I am deeply grateful to Professor Edward Saff for his guidance, for his unwavering support and thoughtful feedback throughout the program. I am grateful to all members of the Mathematics Department at Vanderbilt, both faculty and staff, for creating a welcoming and stimulating environment.
Introduction
The expression (?) corresponds to the Hamiltonian of the collection (x1, . . . ,xN) of particles with energetic pair interaction(·,·) subjected to an external potential q. A discussion of the properties of the Riesz kernel that make it quite remarkable is found in Section 1.4.
Overview of results
The support of the limiting distribution is thus characterized as a certain sublevel set of the external field q(x). An important feature of the minimizations of E(·,;gs, q) is that, like those of the unweighted E(·,;gs,0), they have the optimal order minimal separation distances and (on the support of µq) coverage distance, see sentence below.
Notation and layout
Note that the positive definiteness of the Riesz kernels also implies that the functional in (??) is convex on P(Ω) and therefore has a unique solution. Optimizing (??) forq, which is identically equal to zero, is then equivalent to the problem of approximating the minimizing distribution with discrete measures in the aforementioned Hilbert space.
Riesz kernel
Note that in harmonic=p−2 cases it is sufficient to require that F has 3 continuous derivatives; from5 it follows that the corresponding density of µ is then given by−4π12∆F. I(µ) of a lower semicontinuous kernel is itself lower semicontinuous in the weak topology∗, as a functional on the space of probability measures in Ω.
Hypersingular case s ≥ d
For example, as can be seen from 3, a positive definite kernel such as Is(µ) corresponds to a scalar product on the space of probability measurements on Ω. Clearly, the quadratic form I[µ, ν] can then serve as a scalar product to introduce Hilbert space topology on the vector space of all signed measures of Ω with finite energies.
Γ-convergence
Γ-convergence for the integrable case
In this section we will outline a proof of Γ-convergence in order to compare it with the corresponding proofs in the hypersingular case, see Sections 1.5.2 and 2.5. Ωqdµ is linear in µ, the functional I(µ;gs, q) is lower semicontinuous in µ due to the semicontinuity of gs and q, see1‡.
Γ-convergence for the hypersingular case
To verify the property1Γof the definition of Γ-convergence, we suppose that a sequence{µN} ⊂ P(Ω) converges weakly∗ to µ∈ P(Ω); note that if. Note that bothq and q∗ are continuous; the latter follows from the continuity of the density of µ.
Overview
In the language of Section 1.5, the present chapter contains the proof of the convergence Γ of the function (?) (also in screen (2.1) below) for a hypersingular kernel in the presence of an external field. A motivation for considering this energy expression is that (under mild conditions on the set Ω) for any probability measure μ on Ω that is absolutely continuous with respect to the bounded d-dimensional Hausdorff measure on Ω, there exists a field the exterior is easily described. (x) for which the normalized count measures of the weak minimizers of (2.1)∗ converge to µ (formal definitions are given in the next two subsections).
Main results
A Poppy-seed bagel theorem for (s, d, q)-energy
As with Theorem 1.3, this result holds for the (possibly) larger class of sets Ω satisfying Hd(Ω) =Md(Ω), where Mis the d-dimensional Minkowski content. As an application of Theorem 2.1, we derive a method to construct a series of (s, d, q)- energy-minimizing sets ˆωN such that their normalized counting measures weakly∗ converge to a given distribution.
Separation and covering properties of minimal configurations
Due to this conclusion, the sets {ˆωN}N≥2 for the energy minimization problem (s, d, q) in the entire space Rp are restricted to a compact set, provided that for some compact Ω and a sufficient cube large Cp := [−R, R]p with Ω⊂ Cp, the value C in (2.14) is such that q(x)> C for every xnot inCp. A set Ω ⊂Rp with Hd(Ω)>0 is called d-regular with respect to µ if it has positive constants c0, C0 and a locally finite positive Borel measure µ, such that.
Examples and numerics
Let us construct a sequence of discrete collections{ωˆN}N≥2 weak∗ converging to the probability distribution with density proportional to . S2 is highlighted, as are the positions of the fixed "repulsive charges" that create the external field qd.
Proofs
Proofs of the main theorems
[81, Theorem 7.5] implies namely that it is locally finite because Ω is the Lipschitz image of a compact set inRd. Then, by Theorem 2.1, this sequence converges weakly∗ to dµqC, and it remains to observe that for C > L1 dµqC = dµq, where the two measures are defined in equation (2.5). The desired result is an immediate application of Theorem 2.1 since using equation (2.5) for the external field from (2.9) gives L1 = 0, so the asymptotic distribution is indeed (2.10).
Proofs of separation and covering properties
Combined kernels
To complete the proof of the property 1Γ, we must show that the sum in the RHS of the last inequality approaches the value of S(µ;gs, κ, q). Furthermore, for eachx∈Ωm, 0≤κ(x,x)−κm ≤κm−κm< ε; by taking ε small enough and applying the monotone convergence theorem, we complete the proof of the property 1Γ of Γ-convergence. Furthermore, in the case where contraction ratios of the fractals Ω are all equal, the energy limit Es(Ω, N)/N1+s/d, N ∈ N, along a sequence N ⊂ N can be characterized by the behavior of sequence itself .
Self-similarity and open set condition
It will also be used that if Ω is a self-similar fractal satisfying the open set condition, then 0
Main results
Let Ω ⊂Rp be a self-similar fractal fixed in M similarities with the same shrinkage ratio and M={Mkn : k≥1}. The previous theorem can be slightly extended to show that in the case of equal shrinkage ratiosr1=. If Ω is a self-similar fractal with equal shrinkage ratios, and two sequences N1,N2 ⊂N are such that.
Proofs
The following result indeed establishes a bijection between cluster points of {{logMN} : N ∈N} and those of the set {Es(Ω, N)/N1+s/d : N ∈N}. Thus, bounding the RHS of the previous equation from below will yield the desired result. In the next lemma we write N(k), k ∈N, to denote the k-th element of the sequence N⊂N; we say that Nis is dominated by a set M, if the inequality N(k) In contrast to pseudospectral methods [49], the RBF-FD approach means that in order to obtain a useful approximation of a function or a differential operator, the nodes in expressions like (4.1) must be in the vicinity of pointx, and therefore numerous stencils are constructed throughout the underlying set. The other extreme, which has low regularity, also does not provide a reliable source of nodes, as can be seen in the example of the Halton sequence [53]. In many applications, one must ensure that the distance from a node x to its nearest neighbor behaves approximately as a function of the node's position [52]. In summary, the above expression for weights {wk} is the defining property of the RBF-FD methods with the Gaussian kernel. We now conclude the discussion about the Gauss kernel and look at its new alternative. The remainder of the discussion for the Gaussian kernel above is otherwise applicable without any modification. We will assume that the terms in Esκ for which xj is not among the K nearest nodes toxi are zero, a condition equivalent to truncating κ, provided the nodes are well separated. This further implies that they are quasi-uniform, which for us is the most important property for the purposes of the discussion in section 4.2.1. The value of the exponent is chosen such that s≥d ensures that the functional energy is sufficiently repulsive; from classical potential theory it is known that fors < d the minimum energy configurations are not necessarily uniform, and that their local structure depends on the shape of the domain [74]. The motivation for using an IL in this context is due to the existing results on the discrepancy of ILs. The number of nodes in individual voxels is defined by the function ρ, so the resulting set has piecewise constant density; refining the voxel partition leads to an improved piecewise approximation of the desired (e.g. smooth) density. In practice, the dependency between the number of nodes in the unit cube and the average/minimum distance of the nearest neighbor is tabulated beforehand, and then inverted during the construction of the node set. We can further assume that Ω is contained in the d-dimensional unit cube Cd= [0,1]d (see Figure 4.1 for some of the notations involved); the case of an arbitrary compact set is then followed by the selection of an appropriate bounding cube and the application of scaling and translation. Let {Vm:m∈ D} be a subset where at least one of the neighboring (i.e., sharing a face) voxels has a vertex inside Ω. Vm a reduced and translated version of the IL-point nm (4.8) or the periodic Riesz minimizer of the nm points using nm which defines it. On the other hand, for D1 the distances from zm0 to the nodes from Step 2 are also controlled: derived from that small shape, the nodes in voxelVm have larger insets (depending on cd). The ratio of the second term to the first in (4.13) is bounded by ∇xρ(x)kx−yk/ρ(x) and, provided that the distances from x to its nearest neighbors are close to the value of ρ(x) , is at most ∇xρ(x) for a constant c. Note, the second case in (4.10), which leads to the shrinking of the line step distance, can be thought of as a simplified backward line search;. Left: Probability distribution of the nearest neighbor distances in the set of atmospheric nodes, before (blue) and after (red) executing the repel subroutine. Right: Node locations contributing to the distribution of the ρ(x)/∆(x) ratio beyond the 5 and 95 percentiles. Instead, in Figure 4.9 we show the performance of the generic algorithm, to illustrate complications that can arise when applying it to specialized problems. An implementation of the {Mn} sequence for 1≤n≤200 took 4311 seconds to generate; coordinates of the resulting minimizers as well as the corresponding average separation distances are distributed with the associated codebase [102]. The nearest neighbors were drawn from 1000 nodes of the respective sequence, uniformly distributed over the unit cube. The values shown are averages of the mode numbers for 500 random stencil centersx0; the average was introduced to eliminate the rather unpredictable dependence on x0 and to show the underlying trend. Right: ratios of the mean separation distances to the minimal ones for the same configurations. The second graph in Figure 4.11 shows the relationships between the mean and minimum separation distance h∆ni/∆n for the same range n. We conclude this section by presenting in Figure 4.12 a pair of cross-sections of IL L100 and M100 configurations, which look remarkably similar. T(N) order of Riesz energy asymptotics 8 τ(N) asymptotic scaling factor for the external field term. Full dimension case A sequence of discrete minimum energy configurations that do not converge in the faint star topology. 56] Fornberg, B., Wright, G., and Larsson, E. Some observations concerning interpolants in the limit of flat radial basis functions. Stable computation of differentiation matrices and sparse node stencils based on Gaussian radial basis functions.Formulation of the problem
Choice of method
RBF-FD approximations
Riesz energy
Quasi-Monte Carlo methods
The algorithm
Formulation
Discussion
Sample applications
Final observations and comparisons
Comparisons
Range of applications
Separation properties of sequences {L n } and {M n }
Repelling on implicit surfaces
![Figure 2.1: The exact graph of density dµ q [0,2] is pictured in red, graphs for perturbed values of C 4,1 are in blue.](https://thumb-ap.123doks.com/thumbv2/123dok/11039391.0/33.918.302.612.387.572/figure-exact-graph-density-pictured-graphs-perturbed-values.webp)
![Figure 2.7: Left, empirical density dµ emp of ω e , an approximate 500-point (4, 1, q e (x))-energy minimizer from Example 2.16 (red) overlaid with the graph of dµ q [0,2]e (blue); right, ω e and the graph of external field q e .](https://thumb-ap.123doks.com/thumbv2/123dok/11039391.0/39.918.147.770.105.297/figure-empirical-density-approximate-minimizer-example-overlaid-external.webp)
