Zeitlin's model is a spatial discretization for the 2-D Euler equations on the flat 2-torus or the 2-sphere. Contrary to other discretizations, it preserves the underlying geometric structure, namely that the Euler equations describe Riemannian geodesics on a Lie group. Here we show how to extend Zeitlin's approach to the axisymmetric Euler equations on the 3-sphere. It is the first discretization of the 3-D Euler equations that fully preserves the geometric structure. Thus, this finite-dimensional model admits Riemannian curvature and Jacobi equations, which are discussed.
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We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.
In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, this discretization still results in an unstructured dense coefficient matrix with huge memory requirement and computational complexity. To address this issue, a fast version fractional BCFD algorithm by employing the well-known sum-of-exponentials (SOE) approximation technique is also proposed. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only $\mathcal{O}(MN_{exp})$ operations per iteration without losing any accuracy compared to the direct solvers, where $N_{exp}\ll M$ is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in $\mathcal{O}(MN_{exp})$ memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.
We study the equational theory of the Weihrauch lattice with composition and iterations, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the composition operator $\star$ and its iteration $(-)^\diamond$ , which are true however we substitute (slightly extended) Weihrauch degrees for the variables. We characterize them using B\"uchi games on finite graphs and give a complete axiomatization that derives them. The term signature and the axiomatization are reminiscent of Kleene algebras, except that we additionally have meets and the lattice operations do not fully distributes over composition. The game characterization also implies that it is decidable whether an equation is universally valid. We give some complexity bounds; in particular, the problem is Pspace-hard in general and we conjecture that it is solvable in Pspace.
To solve many problems on graphs, graph traversals are used, the usual variants of which are the depth-first search and the breadth-first search. Implementing a graph traversal we consequently reach all vertices of the graph that belong to a connected component. The breadth-first search is the usual choice when constructing efficient algorithms for finding connected components of a graph. Methods of simple iteration for solving systems of linear equations with modified graph adjacency matrices and with the properly specified right-hand side can be considered as graph traversal algorithms. These traversal algorithms, generally speaking, turn out to be non-equivalent neither to the depth-first search nor the breadth-first search. The example of such a traversal algorithm is the one associated with the Gauss-Seidel method. For an arbitrary connected graph, to visit all its vertices, the algorithm requires not more iterations than that is required for BFS. For a large number of instances of the problem, fewer iterations will be required.
We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.
The aim of this study is to establish a general transformation matrix between B-spline surfaces and ANCF surface elements. This study is a further study of the conversion between the ANCF and B-spline surfaces. In this paper, a general transformation matrix between the Bezier surfaces and ANCF surface element is established. This general transformation matrix essentially describes the linear relationship between ANCF and Bezier surfaces. Moreover, the general transformation matrix can help to improve the efficiency of the process to transfer the distorted configuration in the CAA back to the CAD, an urgent requirement in engineering practice. In addition, a special Bezier surface control polygon is given in this study. The Bezier surface described with this control polygon can be converted to an ANCF surface element with fewer d.o.f.. And the converted ANCF surface element with 36 d.o.f. was once addressed by Dufva and Shabana. So the special control polygon can be regarded as the geometric condition in conversion to an ANCF surface element with 36 d.o.f. Based on the fact that a B-spline surface can be seen as a set of Bezier surfaces connected together, the method to establish a general transformation matrix between the ANCF and lower-order B-spline surfaces is given. Specially, the general transformation is not in a recursive form, but in a simplified form.
Exponential Runge-Kutta methods for semilinear ordinary differential equations can be extended to abstract differential equations, defined on Banach spaces. Thanks to the sun-star theory, both delay differential equations and renewal equations can be recast as abstract differential equations, which motivates the present work. The result is a general approach that allows us to define the methods explicitly and analyze their convergence properties in a unifying way.
Finding eigenvalue distributions for a number of sparse random matrix ensembles can be reduced to solving nonlinear integral equations of the Hammerstein type. While a systematic mathematical theory of such equations exists, it has not been previously applied to sparse matrix problems. We close this gap in the literature by showing how one can employ numerical solutions of Hammerstein equations to accurately recover the spectra of adjacency matrices and Laplacians of random graphs. While our treatment focuses on random graphs for concreteness, the methodology has broad applications to more general sparse random matrices.
When the row and column variables consist of the same category in a two-way contingency table, it is specifically called a square contingency table. Since it is clear that the square contingency tables have an association structure, a primary objective is to examine symmetric relationships and transitions between variables. While various models and measures have been proposed to analyze these structures understanding changes between two variables in behavior at two-time points or cohorts, it is also necessary to require a detailed investigation of individual categories and their interrelationships, such as shifts in brand preferences. This paper proposes a novel approach to correspondence analysis (CA) for evaluating departures from symmetry in square contingency tables with nominal categories, using a power-divergence-type measure. The approach ensures that well-known divergences can also be visualized and, regardless of the divergence used, the CA plot consists of two principal axes with equal contribution rates. Additionally, the scaling is independent of sample size, making it well-suited for comparing departures from symmetry across multiple contingency tables. Confidence regions are also constructed to enhance the accuracy of the CA plot.
We prove that multilevel Picard approximations are capable of approximating solutions of semilinear heat equations in $L^{p}$-sense, ${p}\in [2,\infty)$, in the case of gradient-dependent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations grow at most polynomially in both dimension $d$ and prescribed reciprocal accuracy $\epsilon$.
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