• Lecture notes
• Homework
• Basic course info
• Course description
• Syllabus

Lecture notes

1. Introducing Computational Sensorics, course summary and organization pdf
2. Linear systems review pdf
3. Linear quadratic regulator, controllability, observability, Lyapunov theory, optimal control and dynamic programming, realization theory, linear estimation and the Kalman filter pdf
4. Review of applications of optimization in vector spaces, the main principles, linear spaces, subspaces, linear varieties, convexity, cones, linear independence pdf
5. Normed linear spaces, open and closed sets, convergence, transformations, continuity, the lp and Lp spaces, minimum norm problems, approximation theory pdf
6. Banach spaces, extreme values of functionals and compactness, denseness and separability pdf
7. Hilbert spaces, the Gram-Schmidt procedure, approximation and Fourier series, minimum norm problems, the dual approximation problem, minimum distance to a convex set pdf
8. Hilbert spaces of random variables, the least-squares estimate, the minimum-variance unbiased estimate, recursive estimation pdf
9. Linear functionals, duals of common Banach spaces, extension form of the Banach theorem, alignment and orthogonal complements pdf
10. Minimum norm problems and applications, geometric form of the Hahn-Banach theorem, hyperplanes and convex sets, duality in mimimum norm problems pdf
11. Linear operators, inverse operators, the Banach inverse theorem, adjoint operators, relations between range and nullspace, geometric interpretation of adjoints, optimization in Hilbert space pdf
12. Gateaux and Frechet differentials, optimization of functionals pdf
13. Introduction to inverse problems, the 1D inverse heat conduction problem (IHCP) in finite dimensional spaces, the 1D IHCP in infinite dimensional spaces, iterative regularization method pdf
14. Conjugate gradient methods in finite and infinite dimensional spaces, rate of convergence, preconditioning, the Fletcher-Reeves & Polak-Ribier methods pdf
15. Optimal control theory, local theory of constraint optimization pdf
16. Global theory of constraint optimization pdf
17. Classical regularization methods, Tikhonov and truncated iterative regularization pdf
18. Review of probability theory (ppt), review of statistics (pdf), cross validation (ppt), Bayesian analytics (ppt), Gaussian (ppt), maximum likelihood estimation (ppt)
19. Statistical inversion theory, inverse problems and Bayes formula, likelihood, prior models, posterior distributions, MCMC (ppt), hierarchical models pdf
20. Non-stationary inverse problems, Bayesian filtering, particle filters, spatial priors, Markov fields pdf
21. Numerical aspects of statistical inversion, modeling the prior & the likelihood, discretization effects pdf
22. Model reduction in continuum systems pdf
23. Model reduction in the control of thermal and flow systems (pdf, pdf, pdf, pdf)
24. Review of robust control techniques pdf
25. Reduced-order methods for feedback controller design pdf
26. Advanced regression algorithms pdf, regression/classification with neural networks, other machine learning algorithms
27. Introduction to pattern classification pdf
28. Bayesian decision theory pdf
29. Neural networks, back propagation and advanced optimization pdf
30. Introduction to support vector machines pdf
31. Reinforcement learning pdf
32. Quantification of sensor data, limits of detection, calibration, interferences, sampling, verification pdf
33. Sensors and sensor networks, adaptive and reconfigurable sensor networks pdf
34. Information theory fundamentals, links with modeling data across length scales pdf

Homework

• Homework 1 pdf

• Homework 1 solutions pdf

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Basic course info

Lectures: Tues./Thurs. 2:55 -- 4:10, Upson 102.

Professor: Nicholas Zabaras, 188 Frank H. T. Rhodes Hall, (607) 255-9104, zabaras@cornell.edu

Office hours: Tuesdays 3:00 -- 4:00 p.m.; Fridays 3:00 -- 4:00 p.m.

References: The course lectures will become available on the course web site. For in depth study, a list of articles from the current literature will also be provided to enhance the material of the lectures. There is no required text for this course. Some important books that can be used for general reading in the subject areas of the course include the following:

References on inverse problems:

• S. Andreas Kirsch, An Introduction to the Mathematical Theory of Inverse Problems
• D. Colton, H. W. Engl, A. K. Louis, Surveys on Solution Methods for Inverse Problems

References on optimal control:

• J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations
• D. Luenberger, Optimization by vector space methods
• P. Neittaanmaki and D. Tiba, Optimal control of nonlinear parabolic systems
• Xunjing Li, J. Yong, Hsun-Ching Li, Optimal Control Theory for Infinite Dimensional Systems

References on advanced analysis of feedback:

• M. Vidyasagar, Nonlinear Systems Analysis, second edition, Prentice Hall, 1992
• S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design
• Shankar Sastry, Nonlinear Systems: Analysis, Stability and Control
• G. E. Dullerud, F. Paganini, A Course in Robust Control Theory : A Convex Approach

References on Bayesian statistics & computation:

• P. Lee, Bayesian Statistics: An Introduction (Arnold Publication)
• J. P. Kaipio and E. Somersalo, Computational and Statistical Methods for Inverse Problems (forthcoming)
• W. R. Gilks, et al., Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC, 1995

References on statistical learning techniques:

• T. Kailath, A. Sayed and B. Hassibi, Linear Estimation
• T. Hastie, R. Tibshirani, J. H. Friedman The Elements of Statistical Learning: Data Mining, Inference, and Prediction
• D. Denison, C. Holmes, B. Mallick, A. Smith, Bayesian Methods for Nonlinear Classification and Regression

References on data mining and machine learning:

• T. M. Mitchell, Machine Learning
• R. Duta, P. E. Hart and D. G. Stork, Pattern Classification
• J. Han and M. Kamber, Data Mining: Concepts and Techniques
• D. Pyle, Data Preparation for Data Mining
• N. Cristianini and J. Shawe-Taylor, Support Vector Machines

References on data compression and quantization:

• K. Sayood and M. Morgan Kauffman, Introduction to Data Compression, Second Edition, 2000
• A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Kluwer Academic Press, 1992

Homework: assigned every other Thursday and due the following Thursday in the beginning of the class. You are allowed, even encouraged, to work on the homework in small groups, but you must write up your own homework to hand in.

Term project: A project is required in mathematical or computational aspects of the course in an area of mutual interest. Students are encouraged to investigate aspects of computational sensorics not covered in class. A written report is required (to be posted on the course web site) as well as a class presentation.

Grading: Homework 50% and Project 50%.

Prerequisites: Advanced mathematical background and previous exposure to continuum systems (fluid flow, thermal transport, etc.). Some earlier exposure to optimal control is desirable but not required.

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Course description

The course will be useful to graduate students interested in modeling and advanced experimentation of physical systems. It will emphasize the tools needed to couple the real time acquisition and analysis of large amount of data of dynamical continuum quantities with control algorithms based on reduced-order models derived from a more rigorous design analysis. Such dynamic coupling of rigorous analysis and design mathematical techniques with controlled experimentation will intrinsically account for uncertainties in modeling the physical phenomena and can be used to design innovative continuum processes and systems. This course will present a number of computational sensorics tools using fluid flow and thermal transport systems as examples. However, each student will be encouraged to work as part of the course project in other physical processes or systems and emphasize mathematical/computational topics of the design and control of continuum systems that were not discussed in depth in class.

Catalog description: Examples of industrial control of continnum systems; Mathematical preliminaries; Finite element approach to partial differential equations; Inverse problems and inverse problem solving; Optimal control problems; Numerical analysis of distributed control problems; Reduced-order models for continuum systems; Feedback laws for continuum systems; Robust control and uncertainty; Data mining of continuum systems and models; Data compression techniques; Advanced adaptive sensing and actuation of continuum fields.

Course objectives: This course provides advanced graduate students with a single source introduction to the mathematical, computational and information technologies tools available for the design and robust control of systems and processes governed by partial differential equations.

Intended audience: Graduate Students in Engineering, Physical and Atmospheric Sciences, Computer Science, and Applied Mathematics.

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Syllabus

1. Motivating examples
   • Smooth nonlinear parabolic and elliptic problems
   • Industrial examples
2. Mathematical preliminaries
   • Function spaces, convexity and operator theory
   • Some useful results from partial differential equations
3. Finite element approach to partial differential equations
   • Approximations for parabolic and elliptic problems
4. Inverse problems
   • Tikhonov regularization techniques
   • Continuum adjoint & sensitivity techniques
   • Iterative regularization methods
   • Functional optimization techniques
5. Control problems
   • Linear control systems
   • Nonlinear differentiable control problems
   • Optimality conditions, controllability
6. Numerical analysis of control problems
   • Approximations for convex control problems
   • Distributed control problems governed by variational inequalities
   • Optimality conditions & controllability
7. Reduced-order models for continuum systems
   • Proper orthogonal decomposition, Karhunen-Loeve basis
   • Reduced modeling based on Voronoi-tesselations
   • Reduced-order modeling of adjoint and sensitivity fields
8. Feedback laws for continuum systems
   • Solving Hamilton-Jacobi equations
   • Sub-optimal and ad hoc feedback laws
   • The design-then-approximate versus the approximate-then-design approach to controller design
   • Designing locally optimal feedback laws using linear low-order state models
9. Applications to fluid flow control
   • Optimal control of viscous incompressible flows: Boundary value control and shape control problems
   • Optimal control of inviscid compressible flows
   • Feedback control of viscous incompressible flows
   • Instability control in transitional flows
10. Robust control and uncertainty
    • Stochastic differential equations (SDEs)
    • Response surfaces and interval analyses
    • Inverse design and parameter estimation problems using SDEs
11. Data mining of continuum systems and models
    • Pattern classification: Bayesian decision theory and parameter estimation, nonparameteric techniques, multilayer neural networks, nonmetric methods
    • Predictive modeling, data classification and regression techniques
    • Machine learning, unsupervised learning and clustering
    • Classification and regression techniques for dynamical models
    • Virtual environments
    • Scaling clustering algorithms for mining association rules
    • Generalized search trees for database systems
    • Information retrieval and distributed databases
12. Data compression and quantization techniques
    • Lossless and lossy compression
    • Fundamentals of quantization
    • Predictive coding
13. Advanced sensing of continuum fields
    • Particle velocimetry as an example
    • Adaptive sensing and actuation techniques