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Structure preserving schemes
(2014)SAM Research ReportWe present two novel structure preserving numerical schemes for the Euler equations of hydrodynamics. The first method is concerned with the exact preservation of certain hydrostatic equilibria. This is achieved by a hydrostatic preserving reconstruction procedure and a well-balanced discretization of the gravitational source term. The second method treats the deficiency of angular momentum conservation in standard Eulerian Godunov-type ...Report -
On the approximation of rough functions with deep neural networks
(2020)SAM Research ReportDeep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its high-order accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural ...Report -
On the conservation of energy in two-dimensional incompressible flows
(2020)SAM Research ReportWe prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral Viscosity numerical approximation, respectively. We characterize this conservation of energy in terms of a uniform decay of the so-called structure function, allowing us to extend existing results on energy ...Report -
Numerical approximation of statistical solutions of scalar conservation laws
(2017)SAM Research ReportReport -
Error estimates for DeepOnets: A deep learning framework in infinite dimensions
(2021)SAM Research ReportDeepOnets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepOnets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepOnets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction ...Report -
Computation of measure valued solutions for the incompressible Euler equations
(2014)Research ReportWe combine the spectral (viscosity) method and ensemble averaging to propose an algorithm that computes admissible measure valued solutions of the incompressible Euler equations. The resulting approximate young measures are proved to converge (with increasing numerical resolution) to a measure valued solution. We present numerical experiments demonstrating the robustness and efficiency of the proposed algorithm, as well as the appropriateness ...Report -
Entropy stable schemes on two-dimensional unstructured grids
(2014)Research ReportWe propose an entropy stable high-resolution finite volume scheme to approximate systems of two-dimensional symmetrizable conservation laws on unstructured grids. The scheme is constructed using a judicious combination of entropy conservative fluxes and entropy-stable numerical dissipation operators. High resolution is achieved based on a piecewise linear reconstruction procedure satisfying a suitable sign property. The proposed scheme ...Report -
Entropy-stable space-time DG schemes for non-conservative hyperbolic systems
(2017)SAM Research ReportReport