arXiv subjects

Chris Siefert

Publications and source records attributed to Chris Siefert.

4 recordsLinked to original sources

arXiv2026

Improving Smoothed Aggregation AMG Robustness on Stretched Mesh Applications

Strength-of-connection algorithms play a key role in algebraic multigrid (AMG). Specifically, they determine which matrix nonzeros are classified as weak and so ignored when coarsening matrix graphs and defining interpolation sparsity patterns. The general goal is to encourage coarsening only in directions where error can be smoothed and to avoid coarsening across sharp problem variations. Unfortunately, developing robust and inexpensive strength-of-connection schemes is challenging. The classification of matrix nonzeros involves four aspects: (a) choosing a strength-of-connection matrix, (b) scaling its values, (c) choosing a criterion to classify scaled values as strong or weak, and (d) dropping weak entries which includes adjusting matrix values to account for dropped terms. Typically, smoothed aggregation AMG uses the linear system being solved as a strength-of-connection matrix. It scales values symmetrically using square-roots of the matrix diagonal. It classifies based on whether scaled values are above or below a threshold. Finally, it adjusts matrix values by modifying the diagonal so that the sum of entries within each row of the dropped matrix matches that of the original. While these procedures can work well, we illustrate failure cases that motivate alternatives. The first alternative uses a distance Laplacian strength-of-connection matrix. The second centers on non-symmetric scaling. We then investigate alternative classification criteria based on identifying gaps in the values of the scaled entries. Finally, an alternative lumping procedure is proposed where row sums are preserved by modifying all retained matrix entries (as opposed to just diagonal entries). A series of numerical results illustrates trade-offs demonstrating in some cases notably more robust convergence on matrices coming from linear finite elements on stretched meshes.

math.NA↗

arXiv2026

Efficient local classification of parity-based material topology

Although the classification of crystalline materials can be generally handled by momentum-space-based approaches, topological classification of aperiodic materials remains an outstanding challenge, as the absence of translational symmetry renders such conventional approaches inapplicable. Here, we present a numerically efficient real-space framework for classifying parity-based $\mathbb{Z}_2$ topology in aperiodic systems based on the spectral localizer framework and the direct computation of the sign of a Pfaffian associated with a large sparse skew-symmetric matrix. Unlike projector-based or momentum-space-based approaches, our method does not rely on translational symmetry, spectral gaps in the Hamiltonian's bulk, or gapped auxiliary operators such as spin projections, and instead provides a local, energy-resolved topological invariant accompanied by an intrinsic measure of topological protection. A central contribution of this work is the development of a scalable sparse factorization algorithm that enables the reliable determination of the Pfaffian's sign for large sparse matrices, making the approach practical to realistic physical materials. We apply this framework to identify the quantum spin Hall effect in quasicrystalline class AII systems, including gapless heterostructures, and to diagnose fragile topology in a large $C_2 \mathcal{T}$-symmetric photonic quasicrystal. Overall, our results demonstrate that the spectral localizer, combined with efficient sparse numerical methods, provides a unified and robust tool for diagnosing parity-based topological phases in aperiodic electronic, photonic, and acoustic materials where conventional band-theoretic indexes are inapplicable.

physics.comp-ph↗

arXiv2021

Smooth Aggregation for Difficult Stretched Mesh and Coefficient Variation Problems

Four adaptations of the smoothed aggregation algebraic multigrid (SA-AMG) method are proposed with an eye towards improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill-in within the coarse discretization matrices and can also give rise to sub-optimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi-like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.

math.NA↗

arXiv2013

Spatially Adaptive Stochastic Methods for Fluid-Structure Interactions Subject to Thermal Fluctuations in Domains with Complex Geometries

We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid-structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation-dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs-Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with $O(N)$ computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications.

cond-mat.mes-hall↗