# Registered Data

Contents

- 1 [CT101]
- 1.1 [00858] An adaptive spectral method for oscillatory second-order linear ODEs with frequency-independent cost
- 1.2 [01151] Structure-Preserving Neural Networks for Hamiltonian Systems
- 1.3 [00013] Singularly perturbed problems on a graph
- 1.4 [00460] A Multigrid Method for Many-Electron Schrodinger Equations with ACE
- 1.5 [02103] Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems

# [CT101]

**Session Time & Room****Classification**

## [00858] An adaptive spectral method for oscillatory second-order linear ODEs with frequency-independent cost

**Session Time & Room**:__3D__(Aug.23, 15:30-17:10) @__E606__**Type**: Contributed Talk**Abstract**: I will introduce an efficient method for solving 2nd order, linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. Within a marching scheme, the solution is generated either via a nonoscillatory phase function (computed by defect correction), or spectral collocation, whichever is more efficient for the current timestep. With numerical experiments I will show that our algorithm outperforms other state-of-the-art oscillatory solvers and has a frequency-independent runtime.**Classification**:__65Lxx__,__34E05__,__65L60__,__34-04__,__65Gxx__**Format**: Talk at Waseda University**Author(s)**:**Fruzsina Julia Agocs**(Center for Computational Mathematics, Flatiron Institute)- Alex Harvey Barnett (Center for Computational Mathematics, Flatiron Institute)

## [01151] Structure-Preserving Neural Networks for Hamiltonian Systems

**Session Time & Room**:__3D__(Aug.23, 15:30-17:10) @__E606__**Type**: Contributed Talk**Abstract**: When solving Hamiltonian systems using numerical integrators, preserving the symplectic structure is crucial. We analyze whether the same is true if neural networks (NN) are used. In order to include the symplectic structure in the NN's topology we formulate a generalized framework for two well-known NN topologies and discover a novel topology outperforming all others. We find that symplectic NNs generalize better and give more accurate long-term predictions than physics-unaware NNs.**Classification**:__65Lxx__,__68T07__,__85-08__**Format**: Talk at Waseda University**Author(s)**:**Philipp Horn**(Eindhoven University of Technology)- Barry Koren (Eindhoven University of Technology)
- Veronica Saz Ulibarrena (Leiden University)
- Simon Portegies Zwart (Leiden University)

## [00013] Singularly perturbed problems on a graph

**Session Time & Room**:__3D__(Aug.23, 15:30-17:10) @__E606__**Type**: Contributed Talk**Abstract**: In this talk, a singularly perturbed convection diffusion problems on a graph domain will be discussed. Initially, the problem is designed on a simple graph i.e k-star graph. On the common vertex, the continuity and the Kirchhoff's conditions will be discussed along with their complexity. The problem may be extended to a general graph with many vertices and edges. Some tests problems will be discussed based on upwind finite difference methods using piece-wise Shishkin meshes. Error estimates and the order of convergence are to be discussed.**Classification**:__65Lxx__,__65Mxx__**Format**: Online Talk on Zoom**Author(s)**:**Vivek Kumar Aggarwal**(Delhi Technological University)

## [00460] A Multigrid Method for Many-Electron Schrodinger Equations with ACE

**Session Time & Room**:__3D__(Aug.23, 15:30-17:10) @__E606__**Type**: Contributed Talk**Abstract**: We parameterize the many-electron wave functions by atomic cluster expansion $($ACE$)$ approach and calculate ground-state energies and electron densities of some molecule systems within the variational Monte Carlo framework. Compared with the neural-network-based representations, the novelty of our method lies in $($i$)$ a convenient and accurate linear polynomial expansion; $($ii$)$ a hierarchical structure that applies naturally to a multigrid variation; and $($iii$)$ possibly revealing the correlation of the system by increasing the body-order.**Classification**:__35Q40__,__65N25__,__65N35__,__81Q05__**Format**: Talk at Waseda University**Author(s)**:**Dexuan Zhou**(Beijing Normal University)

## [02103] Global in Time Weak Solutions to Singular 3D Quasi-Geostrophic Systems

**Session Time & Room**:__3D__(Aug.23, 15:30-17:10) @__E606__**Type**: Contributed Talk**Abstract**: Geophysicists have studied 3D Quasi-Geostrophic systems extensively. These systems describe stratified flows in the atmosphere on a large time scale and are widely used for forecasting atmospheric circulation. They couple an inviscid transport equation in $\mathbb{R}_{+}\times\Omega$ with an equation on the boundary satisfied by the trace, where $\Omega$ is either $2D$ torus or a bounded convex domain in $\mathbb{R}^2$. In this talk, we show the existence of global in time weak solutions to a family of singular 3D quasi-geostrophic systems with Ekman pumping, where the background density profile degenerates at the boundary. The proof is based on the construction of approximated models which combine the Galerkin method at the boundary and regularization processes in the bulk of the domain. The main difficulty is handling the degeneration of the background density profile at the boundary.**Classification**:__35Q35__,__76D03__**Format**: Online Talk on Zoom**Author(s)**:**Yiran Hu**(University of Texas at Austin)