Constrained Leja points and the numerical solution of the constrained energy problem

In this paper we investigate the asymptotic behavior of the Riesz s-energy of the first N points of a greedy s-energy sequence on the unit circle, for all values of s in the range 0 ≤ s < ∞ (identifying as usual the case s = 0 with the logarithmic energy). In the context of the unit circle, greedy s-energy sequences coincide with the classical Leja sequences constructed using the logarithmic potential. We obtain first-order and second-order asymptotic results. The key idea is to express the Riesz s-energy of the first N points of a greedy s-energy sequence in terms of the binary representation of N . Motivated by our results, we pose some conjectures for general sequences on the unit circle.

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“…[1,3,4,18,20]. Not many works have been devoted to the study of the energy and distribution of Leja sequences; some of these are [7,15,5,12,13].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Asymptotics of the Energy of Sections of Greedy Energy Sequences on the Unit Circle, and Some Conjectures for General Sequences

López-García

Wagner

2015

Comput. Methods Funct. Theory

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“…To our knowledge, very little is known about this problem in dimensions greater than one. In 1D, a similar numerical problem has been addressed by [18] in the weighted case (using an approach involving iterated Balayage), and by [35], [19] in the unweighted case. Even in the 1D situation, there are various assumptions on the Riesz measure corresponding to the weight V , which would be of interest to relax.…”

Section: Simulation Of the Hole Event And Numerical Aspectsmentioning

confidence: 99%

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

Ghosh

Nishry

2018

Constr Approx

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We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities, and the determination of the most likely configurations of particles that have large holes.As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) β-Ginibre ensembles. We also discuss some possible directions for future investigations.

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“…In view of the behavior of our adaptive rational Arnoldi method for the above symmetric matrices, we believe that the convergence can be asymptotically (that is, for a sequence of symmetric matrices growing larger in size and having a joint limit eigenvalue distribution) compared to min-max rational functions with poles on Γ and zeros on Σ being constrained Leja points in the sense of [11]. The constraint for the zeros is given by the interlacing property of Ritz values associated with symmetric matrices (see, e.g., [7]).…”

Section: 1mentioning

confidence: 99%

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

Güttel

Knizhnerman

2013

Bit Numer Math

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Abstract. Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our blackbox method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix.

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Constrained Leja points and the numerical solution of the constrained energy problem

Cited by 11 publications

References 7 publications

Asymptotics of the Energy of Sections of Greedy Energy Sequences on the Unit Circle, and Some Conjectures for General Sequences

Asymptotics of the Energy of Sections of Greedy Energy Sequences on the Unit Circle, and Some Conjectures for General Sequences

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases

A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions

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