Very recently, Qi and Cui extended the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts and proved that they have Perron eigenpair and Perron-Frobenius eigenpair. The Collatz method was also extended to find Perron eigenpair. Qi and Cui proposed two conjectures. One is the k-order power of a dual number matrix tends to zero if and only if the spectral radius of its standard part less than one, and another is the linear convergence of the Collatz method. In this paper, we confirm these conjectures and provide theoretical proof. The main contribution is to show that the Collatz method R-linearly converges with an explicit rate.
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We develop a novel discontinuous Galerkin method for solving the rotating thermal shallow water equations (TRSW) on a curvilinear mesh. Our method is provably entropy stable, conserves mass, buoyancy and vorticity, while also semi-discretely conserving energy. This is achieved by using novel numerical fluxes and splitting the pressure and convection operators. We implement our method on a cubed sphere mesh and numerically verify our theoretical results. Our experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilization.
The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method arXiv:1910.03080 has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies BKW solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.
This paper presents a new algorithm for generating random inverse-Wishart matrices that directly generates the Cholesky factor of the matrix without computing the factorization. Whenever parameterized in terms of a precision matrix $\Omega=\Sigma^{-1}$, or its Cholesky factor, instead of a covariance matrix $\Sigma$, the new algorithm is more efficient than the current standard algorithm.
Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in physics and engineering. Many such problems have alternative formulations as integral equations that are mathematically more tractable than their PDE counterparts. However, the integral equation formulation poses a challenge in solving the dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and compute costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named ``strong recursive skeletonization factorization.'' The analysis of its parallel scalability applies generally to a class of existing methods that exploit the so-called strong admissibility. Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes data movement. Given a compression tolerance, our method constructs an approximate factorization of a discretized integral operator (dense matrix), which can be used to solve linear systems efficiently in parallel. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple right-hand sides. Large-scale numerical experiments are presented to demonstrate the performance of our implementation using the Julia language.
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation, and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided.
Most identification methods of unknown parameters of linear regression equations (LRE) ensure only boundedness of a parametric error in the presence of additive perturbations, which is almost always unacceptable for practical scenarios. In this paper, a new identification law is proposed to overcome this drawback and guarantee asymptotic convergence of the unknown parameters estimation error to zero in case the mentioned additive perturbation meets special averaging conditions. Theoretical results are illustrated by numerical simulations.
If the Stokes equations are properly discretized, it is known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa algorithm. Despite recent progress in developing efficient iterative methods for solving the Stokes problem, the Uzawa algorithm remains popular in science and engineering, especially when accelerated by Krylov subspace methods. However, in complex geometries, the Schur complement matrix can become severely ill-conditioned, having a significant portion of non-unit eigenvalues. This makes the established Uzawa preconditioner inefficient. To explain this behaviour, we examine the Pressure Schur Complement formulation for the staggered finite-difference discretization of the Stokes equations. Firstly, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement matrix. Secondly, we demonstrate that its condition number increases with increasing the surface-to-volume ratio of the flow domain. As an alternative to the Uzawa preconditioner, we propose using the diffusive SIMPLE preconditioner for geometries with a large surface-to-volume ratio. We show that the latter is much more fast and robust for such geometries. Furthermore, we show that the usage of the SIMPLE preconditioner leads to more accurate practical computation of the permeability of tight porous media. Keywords: Stokes problem, tight geometries, computing permeability, preconditioned Krylov subspace methods
A second-order finite volume scheme is proposed and analyzed for a 2X2 system of non-linear partial differential equations. These equations model the dynamics of growing sandpiles created by a vertical source on a flat, bounded rectangular table in multiple dimensions. The well-balancedness of the scheme is ensured through a modified limitation approach allowing the scheme to reduce to well-balanced first-order scheme near the steady state while maintaining the second-order accuracy away from it. The well-balanced property of the scheme is proven analytically in one dimension and demonstrated numerically in two dimensions. It is also shown through the numerical experiments that the second-order scheme reduces the finite time oscillations, takes fewer time iterations for achieving the steady state and gives sharper resolutions of the physical structure of the sandpile, as compared to the first-order schemes existing in the literature.
In this paper we propose a variant of enriched Galerkin methods for second order elliptic equations with over-penalization of interior jump terms. The bilinear form with interior over-penalization gives a non-standard norm which is different from the discrete energy norm in the classical discontinuous Galerkin methods. Nonetheless we prove that optimal a priori error estimates with the standard discrete energy norm can be obtained by combining a priori and a posteriori error analysis techniques. We also show that the interior over-penalization is advantageous for constructing preconditioners robust to mesh refinement by analyzing spectral equivalence of bilinear forms. Numerical results are included to illustrate the convergence and preconditioning results.
Differential geometric approaches are ubiquitous in several fields of mathematics, physics and engineering, and their discretizations enable the development of network-based mathematical and computational frameworks, which are essential for large-scale data science. The Forman-Ricci curvature (FRC) - a statistical measure based on Riemannian geometry and designed for networks - is known for its high capacity for extracting geometric information from complex networks. However, extracting information from dense networks is still challenging due to the combinatorial explosion of high-order network structures. Motivated by this challenge we sought a set-theoretic representation theory for high-order network cells and FRC, as well as their associated concepts and properties, which together provide an alternative and efficient formulation for computing high-order FRC in complex networks. We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. Crucially, our representation theory reveals previous computational bottlenecks and also accelerates the computation of FRC. As a consequence, our findings open new research possibilities in complex systems where higher-order geometric computations are required.
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