기초계산과학공학

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기초계산과학공학

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페이스북 트위터 카카오톡 네이버밴드 링크복사

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교육학 >특수교육 >특수교육학
강의학기

2011년 2학기

조회수

9,490

This course is intended for graduate students (or undergraduate) who need a rapid and uncomplicated introductions to the field of applied mathematics involving computational linear algebra and differential equations. The lecture has two themes-how to understand equations, and how to solve them . This course include numerical linear algebra(QR,SVD, singular system), Newton’s method for minimization, Equilibrium and stiffness matrix, Least squares, Nonlinear problems, Covariances and Recursive Least squares, Differential equations and finite elements, Finite Difference and Fast Poisson, Boundary value problems in Elasticity and Solid mechanics

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Applied Linear Algebra

차시별 강의

1.	Introduction of the course, Applied Linear Algebra	Four Special Matrices
2.	Applied Linear Algebra	Four Special Matrices Differences, Derivatives, Boundary condition
	Applied Linear Algebra	Differences, Derivatives, Boundary condition
3.	Applied Linear Algebra	Differences, Derivatives, Boundary condition Elimination leads to K = LDL^T
	Applied Linear Algebra	Elimination leads to K = LDL^T Inverse and Delta function
4.	Applied Linear Algebra	Inverse and Delta function
5.	Applied Linear Algebra	Inverse and Delta function Eingenvalues and Eigenvectors
	Applied Linear Algebra	Eingenvalues and Eigenvectors
6.	Applied Linear Algebra (Eingenvalues and Eigenvectors)	Part 1 : Ax = \lambda x and A^k x = \lambda^k x and Diagonalizing A
7.	Applied Linear Algebra (Eingenvalues and Eigenvectors)	Part 2 : Eigenvectors for Derivatives and Differences
	Applied Linear Algebra (Eingenvalues and Eigenvectors, Positive Definite Matrices)	Part 2 : Eigenvectors for Derivatives and Differences, What is Positive Definite?
8.	Applied Linear Algebra (Positive Definite Matrices)	Examples and Energy-based Definition, Positive definiteness, Minimum Problem in n Dimensions, Test for a Minmimum, Newtons Method
	Applied Linear Algebra (Positive Definite Matrices, Numerical Linear Algebra: LU, QR, SVD)	Three Essential Factorizations, Orthogonal Matrices, Orthogonalization A = QR, SVD
9.	Applied Linear Algebra (Numerical Linear Algebra: LU, QR, SVD)	SVD, The pseudoinverse, Condition Numbers and Norms
10.	Applied Linear Algebra (Numerical Linear Algebra: LU, QR, SVD)	Condition Numbers and Norms, Mass Spring System, Stiffness Matrix and Solution
	A Framework for Applied Mathematics(Equilibrium and Stiffness Matrix)	Fixed End and Free End, Minimum Principles
11.	A Framework for Applied Mathematics (Oscillation by Newtons Law)	One Mass and One Spring, Key Example: Motion Around a Circle (Four Finite Difference Methods)
12.	A Framework for Applied Mathematics (Oscillation by Newtons Law)	Key Example: Motion Around a Circle, Line of Masses and Springs
	A Framework for Applied Mathematics (Oscillation by Newtons Law)	Total Energy is Conserved, Applied Force and Resonance, Explicit Finite Differences, Stability and Instability
13.	A Framework for Applied Mathematics (Oscillation by Newtons Law, Least Squares for Rectangular Matrices)	Explicit Finite Differences, Stability and Instability Linear Algebra Interpretation
14.	A Framework for Applied Mathematics (Least Squares for Rectangular Matrices, Graph Models and Kirchhoffs laws)	Linear Algebra Interpretation, Computation of Least Squares, The Incident Matrix
	A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws)	The Graph Laplacian Matrix A^TA, Inputs b, f and Matrices A, C, A^T, Assembling the Matrix K = A^TC*A
15.	A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws, Networks and Transfer Functions)	KKT matrix, Impedance
16.	A Framework for Applied Mathematics (Networks and Transfer Functions)	Time Domain versus Frequency Domain, Transient Response and the Transfer Function, Underdamping and Overdamping
17.	A Framework for Applied Mathematics (Graph Models and Kirchhoffs laws, Networks and Transfer Functions)	KKT matrix, Impedance
	A Framework for Applied Mathematics (Networks and Transfer Functions)	Phase, Time Domain versus Frequency Domain, Transient Response and the Transfer Function
18.	A Framework for Applied Mathematics (Networks and Transfer Functions)	Time Domain versus Frequency Domain, Transient Response and the Transfer Function, Underdamping and Overdamping
19.	Boundary Value Problems (Differential Equations and Finite Elements)	The Framework A^TCA for a Hanging Bar, General Solution and Examples, The Transpose A = d/dx
	Boundary Value Problems (Differential Equations and Finite Elements)	Galerkins Method, Comparison with Finite Difference, More accurate Finite Elements
20.	Boundary Value Problems (Differential Equations and Finite Elements)	Galerkins Method, Comparison with Finite Difference, More accurate Finite Elements
21.	Boundary Value Problems (Cubic splines and Forth-Order Equations, Gradient and Divergence)	Cubic Finite Elements, Finite Difference for (cu)=f, Gradient, Divergence
22.	Boundary Value Problems (Gradient and Divergence)	Gradient, Divergence, Curl, Two important identities
23.	Boundary Value Problems (Gradient and Divergence)	Two important identies, Converse of the Two Identities, Harmonic Function
	Boundary Value Problems (Laplace Equation)	Complex Plane, Analytic function, Derivative of complex valued function, Cauchy-Riemann Equation
24.	Boundary Value Problems (Laplace Equation)	Polar Coordinates: Laplace Equation in a Circle
25.	Boundary Value Problems (Laplace Equation, Finite Differences and Fast Poisson Solver)	Polar Coordinates: Laplace Equation in a Circle, Poissons Equation in a Square
26.	Boundary Value Problems (Finite Differences and Fast Poisson Solver, The Finite Element Method)	Elimination and Fill-in, Solver Using Eigenvalues, Fast Poisson Solvers, Trial and Test Functions: Galerkins method
27.	Boundary Value Problems (The Finite Element Method)	Pyramid Functions, Element Matricers and Element Vectors
28.	Boundary Value Problems(The Finite Element Method)	Global Matrix K and F frome K_e and F_e, Element Matrix in Two Dimension
	Boundary Value Problems(The Finite Element Method)	Element Matrix in Two Dimension, Quadrilateral Elements