr. Gabriele Santin