一般的な輸送コストを伴う JKO プログラム
Cale Rankin1, Ting-Kam Leonard Wong2
1Department of Mathematics, Monash University, Victoria, Australia.
Calculus of variations and partial differential equations
|February 20, 2026
まとめ
JKOのワッサータイン・グラデントフローのスキームを,マニホールドにおける一般輸送コストを用いて修正しました. この修正されたスキームはリーマンのフォッカー・プランク方程式に収束し,計算上の利点を提供します.
科学分野:
- 数学的分析について.
- 微分幾何学による微分幾何学
- ストキャスティック・プロセス ストキャスティック・プロセス
背景:
- JKOスキームは,ワッサースタインのグラデントフローの時間ディスクリテーションです.
- ワースタータイン距離は,マニホールドでは計算的に集約的です.
- フォッカー・プランク方程式は,拡散プロセスをモデル化します.
研究 の 目的:
- マニホールド上の任意の輸送コストを使用してJKOスキームを一般化します.
- リーマンのフォッカー・プランク方程式への収束を確立するために.
- リーマン距離の計算上の代替案を探求する.
主な方法:
- ワッサースタイン距離を一般輸送コストに置き換えることで,JKOスキームの変更.
- コスト関数のヘッセン条件下での収束特性の分析.
- コンパクトで完全なリーマン多様体に関するフォッカー=プランク方程式の適用.
主要な成果:
- コストがリーマンメトリックを誘導するとき,リーマンのフォッカー・プランク方程式に変更されたJKOスキームの収束.
- ノイマン境界条件と完全なリーマン多様体を持つコンパクトサブマニフォールドへの適用性の実証.
- コストとしてブレグマン分岐を用いたヘッセン多様体への成功応用.
結論:
- 一般化されたJKOスキームは,マニホールド上のグラデントフローをディスクレート化するための柔軟なフレームワークを提供します.
- このアプローチは,リーマン距離が難解な場合に計算上の利点を提供します.
- この方法は,最適な輸送,幾何学分析,数学的方法を効果的に結びつけています.
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