Will Pazner

Will Pazner

In this work, we construct high-order finite element spaces for the $L^2$ de Rham complex on triangular meshes amenable to low-order-refined preconditioning. The spaces are constructed using the Duffy transformation, by pulling back appropriately chosen polynomial spaces defined on the unit square; in addition to piecewise polynomials, these spaces also contain certain rational functions, and they reduce to the standard Lagrange, Nédélec, and discontinuous finite elements in the lowest-order case. We establish spectral equivalence, independent of the polynomial degree, of the stiffness matrices defined on these spaces with the lowest-order stiffness matrices defined on refined meshes, constructed using a Gauss-Lobatto triangular lattice. Spectral equivalence of the operators is a consequence of norm equivalences in Jacobi-weighted $L^2$ norms, which are established by proving stability of the Jacobi-Gauss-Lobatto interpolation operator in shifted norms. The low-order-refined preconditioners can also be used to precondition the standard piecewise polynomial finite element spaces using a fictitious space approach. The low-order-refined system can in turn be preconditioned effectively using algebraic multigrid methods. The analytical estimates are confirmed by numerical results on a variety of high-order problems, including on mixed meshes and surface meshes.

Stefanie Günther, Will Pazner, Dongping Qi

Based on the continuous interpretation of deep learning cast as an optimal control problem, this paper investigates the benefits of employing B-spline basis functions to parameterize neural network controls across the layers. Rather than equipping each layer of a discretized ODE-network with a set of trainable weights, we choose a fixed number of B-spline basis functions whose coefficients are the trainable parameters of the neural network. Decoupling the trainable parameters from the layers of the neural network enables us to investigate and adapt the accuracy of the network propagation separated from the optimization learning problem. We numerically show that the spline-based neural network increases robustness of the learning problem towards hyperparameters due to increased stability and accuracy of the network propagation. Further, training on B-spline coefficients rather than layer weights directly enables a reduction in the number of trainable parameters.