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Lecturer(s)
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Vodák Rostislav, RNDr. Ph.D.
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Machalová Jitka, doc. RNDr. Ph.D., MBA
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Course content
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Introduction. Compactness, convexity and extremal principles. Minimization of functionals, fundamental theorem of variational calculus, variants. Variational principles in elasticity. Approximation and FEM, computational algorithms, convergence. Minimization with constraints, characterization, optimality conditions. Lagrange multipliers, Kuhn - Tucker theory. Saddle points and duality. Penalty method, augmented Lagrangians, algorithms of Uzawa type. Subgradients, variational inequality of the 1. and 2. kind, approximation of the inequalities. Applications to contact problems, computations, examples.
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Learning activities and teaching methods
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Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook)
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Learning outcomes
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Realize and master variational formulations of boundary problems and their approximations
Application Apply optimization and saddle-point theory to boundary value problems in order to obtain their solution.
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Prerequisites
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Master's degree in mathematics.
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Assessment methods and criteria
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Oral exam
Exam: to know and to understand the subject
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Recommended literature
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Ciarlet, P. G. (1989). Introduction to Numerical Linear Algebra and Optimization. Cambridge University Press, Cambridge.
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I. Ekeland, R. Temam. (1976). Convex Analysis and Variational Problems. North-Holland, Amsterdam.
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J. Cea. (1978). Optimization. Theory and Algorithms, Lecture Notes, Vol.53. Tata Inst. Fund. Research, Bombay.
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J. Haslinger, M. Miettinen, P.D. Panagiotopoulos. (1999). Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications. Kluwer, Dordrecht.
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J. Nečas, I. Hlaváček. (1983). Úvod do matematické teorie pružných a pružně plastických těles. SNTL, Praha.
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J. Nocedal, S.J. Wright. (1999). Numerical Optimization. Sringer-Verlag, New York, Berlin.
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J. P. Aubin. (1979). Applied Functional Analysis. J. Wiley, New York.
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P.E. Gill, W. Murray, M.H. Wright. (1981). Practical Optimization. Academic Press.
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R. Glowinski, P. Le Tallec. (1989). Augmented Lagrangian and Operator-Splitting Methods. SIAM, Philadelphia.
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