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ECE174 ­ Introduction to Linear and Nonlinear Optimization with Applications

2006-2007 Catalog Description: The linear least squares problem, including constrained and unconstrained quadratic optimization and the relationship to the geometry of linear transformations. Introduction to nonlinear optimization. Applications to signal processing, system identification, robotics, and circuit design. Prerequisites: Math 20F with a grade of C- or better Text Books: 1) C.D. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Philadelphia, 2000 (Required) 2) G. Strang, "Linear Algebra and its Applications," 3rd Edition, HBJ, San Diego, 1988 (Recommended). 3) G. Strang and K. Borre, "Linear algebra, geodesy, and GPS," Cambridge, Wellesley, 1997 (Recommended). Course Objectives: The primary goal is to teach students how to think geometrically in signal spaces, and to apply high-level geometric thinking when solving linear and nonlinear inverse problems in such spaces. Course Topics: 1. Overview of (constrained and unconstrained) linear least squares and nonlinear optimization. Discussion of real-world linear and non-linear inverse problems encountered in engineering (drawn from circuit analysis, robotics, communications, and signal processing) that can be solved using these techniques. 2. Signal spaces as abstract vector spaces and matrices as representing linear transformations between signal spaces. Simple signal/vector spaces of functions (sine/cosine and polynomial). 3. Under- and over-constrained linear systems of equations and linear inverse problems. Constrained and unconstrained quadratic optimization of linear systems (the "linear least squares" problem). 4. Solutions to the linear least squares problem and their relationships to the four fundamental subspaces of linear algebra. Discussion of the projection theorem, the adjoint operator, the normal equations, the adjoint equations, generalized inverses, and the Moore-Penrose pseudoinverse. 5. The Singular Value Decomposition (SVD) and its relationship to the four fundamental subspaces of linear algebra. Use of the SVD to solve and linear inverse problem and to obtain the Moore-Penrose pseudoinverse. 6. Quadratic forms as modeling energy, power, and uncertainty quantities encountered in engineering applications. The weighted least squares problem and its relationship to the problem of maximum likelihood (MLE) estimation. 7. Nonlinear least squares theory. Vector differentiation, hessians, and necessary sufficient conditions for an optimization to exist. 8. Linearization and linearization-based nonlinear optimization techniques. Iterative optimization techniques, including steepest descent, Guass's method, and Newton's method. Constrained nonlinear optimization and the method of Lagrange multipliers (time permitting). Class/laboratory schedule: Three hours of classroom lecture by the professor, and one hour of discussion, lead by a graduate student teaching assistant, per week. The professor also provides two hours of regularly scheduled office hour per week. Students are encouraged to work in groups or teams, but all computer projects turned in must be individually analyzed and written up. Evaluation Methods: Students are given homework, which is not graded. Solutions are provided and students are informed that a selected few of the problems from the homework will be chosen at random and put on the exams. There are two exams: a midterm, which is worth 30% of the grade, and a final, which is worth 40% of of the grade. The remaining 30% of the course grade is based on two written reports which students must turn in based on two computer projects. One project is based on an application of linear least squares solutions techniques. In the past these topics are been the following: · · Linear predictive coding: data compression via the use of minimum least squares predictive coding. Optimal design of an FIR notch filter via the use of constrained, weighted least squares optimization.

· System identification for channel equalization and all-pole filter approximation. The second computer project is an application of steepest descent and the Gauss-Newton technique to solve the GPS location and receiver-clock calibration problem using only range and timing information provided by (at least) four line-of-sight satellites. Prepared by: Kenneth Kreutz-Delgado Updated by: Kevin Quest E-mail: [email protected] E-mail address: [email protected] Date: May 16, 2001 Date: May 30, 2007

Appendix IB: Course Syllabi


ECE174 ­ Introduction to Linear and Nonlinear Optimization with Applications

Contribution of course to meeting the professional component: College level mathematics (nonlinear optimization), engineering topics (circuits, robotics) 1 An understanding of the underlying principles of, and an ability to apply knowledge of mathematics, science and engineering to electrical engineering problems: Emphasis: 2 Assessment: 1 This is a major point of the course, and is strongly emphasized throughout the quarter in lecture. Assessed using questions on two exams and two individually written computer project reports. An ability to design and conduct experiments, as well as to analyze and interpret data: Emphasis: 0 Assessment: 0 A knowledge of electrical engineering safety issues: Assessment: 0 Emphasis: 0 An ability to design a system, component, or process to meet desired needs: Emphasis: 0 Assessment: 0 a) An ability to collaborate effectively with others, b) an ability to function on multidisciplinary teams: Emphasis: 5a): 2 5b): 2 Assessment: 5a): 0 5b): 0 The course usually has students from many different engineering departments (e.g., ECE, MAE, Bioeng, SIO) and a crossdisciplinary tone is always taken in the lecture. Students are strongly encouraged to work together on teams when doing homework and the compute projects. They are expressly told that industry values team-workmanship. However students are asked to turn in individual reports and no direct evaluation of team functioning is done. An ability to identify, formulate, and solve engineering problems: Emphasis: 2 Assessment: 1 This is a major theme of the course. Throughout the quarter examples are given which demonstrate how students can identify, formulate, and map engineering problems into the mathematical framework amenable to solution by the tools taught in the class. Assessment of the students' ability to do so is based on exam questions and the two computer projects. An ability to use the techniques, skills, and modern engineering tools necessary for engineering practice, including familiarity with computer programming and information technology: Emphasis: 1 Assessment: 1 The computer projects are done in Matlab and usually require that the students find and process sound files on the Internet. The projects require sophisticated use of Matlab and to the degree that students are proficient with these tools they impact the written report grades. An understanding of professional and ethical responsibility: Assessment: 0 Emphasis :0 An ability to communicate effectively a) in writing, b) verbally, c) with visual means: Emphasis: 9a): 2 9b): 0 9c): 1 Assessment: 9a): 1 9b): 0 9c): 0 Students are informed that they are to write their reports in an succinct, yet effective and compelling manner. They are encourage to utilize graphical representations and descriptions whenever possible in their reports.


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10. The broad education necessary to understand the impact of engineering solutions in a global and societal context: Emphasis: 1 Assessment: 0 Throughout the quarter, arises as a theme when motivating the utility of the methodologies taught. 11. A recognition of the need for, and the ability to engage in, life-long learning: Assessment: 0 Emphasis: 1 This is stressed when explaining the value of learning new tools (such as those taught in the class) throughout a lifetime of professional work. 12. A knowledge of contemporary issues: Emphasis: 1 Assessment: 0 Contemporary technical issues/problems solvable via the methodologies taught in the course are discussed.

Appendix IB: Course Syllabi




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