2020年度 数学特殊講義F   Special courses on advanced topics in Mathematics F

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開講元
数学科
担当教員名
Pozar Norbert 
授業形態
講義
メディア利用
 
曜日・時限(講義室)
-
クラス
-
科目コード
ZUA.E336
単位数
2
開講年度
2020年度
開講クォーター
3Q
シラバス更新日
2020年6月7日
講義資料更新日
-
使用言語
英語
アクセスランキング
media

講義の概要とねらい

"This course will cover the mean curvature flow from the point of view of the level set method and viscosity solutions. In particular, we will study the anisotropic and crystalline mean curvature flows that serve as models of the evolution of crystals. We will take the point of view of the level set method that allows us to find the solution of the flow as a solution a nonlinear parabolic partial differential equation. Since the most natural notion of generalized solutions are the viscosity solutions, we will spend some time on their introduction and cover some basic properties like the comparison principle and stability. The crystalline mean curvature flow requires us to introduce the notion of facets and the crystalline mean curvature via a connection to the total variation energy. Finally, we will discuss a robust numerical method for the anisotropic mean curvature flow.

Evolution of surfaces and curves have many applications in geometry, material science, image processing, and other fields. Among the most important ones are the evolutions driven by the surface energy, for example the curve shortening flow. The aim of this course is to cover one of the most popular mathematical approaches to this problem, with some discussion of the recent results for surface energies with singular dependence on the normal vector to the surface: the crystalline mean curvature flow."

到達目標

"・Be familiar with the mean curvature flow and its anisotropic variants.
・Understand the level set method for tracking geometric flows.
・Understand fundamentals of the theory of viscosity solutions.
・Learn about numerical methods for mean curvature flows.
・Get aquinted with viscosity solutions for the crystalline mean curvature flow."

実務経験のある教員等による授業科目等

-

キーワード

anisotropic and crystalline mean curvature flow, viscosity solutions, minimizing movements, level set method, comparison principle

学生が身につける力(ディグリー・ポリシー)

専門力 教養力 コミュニケーション力 展開力(探究力又は設定力) 展開力(実践力又は解決力)

授業の進め方

通常の講義形式で行う.また,適宜レポート課題を出す.

授業計画・課題

  授業計画 課題
第1回 "The lectures will cover the following topics (the order is tentative): ・mean curvature flows ・level set method ・geometric partial differential equations ・viscosity solutions for geometric PDEs ・comparison principle, stability, existence of solutions ・anisotropic and crystalline mean curvature flows ・total variation flow ・facets, notion of crystalline mean curvature, examples of solutions ・viscosity solutions for the crystalline mean curvature flow ・discretization of the anisotropic mean curvature flow: minimizing movements, Chambolle's algorithm, total variation minimization algorithm " 講義中に指示する.

教科書

使用しない

参考書、講義資料等

"Giga, Y., Surface evolution equations: A level set approach, Birkhauser Verlag, Basel, 2006 (For those who want to learn more but not required)
Other course material will be announced in the class."

成績評価の基準及び方法

レポート課題(100%)による.

関連する科目

  • MTH.C351 : 函数解析

履修の条件(知識・技能・履修済科目等)

なし

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