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.
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.
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, L1 norm minimization, sparse modeling, incomplete data
Intercultural skills | Communication skills | Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
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- | - | ✔ | ✔ | ✔ |
Two lectures are given in every week.
Course schedule | Required learning | |
---|---|---|
Class 1 | Role of Functional Analysis in Engineering | Explain about the role of functional analysis in engineering |
Class 2 | Convergence of real number sequence and Vector space | Explain about the definitions of vector space and convergence of real number sequence. |
Class 3 | Metric space and Complete metric space | Explain about metric space and complete metric space |
Class 4 | Open set and closed set | Explain definitions and properties of open and closed sets. |
Class 5 | Contraction mapping theorem with applications to functional equations | Explain about the contraction mapping theorem and its applications to functional equations. |
Class 6 | Normed space and Bounded linear operator | Explain about normed space and bounded linear operator. |
Class 7 | Inner product space and Parallelogram law, Banach space and Hilbert space | Explain about inner product space, parallelogram law, Banach space and Hilbert space. |
Class 8 | Projection theorems in Hilbert space | Explain about the projection theorems in Hilbert space. |
Class 9 | Orthogonal projection onto linear variety and Normal equation | Explain about the orthogonal projection onto linear variety and the role of normal equation. |
Class 10 | Generalized inverse and Singular value decomposition | Explain about Generalized inverse and Singular value decomposition |
Class 11 | Singular value decomposition and Image processing | Explain about the image processing and image compression using the SVD. |
Class 12 | Noise and Regularization | Explain the relationship between signal containing noise and regularization method. |
Class 13 | Iterative image reconstruction | Explain the image reconstruction method by the iterative reconstruction method such as ART. |
Class 14 | Norm minimization and Sparce modeling | Explain the relationship between norm minimization and sparse modeling. |
Class 15 | Medical image reconstruction from incomlete data set | Explain the medical image reconstruction method from incomplete observation data set. |
I. Yamada, Kougaku no tameno Kansuu kaiseki, Saiensu co ltd, 2009.
D.G.Luenburger, Optimization by Vector Space Mathods, Wiley, 1997.
C.W. Groetsche, Inverse Problems in the Mathematical Sciences, Springer, 1993.
Grading is made based on 1st exam for functional analysis and 2 exam for inverse problems.
As a general rule, we accept only applications from students in the department of Information and communications Engineering.