Inverse iteration for $p$-ground states

We present two novel methods for approximating minimizers of the abstract Rayleigh quotient Φ(u)/ u p . Here Φ is a strictly convex functional on a Banach space with norm · , and Φ is assumed to be positively homogeneous of degree p ∈ (1, ∞). Minimizers are shown to satisfy ∂Φ(u) − λJ p (u) ∋ 0 for a certain λ ∈ R, where J p is the subdifferential of 1 p · p . The first approximation scheme is based on inverse iteration for square matrices and involves sequences that satisfyThe second method is based on the large time behavior of solutions of the doubly nonlinear evolutionand more generally p-curves of maximal slope for Φ. We show that both schemes have the remarkable property that the Rayleigh quotient is nonincreasing along solutions and that properly scaled solutions converge to a minimizer of Φ(u)/ u p . These results are new even for Hilbert spaces and their primary application is in the approximation of optimal constants and extremal functions for inequalities in Sobolev spaces.

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“…This result first verified in our previous work [21] and was motivated by the paper of R. Biezuner, G. Ercole, and E. Martins [9].…”

Section: Inverse Iterationsupporting

confidence: 77%

Approximation of the least Rayleigh quotient for degree p homogeneous functionals

Hynd

Lindgren

2017

Journal of Functional Analysis

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“…Our results in this paper complement those of [15]. In the first part of that paper the authors extend their own results presented in [14] to an abstract setting, aiming to approximate the least Rayleigh quotient Φ(u)/ u p Y where, according to our notation, Φ : X → [0, ∞] is a functional satisfying certain properties (among them, strict convexity and positive homogeneity of degree p > 1) and X := {u ∈ Y : Φ(u) < ∞} . The authors reduce the problem of minimizing the Rayleigh quotient above to an equivalent subdifferential equation involving the subdifferentials of both functionals Φ and 1 p · p Y .…”

Section: Introductionsupporting

confidence: 88%

“…Indeed, since both w n and w n+1 belong to S Y , by taking v = w n+1 in (14) and using the definition of µ we find…”

Section: The Results Of Convergencementioning

confidence: 99%

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Solving an Abstract Nonlinear Eigenvalue Problem by the Inverse Iteration Method

Ercole

2018

Bull Braz Math Soc, New Series

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Let (X, · X ) and (Y, · Y ) be Banach spaces over R, with X uniformly convex and compactly embedded into Y. The inverse iteration method is applied to solve the abstract eigenvalue problemKeywords: Eigenvalue problems, inverse iteration, quasilinear elliptic equations.with the equality occurring whenever u = tv, for some t ≥ 0;(AB) for each w ∈ Y \ {0} given, there exists at least one u ∈ X \ {0} such thatWe observe from (A1) and (B1) that (1) is homogeneous, that is: if (λ, w) solves (1) the same holds true for (λ, tw) , for all t = 0. Motivated by this intrinsic property of eigenvalue problems, we say that λ is an eigenvalue of (1) and that w is an eigenvector of (1) corresponding to λ or, for shortness, we simply say that (λ, w) is an eigenpair of (1).Hypotheses (A2) and (B2) imply, respectively, that

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“…Ground states and their relations to gradient flows and power methods are well studied in the literature, see, e.g., [11,13,21,[24][25][26]. In particular, they constitute minimizers of the nonlinear Rayleigh quotient…”

Section: Application To Ground Statesmentioning

confidence: 99%

Continuum Limit of Lipschitz Learning on Graphs

Tim

Bungert

2022

Found Comput Math

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Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, for example, of differential operators. A popular strategy here is p-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For $$p<\infty $$ p < ∞ continuum limits of this approach were studied using tools from $$\varGamma $$ Γ -convergence. For the case $$p=\infty $$ p = ∞ , which is referred to as Lipschitz learning, continuum limits of the related infinity Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using $$\varGamma $$ Γ -convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove $$\varGamma $$ Γ -convergence in the $$L^{\infty }$$ L ∞ -topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained $$L^p$$ L p -norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.

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Inverse iteration for $p$-ground states

Cited by 13 publications

References 11 publications

Approximation of the least Rayleigh quotient for degree p homogeneous functionals

Approximation of the least Rayleigh quotient for degree p homogeneous functionals

Solving an Abstract Nonlinear Eigenvalue Problem by the Inverse Iteration Method

Continuum Limit of Lipschitz Learning on Graphs

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