Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

Bungert, Leon; Burger, Martin

doi:10.1007/s00028-019-00545-1

Cited by 28 publications

(52 citation statements)

References 45 publications

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“…If P is a subgradient of a convex homogeneous functional part of the Assumption 1 can be proven [33]. We note that subgradients admit mass preservation.…”

Section: Fractional Calculus Backgroundmentioning

confidence: 95%

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Introducing the p-Laplacian spectra

Cohen

Gilboa

2020

Signal Processing

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In this work we develop a nonlinear decomposition, associated with nonlinear eigenfunctions of the p-Laplacian for p ∈ (1, 2). With this decomposition we can process signals of different degrees of smoothness.We first analyze solutions of scale spaces, generated by γ-homogeneous operators, γ ∈ R. An analytic solution is formulated when the scale space is initialized with a nonlinear eigenfunction of the respective operator. We show that the flow is extinct in finite time for γ ∈ [0, 1).A main innovation in this study is concerned with operators of fractional homogeneity, which require the mathematical framework of fractional calculus.The proposed transform rigorously defines the notions of decomposition, reconstruction, filtering and spectrum. The theory is applied to the p-Laplacian operator, where the tools developed in this framework are demonstrated.

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“…If P is a subgradient of a convex homogeneous functional part of the Assumption 1 can be proven [33]. We note that subgradients admit mass preservation.…”

Section: Fractional Calculus Backgroundmentioning

confidence: 95%

“…Extinction time for arbitrary initial conditions. The corresponding proof can be found in [33] (Theorem 2.13).…”

Section: Shape Preserving Flows For Homogeneous Operatorsmentioning

confidence: 99%

Introducing the p-Laplacian spectra

Cohen

Gilboa

2020

Signal Processing

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“…Theorem 2 (Convergence of Minimizers) Let Ω ⊂ R d be a domain satisfying (11), let the kernel fulfill (K1)-(K3), let the constraint sets O n , O satisfy (3), and (s n ) n∈N ⊂ (0, ∞) be a null sequence which satisfies the scaling condition (8). Then any sequence…”

Section: Assumptions and Main Resultsmentioning

confidence: 99%

“…Still, in the recent work [12] we successfully used comparison principle techniques to show rates of convergence for solutions of the graph infinity Laplacian equation ( 15) in a much more general setting than the one considered in [14]. For showing this it suffices to use quantitative versions of the weak scaling assumption (8) and the domain regularity condition (11).…”

Section: Related Workmentioning

confidence: 99%

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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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Pointwise eigenvector estimates by landscape functions: Some variations on the Filoche–Mayboroda–van den Berg bound

Mugnolo

2023

Mathematische Nachrichten

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Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including p‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments.

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Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

Cited by 28 publications

References 45 publications

Introducing the p-Laplacian spectra

Introducing the p-Laplacian spectra

Continuum Limit of Lipschitz Learning on Graphs

Pointwise eigenvector estimates by landscape functions: Some variations on the Filoche–Mayboroda–van den Berg bound

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