Development of effective and accurate non-conventional solution methods for shape inverse problems: theory and numerics
| Project/Area Number |
23K13012
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Kanazawa University |
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Principal Investigator |
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| Project Period (FY) |
2023-04-01 – 2027-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2026: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2025: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2024: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2023: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | inverse geometry problem / ADMM / shape optimization / shape identification / Shape inverse problems |
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| Outline of Research at the Start |
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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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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Report
(1 results)
Research Products
(15 results)
