scholarly journals Point vortex equilibria on the sphere via Brownian ratchets

2008 ◽  
Vol 465 (2102) ◽  
pp. 437-455 ◽  
Author(s):  
Paul K Newton ◽  
Takashi Sakajo
Keyword(s):  
Equilibrium Configuration ◽  
Null Space ◽  
Relative Equilibrium ◽  
Singular Values ◽  
Singular Value ◽  
Point Vortices ◽  
Basis Set ◽  
Frobenius Norm ◽  
Equilibrium Configurations ◽  
Smallest Singular Value

We describe a Brownian ratchet scheme that we use to calculate relative equilibrium configurations of N point vortices of mixed strength on the surface of a unit sphere. We formulate it as a problem in linear algebra, A Γ =0, where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all inter-vortical distances on the sphere remain constant and Γ ∈ N is the (unit) vector of vortex strengths that must lie in the null space of A . Existence of an equilibrium is expressed by the condition det( A T A )=0, while uniqueness follows if Rank( A )= N −1. The singular value decomposition of A is used to calculate an optimal basis set for the null space, yielding all values of the vortex strengths for which the configuration is an equilibrium and allowing us to decompose the equilibrium configuration into basis components. To home in on an equilibrium, we allow the point vortices to undergo a random walk on the sphere and, after each step, we compute the smallest singular value of the configuration matrix, keeping the new arrangement only if it decreases. When the smallest singular value drops below a predetermined convergence threshold, the existence criterion is satisfied and an equilibrium configuration is achieved. We then find a basis set for the null space of A , and hence the vortex strengths, by calculating the right singular vectors corresponding to the singular values that are zero. We show a gallery of examples of equilibria with one-dimensional null spaces obtained by this method. Then, using an unbiased ensemble of 1000 relative equilibria for each value N =4→10, we discuss some general features of the statistically averaged quantities, such as the Shannon entropy (using all of the normalized singular values) and Frobenius norm, centre-of-vorticity vector and Hamiltonian energy.

Spectral gradient flow and equilibrium configurations of point vortices

2009 ◽  
Vol 466 (2118) ◽  
pp. 1687-1702 ◽  
Author(s):  
Andrea Barreiro ◽  
Jared Bronski ◽  
Paul K. Newton
Keyword(s):  
Symmetric Matrix ◽  
A Priori ◽  
Relative Equilibrium ◽  
Point Vortex ◽  
Gradient Flow ◽  
Rotational Frequency ◽  
Point Vortices ◽  
Equilibrium Configurations ◽  
The Matrix ◽  
Smallest Singular Value

We formulate the problem of finding equilibrium configurations of N -point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency and translational velocity of the configuration. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices M T M . The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

Construction of point vortex equilibria via Brownian ratchets

2007 ◽  
Vol 463 (2082) ◽  
pp. 1525-1541 ◽  
Author(s):  
Paul K Newton ◽  
George Chamoun
Keyword(s):  
Equilibrium State ◽  
Initial Conditions ◽  
Point Vortex ◽  
Singular Values ◽  
Basis Set ◽  
Frobenius Norm ◽  
Initial State ◽  
Optimal Basis ◽  
Equilibrium Configurations ◽  
Value Decomposition

A theory capable of producing equilibrium configurations of point vortices in the plane, along with a numerical scheme to compute them, is described. The theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation , where A is the (1/2) N ( N −1)× N non-normal configuration matrix obtained by requiring that all intervortical distances remain fixed, and are the N -vortex strengths. For existence of an equilibrium, A must have a non-trivial nullspace. We consider the singular values of A ; when this has one or more zero singular values, the nullspace of A is non-empty and an equilibrium exists for some choice of Γ . New equilibrium configurations are found numerically by randomly depositing N points in the plane, which generically gives rise to a configuration matrix A with empty nullspace. Using the sum of squares of the k smallest singular values of A as a ‘ratchet’, we ‘thermally fluctuate’ the configuration, allowing each point to execute a random walk in the plane, retaining only those configurations which reduce this quantity at the next step. The configuration is thus driven to one with nullspace ( A )= k >0. These converged states are not necessarily nearby their initial configurations, typically they are asymmetric, and often we can drive the same initial state to several different equilibria. A reverse-ratchet method is also described, which can produce initial conditions that would evolve to a specified equilibrium state. Once a converged final state is achieved, the full singular value decomposition of A is used to calculate an optimal basis set for the nullspace of A and thus all allowable Γ . The distribution of the singular values gives important information on the size of each equilibrium state (as measured by Frobenius norm), their distance from each other (spacing and density) and how far a randomly chosen system of N points in the plane is from the nearest equilibrium configuration with a specified rank, as well as its Shannon entropy.

Singular value decomposition for cross‐well tomography

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1655-1661 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Singular Value Decomposition ◽  
Null Space ◽  
Vertical Component ◽  
Singular Values ◽  
Singular Value ◽  
Velocity Estimation ◽  
One Dimensional ◽  
Anisotropic Models ◽  
Value Decomposition ◽  
Frequency Variations

I perform singular value decomposition (SVD) on the matrices that result in tomographic velocity estimation from cross‐well traveltimes in isotropic and anisotropic media. The slowness model is parameterized in four ways: One‐dimensional (1-D) isotropic, 1-D anisotropic, two‐dimensional (2-D) isotropic, and 2-D anisotropic. The singular value distribution is different for the different parameterizations. One‐dimensional isotropic models can be resolved well but the resolution of the data is poor. One‐dimensional anisotropic models can also be resolved well except for some variations in the vertical component of the slowness that are not sensitive to the data. In 2-D isotropic models, “pure” lateral variations are not sensitive to the data, and when anisotropy is introduced, the result is that the horizontal and vertical component of the slowness cannot be estimated with the same spatial resolution because the null space is mostly related to horizontal and high frequency variations in the vertical component of the slowness. Since the distribution of singular values varies depending on the parametrization used, the effect of conventional regularization procedures in the final solution may also vary. When the model is isotropic, regularization translates into smoothness, and when the model is anisotropic regularization not only smooths but may also alter the anisotropy in the solution.

Effective diffraction separation using the improved optimal rank-reduction method

Geophysics ◽  
2022 ◽  
pp. 1-85
Author(s):  
Peng Lin ◽  
Suping Peng ◽  
Xiaoqin Cui ◽  
Wenfeng Du ◽  
Chuangjian Li
Keyword(s):  
Seismic Data ◽  
Reduction Method ◽  
Singular Values ◽  
Vital Role ◽  
Singular Value ◽  
Low Rank ◽  
Small Scale ◽  
Frobenius Norm ◽  
Rank Reduction ◽  
Reduction Methods

Seismic diffractions encoding subsurface small-scale geologic structures have great potential for high-resolution imaging of subwavelength information. Diffraction separation from the dominant reflected wavefields still plays a vital role because of the weak energy characteristics of the diffractions. Traditional rank-reduction methods based on the low-rank assumption of reflection events have been commonly used for diffraction separation. However, these methods using truncated singular-value decomposition (TSVD) suffer from the problem of reflection-rank selection by singular-value spectrum analysis, especially for complicated seismic data. In addition, the separation problem for the tangent wavefields of reflections and diffractions is challenging. To alleviate these limitations, we propose an effective diffraction separation strategy using an improved optimal rank-reduction method to remove the dependence on the reflection rank and improve the quality of separation results. The improved rank-reduction method adaptively determines the optimal singular values from the input signals by directly solving an optimization problem that minimizes the Frobenius-norm difference between the estimated and exact reflections instead of the TSVD operation. This improved method can effectively overcome the problem of reflection-rank estimation in the global and local rank-reduction methods and adjusts to the diversity and complexity of seismic data. The adaptive data-driven algorithms show good performance in terms of the trade-off between high-quality diffraction separation and reflection suppression for the optimal rank-reduction operation. Applications of the proposed strategy to synthetic and field examples demonstrate the superiority of diffraction separation in detecting and revealing subsurface small-scale geologic discontinuities and inhomogeneities.

Relative equilibrium configurations of point vortices on a sphere

2013 ◽  
Vol 18 (4) ◽  
pp. 344-355 ◽  
Author(s):  
Maria V. Demina ◽  
Nikolai A. Kudryashov
Keyword(s):  
Relative Equilibrium ◽  
Point Vortices ◽  
Equilibrium Configurations

scholarly journals Vortex crystals on the surface of a torus

2019 ◽  
Vol 377 (2158) ◽  
pp. 20180344 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Null Space ◽  
Spherical Surface ◽  
Particle Interaction ◽  
Singular Values ◽  
Point Vortices ◽  
Relative Configuration ◽  
Point Configurations ◽  
Toroidal Surface ◽  
The Matrix ◽  
Value Decomposition

Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose–Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation A Γ  = 0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

scholarly journals The smallest singular value of certain Toeplitz-related parametric triangular matrices

2021 ◽  
Vol 9 (1) ◽  
pp. 103-111
Author(s):  
Maryam Shams Solary ◽  
Alexander Kovačec ◽  
Stefano Serra Capizzano
Keyword(s):  
Numerical Experiments ◽  
Singular Values ◽  
Singular Value ◽  
Triangular Matrices ◽  
Matrix Size ◽  
The Matrix ◽  
Lower Triangular ◽  
Triangular Toeplitz Matrix ◽  
Range Of Values ◽  
Smallest Singular Value

Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a 1, a 2, ..., ap , a 1, ..., ap , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a 1, a 2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a 1, ..., ap . It depends on the asymptotics in µ of the l 2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

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