Signal processing of simplicial complexes

High order finite volume approximations of differential operators on nonuniform grids

Realizing two-dimensional, continuous directed fields by vector fields and an algorithm to remove dichotomous ambiguity in a discrete field

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

This is the most important chapter in the entire book. A new method, designated the symbolic vector method, in treating vector analysis will be introduced. The main advantages of this method are the differential expressions of the three key functions in vector analysis are derived based on one basic formula, all of the integral theorems in vector analysis are deduced from one generalized theorem, the commonly used vector identities are found by an algebraic method without pedorming any differentiation, and two differential operators in the curvilinear coordinate system are introduced. They are designated as the divergence operator and the curl operator, which are distinct from the operator used for the gradient. The technical meaning of the words “divergence,” “curl”, and “gradient” will be explained shortly. It should be pointed out that the nomenclature for some technical terms introduced in this chapter is different from the original one used in [18).

Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection‐diffusion‐reaction equation the diffusion coefficient is a symmetric tensor of second order.

Smooth vector fields on can be decomposed into the sum of a gradient vector field and divergence-free (solenoidal) vector field under suitable hypotheses. This is called the Helmholtz–Hodge decomposition (HHD), which has been applied to analyze the topological features of vector fields. In this study, we apply the HHD to study certain types of vector fields. In particular, we investigate the existence of strictly orthogonal HHDs, which assure an effective analysis. The first object of the study is linear vector fields. We demonstrate that a strictly orthogonal HHD for a vector field of the form F(x) = Ax can be obtained by solving an algebraic Riccati equation. Subsequently, a method to explicitly construct a Lyapunov function is established. In particular, if A is normal, there exists an easy solution to this equation. Next, we study planar vector fields. In this case, the HHD yields a complex potential, which is a generalization of the notion in hydrodynamics with the same name. We demonstrate the convenience of the complex potential formalism by analyzing vector fields given by homogeneous quadratic polynomials.

The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods

Atrial Fibrillation (AF) is the most common supraventricular arrhythmia and has different underlying activation mechanism, including functional rotors (FR) and ectopic foci (EF).In this work we propose an approach for locating FR and EF in potential maps, which were tested with mathematical phantoms.12 phantoms were created (128x128 array, 4 s, Fs 500 Hz), simulating the motion of: FR (4 maps), EF (4 maps) and superpositions of these (4 maps).These were downsampled to different grids from 16x16 to 8x8, simulating electrode acquisition.Noise (SNR from 2 to 60) was added.To locate the mechanism, the signals were filtered and interpolated.Farneback optical flow was applied to compute the motion vector field (MVF).The MVF was normalized and its temporal average was calculated.Finally, we computed curl and divergence, by using a x and y oriented 11 11 Sobel filter as a estimation of the of the partial derivatives.The location of extrema in the curl and divergence maps were used for locating FR and EF respectively.Method robustness was tested by comparing the algorithm performance on different grid sizes and SNR.The mechanism was considered to be detected accurately if the position was within a normalized error of 5% from its respective phantom location.The results showed that our approach was able to locate both mechanisms, but revealed a dependency on spatial resolution.

While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finite-difference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vector fields on arbitrary tetrahedral grids. We introduce a variational, multiscale decomposition of vector fields into three intuitive components: a divergence-free part, a curl-free part, and a harmonic part. We show how our discrete approach matches its well-known smooth analog, called the Helmotz-Hodge decomposition, and that the resulting computational tools have very intuitive geometric interpretation. We demonstrate the versatility of these tools in a series of applications, ranging from data visualization to fluid and deformable object simulation.

An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence

The separation of wave modes for isotropic elastic wavefields is typically done using Helmholtz decomposition. However, Helmholtz decomposition using conventional divergence and curl operators is not satisfactory for anisotropic media and leaves the different wave modes only partially separated. The separation of anisotropic waveflelds requires more sophisticated operators which depend on local material parameters. Wavefleld separation operators for TI (transversely isotropic) models can be constructed based on the polarization vectors evaluated at each point of the medium by solving the Christoffel equation using local medium parameters. These polarization vectors can be represented in the space domain as localized filters, which resemble conventional derivative operators. The spatially‐variable “pseudo” derivative operators perform well in 2D heterogeneous TI media even at places of rapid variation. Wave separation for 3D TI media can be performed in a similar way. In 3D TI media, P and SV waves are polarized only in symmetry planes, and SH waves are polarized orthogonal to symmetry planes. Using the mutual orthogonality property between these modes, we only need to solve for the P wave polarization vectors from the Christoffel equation, and SV and SH wave polarizations can be constructed using the relationship between these three modes. Synthetic results indicate that the operators can be used to separate wavefields for TI media with arbitrary strength of anisotropy.

The divergence and curl operators are fundamental mathematical tools introduced in university-level electromagnetism courses. Although students generally understand the operational use of these concepts, developing a clear physical intuition of what divergence and curl are can be challenging. This difficulty often arises due to the superficial or confusing treatment found in many standard textbooks. The goal of this article is to provide a clear demonstration, based on examples, that two-dimensional visualizations can be misleading when used to interpret these operators. A combination of graphical and analytical (symbolic) representations was found to be essential for developing an accurate physical intuition about divergence and curl. A deeper comprehension of these operators not only improves proficiency in electromagnetism, but also benefits a range of other fields, from fluid dynamics to meteorology and to technical applications.

P and S decomposition is an essential step in isotropic viscoelastic reverse time migration (VRTM). Separation using divergence and curl operators doesn’t preserve the phase or amplitude of the input viscoelastic wavefield, so an alternative decomposition that preserves the vector components of the Pand S-waves in the input wavefield is desired. With a decoupled assumption, we rearrange the memory variables in the elastodynamic equations based on the standard linear solid. The vector components of the propagating Pand S-wave can be can be decomposed in isotropic viscoelastic media without losing accuracy. Synthetic tests show that the Pand S-waves can be decomposed with no distortion of the amplitude or phase, and have the same vector components of particle velocity that exist in the viscoelastic wavefield before decomposition.

UDC 517.987.5; 517.958 The vector field is rectified in the neighborhood of a nonsingular point. By means of local foliation into nonsingular hypersurfaces containing phase curves of the vector field, we obtain a two-dimensional vector quotient field. A quotient field in the phase plane defines the rectified phase pattern and the module (over the ring of first integrals) of invariant measures. The pair that is composed of the phase pattern and the invariant measure of the vector field in the plane has a stationary subaigebra in the algebra of all vector fields. The triple consisting of the initial vector field, its invariant measure, and an element of the stationary subalgebra of the pairs, admits a relative integral invariant, namely, the integral of the measure over sets that are bounded by the phase curves of two vector fields. The relative integral invariant is generated (over the ring of first integrals) by the action of an element of the stationary subalgebra (the product of the Hamiltonian or the potential by the time of the element of the subalgebra). This explicitly asymmetric construction relative to the pair of vector fields becomes symmetric if we consider singular invariant measures and smooth vector fields. Suppose that an element of the subalgebra is a nonsingular vector field at the point being considered. Then we can take it as the initial element for the next triple. Obviously, the iteration of this procedure generates a chain of triples and, respectively, a chain of measures. We are interested in the convergence of the arising measures (the sense of convergence is defined in the article). We say that a chain of triples is subjected to the weight k > 0 (say, k is a positive integer) if, in each triple, the invariant measure generated by the action of the element of the subaigebra is generated by the first integral of the first element of the triple of degree k (The invariant measure of a triple makes it possible to determine two numbers, namely, the successive weight, i.e., the order of singularity of the measure generated by the action of the element of the subalgebra over the ring of the first integrals of the subalgebra and the preceding weight, which is the same as above, only with the replacement of the element of the subalgebra by the initial vector field.). The order of singularity of an invariant measure is the weight of the measure relative to the action for the element of the subalgebra. Theorem. The orders of singularity of the invariant measures of a chain of triples subjected to the weight k correspond to the orbits of the Feigenbaum mapping.

Topological physics requires stable methods to measure polarization singularities in optical vector fields. Here, the use of a periodic plasmonic metasurface is proposed for computation of the divergence of vectorial paraxial beams. We design such an optical device to perform spatial differentiation in two directions, parallel and perpendicular to the incidence plane, simultaneously. The divergence operation is achieved by creating constructive interference between two derivative results. We demonstrate that such optical computations provide a direct pathway to identifying specific polarization singularities in vector fields.