2022 Fiscal Year Final Research Report
Research on Preconditioners for Ill-Conditioned Linear Systems
| Project/Area Number |
19K20281
|
|---|
| Research Category |
Grant-in-Aid for Early-Career Scientists
|
| Allocation Type | Multi-year Fund |
|---|
| Review Section |
Basic Section 60090:High performance computing-related
|
|---|
| Research Institution | Kansai University of International Studies (2021-2022)
Chuo University (2019-2020) |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2019-04-01 – 2023-03-31
|
| Keywords | 数値線形代数 / 前処理行列 / 悪条件行列 / 連立一次方程式 / 精度保証付き数値計算 |
|---|
| Outline of Final Research Achievements |
In this research period, a preconditioner for ill-conditioned least squares problem was proposed. The proposed preconditioner uses the QR decomposition of the coefficient matrix of the normal equation of the least squares problem. This preconditioner does not work without rounding errors. However, it is shown numerically that it becomes a preconditioner due to rounding errors.
In a related work, verification methods for sparse non-symmetric linear systems have been proposed. The presentation received the JSIAM 2019 Young Scientist Outstanding Presentation Award was received. A verification method for sparse least squares problems was proposed and received the JSST 2019 Outstanding Presentation Award.
|
| Free Research Field |
精度保証付き数値計算
|
| Academic Significance and Societal Importance of the Research Achievements |
疎な連立一次方程式の精度保証付き数値計算は精度保証付き数値計算における重要な課題として認識されている、LU分解を用いた疎な連立一次方程式の精度保証付き数値計算は多くの問題に適用ができる可能性があり、精度保証付き数値計算の実応用に貢献したと考えられる。また、前処理行列に関しては丸め誤差を行列近似として捉えることにより、新しい前処理行列を構成できることを示した。ただし、行列が悪条件であることが必要となる。そのため、すべての問題に適用できるわけではないので、社会的な意義は大きくない。しかし、悪条件を考える際に新しい方向性を示した、という意味では学術的な価値があると考えられる。
|
