Journal: SAM Research Report

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Seminar for Applied Mathematics, ETH Zurich

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Publications 1 - 10 of 955
  • On universal approximation and error bounds for Fourier Neural Operators
    Item type: Report
    Kovachki, Nikola; Lanthaler, Samuel; Mishra, Siddhartha (2021)
    SAM Research Report
    Fourier 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 FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.
  • On Tensor Products of Quasi-Banach Spaces
    Item type: Report
    Hansen, Markus (2010)
    SAM Research Report
    Due to applications in approximation theory we are interested in tensor products of quasi-Banach spaces. Though a general abstract theory seems not possible beyond basic topological issues because the dual spaces are possibly trivial, we aim at extending some basic notions like crossnorms, reasonable and uniform norms. In the present paper this is done for quasi- Banach spaces with separating duals, and this condition turns out to be the (in a certain sense) minimal requirement. Moreover, we study extensions of the classical injective and p-nuclear tensor norms to quasi Banach spaces. In particular, we give a sufficient condition for the $p$nuclear quasi-norms to be crossnorms, which particularly applies to the case of weighted $l_p$-sequence spaces.
  • Mathematical analysis of electromagnetic plasmonic metasurfaces
    Item type: Report
    Ammari, Habib; Li, Bowen; Zou, Jun (2019)
    SAM Research Report
  • Well-balanced methods for Computational Astrophysics
    Item type: Report
    Käppeli, Roger (2022)
    SAM Research Report
    We review well-balanced methods for the faithful approximation of solutions of systems of hyperbolic balance laws that are of interest to computational astrophysics. Well-balanced methods are specialized numerical techniques that guarantee the accurate resolution of non-trivial steady-state solutions, that balance laws prominently feature, and perturbations thereof. We discuss versatile frameworks and techniques for generic systems of balance laws for finite volume and finite difference methods. The principal emphasis of the presentation is on the algorithms and their implementation. Subsequently, we specialize in hydrodynamics' Euler equations to exemplify the techniques and give an overview of the available well-balanced methods in the literature, including the classic hydrostatic equilibrium and steady adiabatic flows. The performance of the schemes is evaluated on a selection of test problems.
  • Sparse finite elements for stochastic elliptic problems - higher order moments
    Item type: Report
    Schwab, Christoph; Todor, Radu Alexandru (2003)
    SAM Research Report
    We define the higher order moments associated to the stochastic solution of an elliptic BVP in D \subset Rd with stochastic source terms and boundary data. We prove that the k-th moment (or k-point correlation function) of the random solution solves a deterministic problem in Dk \subset Rdk. We discuss well-posedness and regularity in scales of Sobolev spaces with bounded mixed derivatives. We discretize this deterministic k-th moment problem using sparse tensor product FE-spaces and, exploiting a spline wavelet basis, we propose an algorithm of (up to logarithmic terms) the same accuracy and complexity as a multigrid finite element method for the mean field problem in D.
  • Neural and gpc operator surrogates: construction and expression rate bounds
    Item type: Report
    Herrmann, Lukas; Schwab, Christoph; Zech, Jakob (2022)
    SAM Research Report
    Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and gpc operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity.
  • Stability analysis for the method of transport
    Item type: Report
    Morel, Anne-Thérèse; Bodenmann, R. (1996)
    SAM Research Report
    In this paper, we provide analytical stability estimates for the method of transport. We first prove stability of the second-order method of transport applied to the linear advection equation with constant coefficient in one dimension by using the von Neumann method and with the positive operator technique. In a second step, we extend the proof to the linear advection equation with variable coefficient. Finally, we investigate and compare the existing multidimensional schemes from van Leer, Colella, and LeVeque for the linear advection equation with constant coefficients.
  • Multiple Traces Formulation and Semi-Implicit Scheme for Modeling Biological Cells under Electrical Stimulation
    Item type: Report
    Henriquez, Fernando; Jerez-Hanckes, Carlos (2017)
    SAM Research Report
    We model the electrical behavior of several biological cells under external stimuli by extending and computationally improving the semi-implicit multiple traces formulation presented in (Henriquez et al., Numerische Mathematik, 2016). Therein, the electric potential and current for a single cell are retrieved through the coupling of boundary integral operators and non-linear ordinary differential systems of equations. Yet, the low-order discretization scheme presented becomes impractical when accounting for interactions among multiple cells. In this note, we consider multi-cellular systems and show existence and uniqueness of the resulting non-linear evolution problem in finite time. Our main tools are analytic semigroup theory along with mapping properties of boundary integral operators in Sobolev spaces. Thanks to the smoothness of cellular shapes, solutions are highly regular at a given time. Hence, spectral spatial discretization can be employed, thereby largely reducing the number of unknowns. Time-space coupling is achieved via a semi-implicit time-stepping scheme shown to be stable and convergent. Numerical results in two dimensions validate our claims and match observed biological behavior for the Hodgkin-Huxley dynamical model.
  • Modelling the active cochlea as a fully-coupled system of subwavelength Hopf resonators
    Item type: Report
    Ammari, Habib; Davies, Bryn (2019)
    SAM Research Report
  • Div-curl problems and stream functions in 3D Lipschitz domains
    Item type: Report
    Kirchhart, Matthias; Schulz, Erick (2020)
    SAM Research Report
Publications 1 - 10 of 955