Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

Cluster Gauss–Newton method

Parameter estimation problems of mathematical models can often be formulated as nonlinear least squares problems. Typically these problems are solved numerically using iterative methods. The local minimiser obtained using these iterative methods usually depends on the choice of the initial iterate. Thus, the estimated parameter and subsequent analyses using it depend on the choice of the initial iterate. One way to reduce the analysis bias due to the choice of the initial iterate is to repeat the algorithm from multiple initial iterates (i.e. use a multi-start method). However, the procedure can be computationally intensive and is not always used in practice. To overcome this problem, we propose the Cluster Gauss–Newton (CGN) method, an efficient algorithm for finding multiple approximate minimisers of nonlinear-least squares problems. CGN simultaneously solves the nonlinear least squares problem from multiple initial iterates. Then, CGN iteratively improves the approximations from these initial iterates similarly to the Gauss–Newton method. However, it uses a global linear approximation instead of the Jacobian. The global linear approximations are computed collectively among all the iterates to minimise the computational cost associated with the evaluation of the mathematical model. We use physiologically based pharmacokinetic (PBPK) models used in pharmaceutical drug development to demonstrate its use and show that CGN is computationally more efficient and more robust against local minima compared to the standard Levenberg–Marquardt method, as well as state-of-the art multi-start and derivative-free methods.

An Efficient Conjugate Gradient Method for Convex Constrained Monotone Nonlinear Equations with Applications

This research paper proposes a derivative-free method for solving systems of nonlinearequations with closed and convex constraints, where the functions under consideration are continuousand monotone. Given an initial iterate, the process first generates a specific direction and then employsa line search strategy along the direction to calculate a new iterate. If the new iterate solves theproblem, the process will stop. Otherwise, the projection of the new iterate onto the closed convex set(constraint set) determines the next iterate. In addition, the direction satisfies the sufficient descentcondition and the global convergence of the method is established under suitable assumptions.Finally, some numerical experiments were presented to show the performance of the proposedmethod in solving nonlinear equations and its application in image recovery problems.

On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case

In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local ω -continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section.

Convergence Analysis of Gradient Descent for Eigenvector Computation

We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{ (lambda_1/Delta_p)^2 log(1/epsilon), 1/epsilon }), where Delta_p=lambda_p-lambda_p+1>0 represents the generalized positive eigengap and always exists without loss of generality with lambda_i being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of lambda_1. The rate is linear at (lambda_1/Delta_p)^2 log(1/epsilon) if (lambda_1/Delta_p)^2=O(1), otherwise sub-linear at O(1/epsilon). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap Delta_1= lambda_1 - lambda_2=0 but the generalized eigengap Delta_p satisfies (lambda_1/Delta_p)^2=O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.

Application of artificial viscosity in establishing supercritical solutions to the transonic integral equation

The nonlinear singular integral equation of transonic flow is examined in the free-stream Mach number range where only solutions with shocks are known to exist. It is shown that, by the addition of an artificial viscosity term to the integral equation, even the direct iterative scheme, with the linear solution as the initial iterate, leads to convergence. Detailed tables indicating how the solution varies with changes in the parameters of the artificial viscosity term are also given. In the best cases (when the artificial viscosity is smallest), the solutions compare well with known results, their characteristic feature being the representation of the shock by steep gradients rather than by abrupt discontinuities. However, ‘sharp-shock solutions’ have also been obtained by the implementation of a quadratic iterative scheme with the ‘artificial viscosity solution’ as the initial iterate; the converged solution with a sharp shock is obtained with only a few more iterates. Finally, a review is given of various shock-capturing and shock-fitting schemes for the transonic flow equations in general, and for the transonic integral equation in particular, frequent comparisons being made with the approach of this paper.