Registered Data

[CT118]

[00910] An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

  • Session Date & Time : 5B (Aug.25, 10:40-12:20)
  • Type : Contributed Talk
  • Abstract : Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.
  • Classification : 65R20, 65M80, 65N35
  • Author(s) :
    • Fredrik Fryklund (New York University)
    • Sara Pålsson (KTH Royal Institute of Technology)
    • Anna-Karin Tornberg (KTH Royal Institute of Technology)

[01141] On tensor-based training of neural networks

  • Session Date & Time : 5B (Aug.25, 10:40-12:20)
  • Type : Contributed Talk
  • Abstract : In this work by resorting to the continuous ‘model’ of a shallow neural network, we present a novel training approach, that is based on a suitable approximate solution of a Fredholm integral equation of the first kind. Here, we concentrate on least-squares collocation, functional tensor networks and alternating ridge regression. Application of the algorithm to some supervised learning tasks is on par with other state-of-the-art approaches.
  • Classification : 65R20
  • Author(s) :
    • Patrick Gelß ( Zuse Institute Berlin)
    • Aizhan Issagali ( Freie Universität Berlin)
    • Ralf Kornhuber (Freie Universität Berlin)

[02232] A collocation method for third-kind VIEs with nonsmooth solutions

  • Session Date & Time : 5B (Aug.25, 10:40-12:20)
  • Type : Contributed Talk
  • Abstract : In this talk, we consider numerical solution of third-kind Volterra integral equations with nonsmooth solutions. We construct a collocation scheme on a modified graded mesh using a basis of fractional polynomials. For the proposed method, we derive an error estimate in the $L^\infty$-norm, which shows that the optimal order of global convergence can be obtained by choosing appropriate parameter and mesh, even when the exact solution has low regularity. Numerical experiments confirm the theoretical results.
  • Classification : 65R20, 65L05
  • Author(s) :
    • Chengming Huang (Huazhong University of Science and Technology)
    • Zheng Ma (Huazhong University of Science and Technology)

[00904] Quadrature Methods and Error Estimates for Particles in Stokes Flow

  • Session Date & Time : 5B (Aug.25, 10:40-12:20)
  • Type : Contributed Talk
  • Abstract : For axisymmetric particles in Stokes flow, boundary integral methods can be utilized for numerical evaluation of flow velocity on and outside particle surfaces. Precomputation yields a highly efficient and accurate quadrature by expansion $($QBX$)$ method for singular integrals when evaluating on-surface. For evaluation close to the particle surface $($nearly singular integrals$)$, a line interpolation method aided by quadrature error estimates is introduced and compared to QBX in terms of both accuracy and efficiency.
  • Classification : 65Rxx, 65Dxx, 41Axx, 76-XX
  • Author(s) :
    • Pritpal Matharu (KTH Royal Institute of Technology)
    • Anna-Karin Tornberg (KTH Royal Institute of Technology)

[00074] Chaos in multidimensional disordered nonlinear lattices

  • Session Date & Time : 5B (Aug.25, 10:40-12:20)
  • Type : Contributed Talk
  • Abstract : We study the mechanisms of energy transport in multidimensional heterogeneous lattice models, studying in particular the case of the Klein-Gordon model of coupled anharmonic oscillators in one and two spatial dimensions. We perform an extensive numerical investigation of the dynamics of the considered model revealing (i) the effects of the type of the impurity (heterogeneity) parameter on the systems' transport properties and classify the transport mechanisms of the nonlinear versions of the models into various dynamical regimes. (ii) that for it's nonlineaar version, chaotic transport persists and (iii) chaotic hotspots meander in the region of energy concentration supporting the spreading mechanism of energy.
  • Classification : 70K55, 70H07
  • Author(s) :
    • Bob Senyange (Muni University)