As on arXiv
Abstract
This paper proposes a theoretical framework for modeling and optimizing the bounded functions based on the Fourier series approximation and Ricci flow. Specifically, the initial manifold, is approximated using Fourier series approximation in conjunction with the center and boundary sampling procedure introduced in the paper. The manifold is iteratively evolved using an algorithm that involves sampling along geodesic hyper-sphere defined by the Riemannian metric tensor. Thus obtained surrogate manifold is optimized by applying inverse Ricci flow i.e. instead of regularizing the manifold, flow allows for the high curvature regions to blow into finite time singularities. This allows for the singularities to occur at potential global optima assuming the deviation of the manifold at any point is smaller than the optimum. In addition, the error bound is established on the accuracy of the surrogate manifold. Finally, the proposed method is tested on stochastic sampling from five benchmark functions to illustrate the utility of this method.
- Submitted
- Nov 6, 2024
- Last revised
- Mar 12, 2025 (v2)
- Subjects
- math.DG, math.OC
Its neighbourhood
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Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities
Topics Singularities · Shared terms “time singularity”, “finite time”
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Topics Singularities · Shared terms “time singularity”, “finite time”, “proposed”
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Topics Singularities · Shared terms “time singularity”, “finite time”
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Topics Singularities · Shared terms “finite time”
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Topics Singularities · Shared terms “time singularity”, “finite time”
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Topics Ricci flow · Shared terms “stochastic”, “metric tensor”, “proposed”
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Topics Singularities · Shared terms “time singularity”, “finite time”
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Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I
Topics Singularities · Shared terms “time singularity”, “finite time”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.