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Concept# Hilbert's problems

Summary

Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.
Nature and influence of the problems
Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems, such as the 5th, experts have traditionally agreed on a sing

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In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.

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The lattice Green's function method (LGFM) is the discrete counterpart of the continuum boundary element method and is a natural approach for solving intrinsically discrete solid mechanics problems that arise in atomistic-continuum coupling methods. Here, the LGFM is extended to problems in a bounded domain with applied boundary conditions. An efficient controlled coarse-graining method is introduced to significantly reduce the number of atomistic degrees of freedom on the outer boundary, and thus the size of the dense Green's function matrices involved while preserving the high accuracy of the solution. The method is demonstrated on various example problems to show reductions in computational costs and errors versus reference solutions.

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We analyze the adaptive first order algorithm AMSGrad, for solving a constrained stochastic optimization problem with a weakly convex objective. We prove the $\tilde O(t^{-1/2})$ rate of convergence for the squared norm of the gradient of Moreau envelope, which is the standard stationarity measure for this class of problems. It matches the known rates that adaptive algorithms enjoy for the specific case of unconstrained smooth nonconvex stochastic optimization. Our analysis works with mini-batch size of 1, constant first and second order moment parameters, and possibly unbounded optimization domains. Finally, we illustrate the applications and extensions of our results to specific problems and algorithms.

2021