| Course Unit Code | 470-4504/03 |
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| Number of ECTS Credits Allocated | 6 ECTS credits |
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| Type of Course Unit * | Choice-compulsory type B |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | |
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| Semester when the Course Unit is delivered | Winter Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| DOM0015 | Ing. Simona Bérešová, Ph.D. |
| Summary |
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The course introduces various types of iterative methods for solving linear
and nonlinear systems. The lectures focus on the basic ideas, however, it include some latest results in the field. |
| Learning Outcomes of the Course Unit |
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| Students will be able to use various types of iterative methods for solving linear and nonlinear alebraic systems. He will become acquainted with basic ideas as well as some recent results in the field. |
| Course Contents |
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Lectures:
Systems of equations arising from mathematical modelling in engineering.
Properties of systems arising from finite element methods.
Classical iterative methods. Richardson, Jacobi, Gauss-Seidel iterative methods. Convergence studies.
Multigrid methods.
Method of conjugate gradients. Fundamentals. Implementation.
Global properties and convergence rate estimates based on the condition number.
Preconditioning. Preconditioned conjugate gradients method. Incomplete factorization.
Solution to nonsymmetric systems. GMRES.
Solution to nonlinear systems. Properties of nonlinear operators. Newton method. Local convergence. Inexact Newton method. Damping and global convergence.
Implementation of iterative methods on parallel computers. Domain decomposition methods.
Comparison of direct and iterative methods. Solution to large-scale systems.
Tutorials:
Systems of equations arising in mathematical modeling in engineering. Assembling the system matrix in the finite element method, properties.
Solution to systems using Richardson, Jacobi, and Gauss-Seidel iterative methods. Multigrid method.
Implementation of conjugate gradient method, rate of convergence.
Implementation of various preconditioners in the conjugate gradients method. Incomplete factorization.
Implementation of GMRES.
Implementation of Newton method and inexact Newton method.
Implementation of iterative methods on parallel computers. Domain decomposition methods.
Comparison of direct and iterative methods. Solution to large-scale systems.
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| Recommended or Required Reading |
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| Required Reading: |
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C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM,
Philadelphia 1995, http://www.siam.org/catalog/mcc12/kelley.htm
B. Barrett et al.: Templates for the solution of linear systems, SIAM,
Philadelphia 1993, http://www.siam.org/catalog/mcc01/barrett.htm
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C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM,
Philadelphia 1995, http://www.siam.org/catalog/mcc12/kelley.htm
B. Barrett et al.: Templates for the solution of linear systems, SIAM,
Philadelphia 1993, http://www.siam.org/catalog/mcc01/barrett.htm
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| Recommended Reading: |
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O. Axelsson: Iterative Solution Methods, Cambridge University Press, 1994
Werner C. Rheinboldt: Methods for Solving Systems of Nonlinear Equations,
SIAM, Philadelphia 1998, http://www.siam.org/catalog/mcc02/cb70.htm
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O. Axelsson: Iterative Solution Methods, Cambridge University Press, 1994
Werner C. Rheinboldt: Methods for Solving Systems of Nonlinear Equations,
SIAM, Philadelphia 1998, http://www.siam.org/catalog/mcc02/cb70.htm
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| Planned learning activities and teaching methods |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Credit and Examination | Credit and Examination | 100 (100) | 51 |
| Credit | Credit | 30 | 15 |
| Examination | Examination | 70 | 36 |