| 研究課題/領域番号 |
22K03440
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分12040:応用数学および統計数学関連
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| 研究機関 | 明治大学 |
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研究代表者 |
Ginder Elliott 明治大学, 総合数理学部, 専任教授 (30648217)
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| 研究期間 (年度) |
2022-04-01 – 2025-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,160千円 (直接経費: 3,200千円、間接経費: 960千円)
2024年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2023年度: 1,950千円 (直接経費: 1,500千円、間接経費: 450千円)
2022年度: 1,560千円 (直接経費: 1,200千円、間接経費: 360千円)
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| キーワード | surface PDE / minimizing movements / interfacial dynamics / level set method / approximation methods / Minimizing Movements / Surface PDE / Interfacial dynamics / Curvature flow / Approximation methods |
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| 研究開始時の研究の概要 |
Minimizing movements (MM) の理論とその応用は，平坦空間で広く研究されてきた．この近似解法は，時間が離散化された汎関数の最小化に基づいており，数学的な解析だけでなく，数値計算にも用途がある．本研究は，MMの適用性を曲面上における偏微分方程式（SPDE）の設定にまで拡張することを目指し，更なる応用研究を進めたい．主なる対象として，放物型と双曲型SPDE用MMの数理解析に加え，計算手法の開発を確立し，それらによる解の振る舞いを平坦空間の解の振る舞いを数理的に比較する．また，閾値法(threshold dynamics，TD法)の利用を曲面にまで展開したいと考えている．
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| 研究実績の概要 |
This year, we focused on furthering our surface-type minimizing movments and their application to constrained interfacial motions. We successfully formulated surface MBO and surface HMBO algorithms in the minimizing movement framework and, by extending techniques from the level set method to the surface PDE setting, we generalized these algorithms to incorporate area-constraints. Additionally, our algorithms were formalized as approximation methods, and their numerical error analyses were performed.
Our computational methods, those corresponding to the surface-type minimizing movement for heat-type problems, were used to illustrate the numerical behavior of area-constrained motion of interfaces moving under surface-constrained curvature flow in the two phase setting. Similarly, our wave-type minimizing movement was used to visualize the behavior of area-constrained hyperbolic mean curvature flow on surfaces. These results revealed new novel interfacial motions, and suggest the wide-ranging applicability of the closest point method.
We also integrated the SDVF into our surface-type minimizing movements and conducted numerous experiments involving multiphase geometries. These experiments demonstrated that, as in the flat setting, penalty terms can be employed in surface-type minimizing movements. Moreover, given that computations involving the SDVF are conducted on point clouds, interpolation techniques that precisely locate interfaces were required. In response, we devised and tested a technique to address this challenge.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Our research has progressed according to our original plan. Namely, this year we successfully extended our surface-type minimizing movements to be able to treat area-constrained interfacial motions. We also performed corresponding numerical error analyses, which was one of our main research goals. Also, the computational speed of the numerical methods used to construct the signed distance vector fields were improved. Although they are still computationally heavy, these improvements have enabled us to apply our methods to solve surface PDE on complex geometries. Our work also uncovered a new area for research. In particular, we have found that the notion of "point correspondence" could be added into our HMBO algorithm.
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| 今後の研究の推進方策 |
This year, using the signed distance vector field, we aim to show that surface-type minimizing movements can approximate multiphase area-constrained interfacial motions on surfaces. Taking surface MCF and surface HMCF as primary examples, this will be done by adding auxillary penalties onto energy functionals for each of the area constraints. Although new level set techniques for computing multiphase areas and lengths will be required, we expect that the approaches in the flat setting will carry over to the case of surfaces. We also anticipate that the interpolation methods used within the closest point method will need refining, especially near junctions. Regarding the newly understood point correspondence, we will first inquire about its usage in the flat setting.
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