2022　Functional Analysis and Inverse Problems

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Undergraduate major in Information and Communications Engineering
Instructor(s)
Class Format
Lecture    (HyFlex)
Media-enhanced courses
Day/Period(Room No.)
Mon5-6(W631)  Thr5-6(W631)
Group
-
Course number
ICT.S302
Credits
2
2022
Offered quarter
1Q
Syllabus updated
2022/3/16
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

To find solution strategies for various inverse problems in modern data sciences such as signal processing, image processing, pattern recognition and machine learning, the unified mathematical perspective built through Functional Analysis will certainly serve as helpful guide. Starting from definitions of convergence of real number sequence and vector space which serve as prerequisites of Functional Analysis, this lecture surveys its central ideas, e.g., in Metric space, Normed space, Inner product space, Banach space and Hilbert space, together with their applications to typical inverse problems.

Student learning outcomes

Through the lectures, the students will be able to:
1) understand mathematical meanings of spaces, convergences and operators and apply these to real world problems.
2) build mathematical perspectives to grasp many real world inverse problems in unified ways.

Keywords

Metric space, Complete metric space, Open set, Closed set, Contraction mapping theorem, Normed space, Bounded linear operator, Inner product space, Parallelogram law, Banach space, Hilbert space, Projection theorem, Orthogonal projection onto linear variety, Normal equation, Generalized inverse, Singular value decomposition, regularization, Iterative image reconstruction, Constraint, L1 norm minimization, sparse modeling, incomplete data

Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills ✔ Critical thinking skills ✔ Practical and/or problem-solving skills

Class flow

Two lectures are given in every week.

Course schedule/Required learning

Course schedule Required learning
Class 1 Role of functional analysis in engineering and review of first-year mathematics Explain about the reasons for studying functional analysis the role of functional analysis in engineering
Class 2 Metric space and Complete metric space Explain about metric space and complete metric space
Class 3 Open set and closed set Explain definitions and properties of open and closed sets.
Class 4 Contraction mapping theorem with applications to functional equations Explain about the contraction mapping theorem and its applications to functional equations.
Class 5 Normed space and Bounded linear operator Explain about normed space and bounded linear operator.
Class 6 Inner product space and Parallelogram law, Banach space and Hilbert space Explain about inner product space, parallelogram law, Banach space and Hilbert space.
Class 7 Projection theorems in Hilbert space Explain about the projection theorems in Hilbert space.
Class 8 Orthogonal projection, Generalized inverse, Fourier series expansion -revisited Explain about Orthogonal projection, Generalized inverse, Fourier series expansion.
Class 9 Singular value decomposition and Image processing Explain about Singular value decomposition, the image processing and image compression using the SVD.
Class 10 Noise and Regularization Explain the relationship between signal containing noise and regularization method.
Class 11 Iterative image reconstruction Explain the image reconstruction method by the iterative reconstruction method such as ART.
Class 12 Image Reconstruction Problem with Constraints Explain the image reconstruction method with constraints.
Class 13 Norm minimization and Sparce modeling Explain the relationship between norm minimization and sparse modeling.
Class 14 Medical image reconstruction from incomlete data set Explain the medical image reconstruction method from incomplete observation data set.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

I. Yamada, Kougaku no tameno Kansuu kaiseki, Saiensu co ltd, 2009.

Reference books, course materials, etc.

D.G.Luenburger, Optimization by Vector Space Mathods, Wiley, 1997.
C.W. Groetsche, Inverse Problems in the Mathematical Sciences, Springer, 1993.

Assessment criteria and methods

Grading is made based on the 1st report for functional analysis and the 2nd report for inverse problems.

Related courses

• LAS.M102 ： Linear Algebra I / Recitation
• LAS.M101 ： Calculus I / Recitation
• ICT.S206 ： Signal and System Analysis
• ICT.S210 ： Digital Signal Processing
• ICT.M316 ： Numerical Analysis (ICT)
• ICT.C201 ： Introduction to Information and Communications Engineering
• ICT.H504 ： Medical Image Processing
• ICT.H421 ： Medical Imaging Systems
• ICT.S414 ： Advanced Signal Processing (ICT)

Prerequisites (i.e., required knowledge, skills, courses, etc.)

As a general rule, we accept only applications from students in the department of Information and communications Engineering.