| 研究課題/領域番号 |
23K13012
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| 研究種目 |
若手研究
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| 配分区分 | 基金 |
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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 | 金沢大学 |
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研究代表者 |
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| 研究期間 (年度) |
2023-04-01 – 2027-03-31
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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 |
3,120千円 (直接経費: 2,400千円、間接経費: 720千円)
2026年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2025年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2024年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2023年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
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| キーワード | inverse geometry problem / ADMM / shape optimization / shape identification / Shape inverse problems |
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| 研究開始時の研究の概要 |
This research aims to effectively and accurately solve time-dependent inverse geometry problems with complex geometries and under noisy data by developing non-conventional solution methods. The analysis of the problems at long time horizons and rigorous mesh sensitivity analysis are carried out to support the method's development. A wide range of analytical methods such as numerical analyses and tools from optimal control theory are required. Important applications from small to large scale problems such as identification of tumor shapes and exploration of geological resources are expected.
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| 研究実績の概要 |
During the last fiscal year, as part of my research plan, I had three papers published, one accepted paper, and two submitted manuscripts, all in highly respected peer-reviewed journals. The results were presented in three international conferences and three local scientific meetings.
The first published paper addresses an inverse problem within the context of the stationary advection-diffusion problem. The second published paper examines a novel and stable shape optimization method for free surface problems with Stokes flow, achieved through the coupling of boundary data as a complex Robin-type boundary condition. The accepted paper establishes results on existence, stability analysis, and inversion via multiple measurements for boundary shape reconstruction, providing more accurate reconstructions of the unknown shapes. The two submitted papers explore the application of the new coupled complex boundary method to obstacle detection in Stokes fluid flow and present the development of a novel robust alternating direction method of multipliers for solving geometric inverse problems in a shape optimization setting. These papers align closely with the objectives outlined in Project (A) of the proposal.
The other published paper introduces a comoving mesh method for multi-dimensional moving boundary problems which plays a crucial role in developing shape optimization methods for time-dependent problems. The results of this study are directly relevant to the theme of the proposed research.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
Several objectives specified in the proposal have been accomplished through the papers that have been published and submitted. The findings within these papers suggest that dealing with the more demanding elements of the proposal, which encompass time-dependent cases of the model equations, could present more difficulties. Nonetheless, they also underscore the existence of several captivating nuanced issues that demand comprehension before addressing the more challenging objectives. At present, my attention is directed towards identifying and resolving these finer issues related to gradient flows within stationary contexts.
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| 今後の研究の推進方策 |
Instead of directly addressing the more challenging aspects of the proposal, my current focus lies in delving into the intriguing questions surrounding gradient flows, which originated from the initial phase of the proposal. These inquiries have already been addressed to some extent within the context of stationary problems. I anticipate that the findings of these investigations will offer valuable insights for navigating the ambitious components of the proposal in subsequent stages.
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