Contents

General Information

References

Prerequisite

  • Real and complex analysis

Homeworks

  • Homework 1: Return by March 20, 2026 (Friday) 23:59
  • Homework 2: Return by March 27, 2026 (Friday) 23:59
  • Homework 3: Return by April 10, 2026 (Friday) 23:59
  • Homework 4: Return by April 24, 2026 (Friday) 23:59
  • Homework 5: Return by May 8, 2026 (Friday) 23:59

Schedule

  • The lectures are on Friday (09:10-12:00) at 志希070116.
Time Room Activities
27.2.2026 09:10-12:00 志希070116 Week 1 - no class: compensation holiday (peace memorial day)
6.3.2026 09:10-12:00 志希070116 Week 2: Preliminaries
13.3.2026 09:10-12:00 志希070116 Week 3: Weak derivatives and distribution derivatives
20.3.2026 09:10-12:00 志希070116 Week 4: Definition and elementary properties of the Sobolev spaces [Return Homework 1 by 23:59]
27.3.2026 09:10-12:00 志希070116 Week 5: [Return Homework 2 by 23:59]
3.4.2026 09:10-12:00 志希070116 Week 6 - no class: compensation holiday (children’s day)
10.4.2026 09:10-12:00 志希070116 Week 7: Solving elliptic PDE for small wave number, the maximum principle [Return Homework 3 by 23:59]
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Li, Bo-Jyun
Title. An introduction of Ito's calculus
Abstract. This talk explores the construction of the Ito integral, beginning with the transition from Riemann-Stieltjes integration to the stochastic domain. While classical integration requires the integrator to be of bounded variation, Brownian motion violates this condition, necessitating a new framework for stochastic integration. We construct the Ito integral starting from elementary functions and extend it to adapted, square-integrable processes through the fundamental Ito Isometry and the completeness of \(L^2\) spaces. This process establishes the integral as a martingale transform that maintains linearity and continuity. The discussion concludes by highlighting how the reliance on left-endpoint evaluation distinguishes Ito's calculus from deterministic frameworks, particularly through the emergence of a quadratic variation term in the stochastic integration-by-parts formula.
17.4.2026 09:10-12:00 志希070116 Week 8: Solving elliptic PDE: Eigenvalue problem and Fredholm alternative
24.4.2026 09:10-12:00 志希070116 Week 9: [Return Homework 4 by 23:59]
1.5.2026 09:10-12:00 志希070116 Week 10 - no class: labor day
8.5.2026 09:10-12:00 志希070116 Week 11: [Return Homework 5 by 23:59]
click me to see the title and abstract of today's first talk (30 minutes)
Speaker. Kao, An-Hsien
Title. Restart and Deflation in Golub-Kahan Bidiagonalization
Abstract. The Golub-Kahan Bidiagonalization (GKB) process faces two practical difficulties: progressive loss of orthogonality among Lanczos vectors, and prohibitive memory growth as the subspace expands. Reorthogonalization and thick restart address these issues by restoring orthogonality and limiting subspace size. When additional singular triplets are needed beyond an existing partial SVD, explicit deflation projects out already-converged singular vectors from the matrix-vector product, directing subsequent GKB iterations toward the remaining spectrum.​​​​​​​​​​​​​​​​
15.5.2026 09:10-12:00 志希070116 Week 12
22.5.2026 09:10-12:00 志希070116 Week 13 - no class: Attending conference (2026 NCTS Workshop on Mathematics of Living Systems)
29.5.2026 09:10-13:00 志希070116 Week 14
5.6.2026 09:10-13:00 志希070116 Week 15
12.6.2026 09:10-13:00 志希070116 Week 16

Completion

  • The course can be taken for credit by attending the lectures, returning written solutions (60%) in \(\LaTeX\) and giving (at least) 2 presentations (each 20%).