In this lecture, students will study the theorems of functional analysis deeply and learn how to apply them to numerical analysis.
The studentes are expected to know the basic concepts concerning topological space, Banach space, Hilbert space, Lebesgue integral, etc. So the lecture will be concentrated on studying how to apply the fundamental theories of functional analysis to the interpolation theorems or numerical analysis of PDE.
The studentes are expected to know the basic concepts concerning topological space, Banach space, Hilbert space, Lebesgue integral, etc. So the lecture will be concentrated on studying how to apply the fundamental theories of functional analysis to the interpolation theorems or numerical analysis of PDE.
To understand how to apply the fundamental theories of functional analysis to the interpolation theorems or numerical analysis.
- To learn the relation between functional and numerical analysis, and understand how to jusify approximate solutions which are calcurated numerically.
- To learn the uniform bounded principle and its applications, and understand the theorems of convergence from the viewpoint of functional analysis.
- To learn applications of the uniform bounded principle to PDE, and understand the error estimate or stability theories of numerical solution.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1． | Functional analysis and numerical analysis (introduction). | Review of the undergraduate mathematics (derivatives/integrals, numerical analysis, and functional analysis). | 190minutes |
| 2． | Newton method for infinite-dimensional spaces (preliminaries; existence lemma of inverse operator). | Review of the undergraduate mathematics (derivative, matrices and infinite series). | 190minutes |
| 3． | Newton method for infinite-dimensional spaces (preliminaries; theorem of finite increments) . | Review of the undergraduate mathematics (mean-value theorem). | 190minutes |
| 4． | Newton method for infinite-dimensional spaces (Kantrovich's theorem) . | Review of the undergraduate mathematics (Newton's method on finite dimensional spaces). | 190minutes |
| 5． | Review on topological and normed spaces. | Review of the undergraduate mathematics (sets and topology). | 190minutes |
| 6． | Numerical analysis on Banach spaces of finite dimension. | Review of the undergraduate mathematics (linear algebra). | 190minutes |
| 7． | Numerical analysis on Banach spaces of infinite dimension. | Review of the undergraduate mathematics (functional analysis). | 190minutes |
| 8． | Operator norms. | Review of the undergraduate mathematics (functional analysis). | 190minutes |
| 9． | Uniform bounded principle. | Review of the previous lecture. | 190minutes |
| 10． | Theory of sequence transformation. | Review of the previous lecture. | 190minutes |
| 11． | Application of uniform bounded principle to sequence transformation. | Review of the previous lecture. | 190minutes |
| 12． | Application to numerical integration. | Review of the previous lecture. | 190minutes |
| 13． | Application to interpolation theories. | Review of the previous lecture. | 190minutes |
| 14． | Application to PDE. | Review of the all of lectures. | 190minutes |
| Total. | - | - | 2660minutes |
| Final examination | Regular assignments | Total. | |
|---|---|---|---|
| 1. | 20% | 20% | 40% |
| 2. | 30% | 10% | 40% |
| 3. | 10% | 10% | 20% |
| Total. | 60% | 40% | - |
By the final examination (60%) and two or three regular assignments (40%).
More precisely, by the answers of the followings:
questions on fundamentals of Banach and Frechet spaces, and their relation to numerical analysis (40%),
questions on the principle of uniform boundedness and its foundations (40%),
questions on some applications of the principle of uniform boundedness (20%).
Answer questions on the foundations of Banach and Frechet spaces and their relation to numerical analysis (40 points).
As described above, the final examination (written test) will be 60% and the regular assignments (reports) 40% in the evaluation. But, if it becomes difficult to give the final examination in the classroom, the final assignment (report; it may be divided into two or three assignments) will be substituted for the final examination (to be announced at the beginning of the class or when the situation suddenly changes).
More precisely, by the answers of the followings:
questions on fundamentals of Banach and Frechet spaces, and their relation to numerical analysis (40%),
questions on the principle of uniform boundedness and its foundations (40%),
questions on some applications of the principle of uniform boundedness (20%).
Answer questions on the foundations of Banach and Frechet spaces and their relation to numerical analysis (40 points).
As described above, the final examination (written test) will be 60% and the regular assignments (reports) 40% in the evaluation. But, if it becomes difficult to give the final examination in the classroom, the final assignment (report; it may be divided into two or three assignments) will be substituted for the final examination (to be announced at the beginning of the class or when the situation suddenly changes).
Students are expected to understand very well about calculus, linear algebra and numerical analysis studied in undergraduate
course. Moreover, it is required that the subjects Set theory and topology, Fundamentals of mathematical analysis, Functional
analysis (or similar ones) have been already taken.
- Non-social and professional independence development course
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | N/A |
Last modified : Fri Mar 18 23:23:43 JST 2022