Surface PDE: a minimizing movement approach
Project/Area Number |
22K03440
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Research Category |
Grant-in-Aid for Scientific Research (C)
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Allocation Type | Multi-year Fund |
Section | 一般 |
Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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Research Institution | Meiji University |
Principal Investigator |
Ginder Elliott 明治大学, 総合数理学部, 専任教授 (30648217)
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Project Period (FY) |
2022-04-01 – 2025-03-31
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Project Status |
Granted (Fiscal Year 2022)
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Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2024: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2023: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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Keywords | surface PDE / minimizing movements / interfacial dynamics / level set method / approximation methods / Minimizing Movements / Surface PDE / Interfacial dynamics / Curvature flow / Approximation methods |
Outline of Research at the Start |
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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Outline of Annual Research Achievements |
Our research focused on developing minimizing movments for surface-constrained partial differential equations. The corresponding approximation methods were successfully realized by incorporating the closest point method into a functional minimization scheme. Using our surface-type minimizing movements, we were able to effectively simulate mean curvature flow and hyperbolic mean curvature flow of interfaces on surfaces. These methods represent generalizations of the MBO (Merriman, Bence, and Osher) and HMBO (Hyperbolic Merriman, Bence, and Osher) algorithms, specifically tailored for the surface-constrained setting. In addition, we designed a surface-constrained signed distance vector field (SDVF) for describing phase geometries on surfaces in multiphase settings. We further implemented the numerical algorithms that enable the application of the SDVF to computational problems. Regarding our approximation method that combines the closest point method and minimizing movements, numerical error analyses were conducted for the heat and wave equations on surfaces, under various conditions. Convergence of our surface-type minimizing movement, with respect to the spatial discretization, was also examined. Our the results revealed that the numerical solution converges to the exact solution.
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Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Over the past year, our research on minimizing movements for surface partial differential equations has made significant progress, closely aligning with our original plan and encountering minimal obstacles. Our primary goal was to extend the applicability of the closest point method by incorporating it into the setting of minimizing movements.
By addressing this task early on, we effectively initiated our research and established momentum, allowing it to commence smoothly and efficiently. Notably, numerical tests and analyses played a vital role in confirming and bolstering our findings, providing robust evidence to support our research outcomes.
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Strategy for Future Research Activity |
Looking ahead, the roadmap for our research included various tasks aimed at enhancing and refining our approximation methods. First of all, we endeavor to reduce the computational time of our methods--efficient, and accurate algorithms will enable new simulations and applications. Second, we will perform a numerical analysis of area preserving motions. This analysis will allow us to quantify and understand the numerical errors associated with our approximation methods, and to facilitate their refinement. Finally, it is necessary to evaluate the performance of our methods on complex geometries. These tests will access the robustness, stability, and accuracy of our methods on complex surface geometries. In turn, this will help to identify limitations and areas for improvement.
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Report
(1 results)
Research Products
(12 results)