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Bound states in the continuum and Fano resonances in subwavelength resonator arrays
(2021)SAM Research ReportWhen wave scattering systems are subject to certain symmetries, resonant states may decouple from the far-field continuum; they remain localized to the structure and cannot be excited by incident waves from the far field. In this work, we use layer-potential techniques to prove the existence of such states, known as bound states in the continuum, in systems of subwavelength resonators. When the symmetry is slightly broken, this resonant ...Report -
Rigid Sets and Coherent Sets in Realistic Ocean Flows
(2021)SAM Research ReportThis paper focuses on the extractions of Lagrangian Coherent Sets from realistic velocity fields obtained from ocean data and simulations, each of which can be highly resolved and non volume-preserving. Two classes of methods have emerged for such purpose: those relying on the flow map diffeomorphism associated with the velocity field, and those based on spectral decompositions of the Koopman or Perron-Frobenius operators. The two classes ...Report -
On generalization error estimates of physics informed neural networks for approximating dispersive PDEs
(2021)SAM Research ReportPhysics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for several dispersive PDEs.Report -
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On the well-posedness of Bayesian inversion for PDEs with ill-posed forward problems
(2021)SAM Research ReportWe study the well-posedness of Bayesian inverse problems for PDEs, for which the underlying forward problem may be ill-posed. Such PDEs, which include the fundamental equations of fluid dynamics, are characterized by the lack of rigorous global existence and stability results as well as possible non-convergence of numerical approximations. Under very general hypotheses on approximations to these PDEs, we prove that the posterior measure, ...Report -
Well-posedness of Bayesian inverse problems for hyperbolic conservation laws
(2021)SAM Research ReportWe study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein ...Report -
On universal approximation and error bounds for Fourier Neural Operators
(2021)SAM Research ReportFourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the ...Report -
Electrostatic Force Computation with Boundary Element Methods
(2021)SAM Research ReportBoundary element methods are a well-established technique for solving linear boundary value problems for electrostatic potentials. In this context we present a novel way to approximate the forces exerted by electrostatic fields on conducting objects. Like the standard post-processing technique employing surface integrals derived from the Maxwell stress tensor the new approach solely relies on surface integrals, but, compared to the former, ...Report