An algorithmic exploration of the existence of high-order summation by parts operators with diagonal norm

This paper explores a common class of diagonal-norm summation by parts (SBP) operators found in the literature, which can be parameterized by an integer triple $(s,t,r)$ representing the interior order of accuracy ($2s)$, the boundary order of accuracy ($t$), and the dimension of the boundary closure ($r$). There is no simple formula for determining whether or not an SBP operator exists for a given triple of parameters. Instead, one must check that certain compatibility conditions are met: namely that a particular linear system of equations has a positive solution. Partly because of the complexity involved, not much is known about diagonal-norm SBP operators with $2s>10$. By utilizing a new algorithm for answering the question "Does an SBP operator exist for the parameters $(s,t,r)$?", it is possible to explore the existence of SBP operators with high order accuracy, and previously unknown SBP operators with interior order of accuracy as large as $2s=30$ are found. Additionally, a method for optimizing the spectral radius of the SBP derivative is introduced, and the effectiveness of this method is explored through numerical experiment.Comment: Version 3 incorporates some numerical experiments demonstrating the effectiveness of the new operators. Version 2 fixes a mistaken interpretation of previous results. In particular, it is not true, as the previous version stated, that the SBP equations are typically solved as a large nonlinear syste

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies have at most $|E|$ elements and that these elements carry a weight related to their "importance" in relation to the corresponding $p$-modulus problem. When $p=2$, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.Comment: Corrected several typo

Modulus of families of walks on graphs

We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide.Comment: Fixed a minor mistake and updated a referenc

Modulus on graphs as a generalization of standard graph theoretic quantities

This paper presents new results for the modulus of families of walks on a graph---a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut.Comment: Updated with referee's comments. To appear in ECG

Numerical Investigation of Metrics for Epidemic Processes on Graphs

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.Comment: 6 pages, 1 figure, 3 tables, In Proceedings of 2015 Asilomar Conference on Signals, Systems, and Computer

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events