Course detail
Mathematics 5 (M)
FAST-CA003Acad. year: 2018/2019
Interpolation and approximation of functions. Numerical solution of algebraic equations and their systems. Numerical derivatives and quadrature. Variance analysis, regression analysis. Numerical solution of stationary and non-stationary boundary and initial problems for differential equations with applications to civil engineering. Direct, sensitivity and inverse problems.
Language of instruction
Czech
Number of ECTS credits
4
Mode of study
Not applicable.
Guarantor
Department
Institute of Mathematics and Descriptive Geometry (MAT)
Learning outcomes of the course unit
The students should understand the basic principles of numerical deterministic and stochastic calculations. They should understand the principle of iteration methods for solving the equation f(x)=0 and systems of linear algebraic equations mastering the calculation algorithms. They should learn how to get the basics of interpolation and approximation of functions to solve practical problems. They should be acquainted with the principles of numerical differentiation to be able to numerically solve boundary problems for ordinary differential equations. They should understand numerical calculations of definite integrals. They should bu also able to apply such knowledge to particular direct, sensitivity and inverse engineering problems.
Prerequisites
Basic knowledge of numerical mathematics, probability and statistics, applied to technical problems.
Co-requisites
Not applicable.
Planned learning activities and teaching methods
Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations - lectures, seminars.
Assesment methods and criteria linked to learning outcomes
Successful completion of the scheduled tests and submission of solutions to problems assigned by the teacher for home work. Unless properly excused, students must attend all the workshops. The result of the semester examination is given by the sum of maximum of 70 points obtained for a written test and a maximum of 30 points from the seminar.
Course curriculum
1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.
Work placements
Not applicable.
Aims
Students will obtain the basic knowledge of numerical mathematics, probability and statistics, applied to technical problems, especially from material engineering.
Specification of controlled education, way of implementation and compensation for absences
Extent and forms are specified by guarantor’s regulation updated for every academic year.
Recommended optional programme components
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
Not applicable.
Recommended reading
Not applicable.
Classification of course in study plans
Type of course unit
Lecture
26 hod., optionally
Teacher / Lecturer
Syllabus
1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.
Exercise
13 hod., compulsory
Teacher / Lecturer
Syllabus
Follows directly particular lectures:
1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.
1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.