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Lecture 41:
Wed 9 Dec |
Recap for final exam
lecture41.pdf: Notes for Lecture 41
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Lecture 40:
Mon 7 Dec |
12: UQ, Parameter estimation for nonlinear models
Bayesian approach: deducing the distribution for a coin toss (Smith, Section 4.8)
MATLAB: cointoss1.m: Deduce distribution with uninformed prior
MATLAB: cointoss2.m: Deduce distribution with a bad prior
lecture40.pdf: Notes for Lecture 40
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Lecture 39:
Fri 4 Dec |
12: UQ, Parameter estimation for nonlinear models
Linearization about the least squares solution gives variance estimates
Smith, Section 7.3
MATLAB: get_shm.m: produce data for damped spring example
MATLAB: shm_est.m: estimate damping, stiffness using lsqnonlin (need Optimization Toolbox)
lecture39.pdf: Notes for Lecture 39
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Lecture 38:
Wed 2 Dec |
12: UQ, Parameter estimation for linear models
Frequentist perspective: variance estimation
Smith, Section 7.2
MATLAB: ls_demo1.m: demo of parameter, variance estimation
MATLAB: ls_demo2.m: averages previous demo over 1000 trials
lecture38.pdf: Notes for Lecture 38
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Lecture 37:
Mon 30 Nov |
12: UQ, Parameter estimation for linear models
Frequentist perspective: least squares gives an unbiased estimator
Smith, Section 7.2
lecture37.pdf: Notes for Lecture 37
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Lecture 36:
Wed 18 Nov |
12: Introduction to uncertainty quantification
Ralph Smith, Uncertainty Quantification
Least squares solution for linear models
lecture36.pdf: Notes for Lecture 36
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Lecture 35:
Fri 20 Nov |
11: Implementing explicit and implicit time-stepping methods
MATLAB: cg_demo1.m: demonstation of the conjugate gradient method
MATLAB: cg_demo2.m: another demonstration for a 2d PDE problem
lecture35.pdf: Notes for Lecture 35
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Lecture 34:
Mon 16 Nov |
11: Finite element solution of the wave equation
MATLAB: wave_fem_exp.m: wave eqn: finite elements, matrix exponential
lecture34.pdf: Notes for Lecture 34
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Lecture 33:
Fri 13 Nov |
10: Wave equation as a first-order system
lecture33.pdf: Notes for Lecture 33
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Lecture 32:
Wed 11 Nov |
10: Wave equation with sinusoidal driving: resonance
MATLAB: resonance.m: resonance in a wave equation
Ernst Chladni (Wikipedia)
Chladni patterns, driven by voice (YouTube)
lecture32.pdf: Notes for Lecture 32
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Lecture 31:
Mon 9 Nov |
10: Solution of the wave equation via the spectral method (Gockenbach, Chapter 7)
MATLAB: wave_demo1.m: wave eqn: spectral method
lecture31.pdf: Notes for Lecture 31
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Lecture 30:
Fri 6 Nov |
9: Forward and backward Euler methods for the heat equation
CFL stability constraint for forward Euler
Unconditional stability of backward Euler
MATLAB: heat_fem_be.m: heat eqn: finite elements + backward Euler
Richard Courant (Wikipedia)
and his mathematical genealogy
Kurt Otto Friedrichs (Wikipedia)
Hans Lewy (Wikipedia)
lecture30.pdf: Notes for Lecture 30
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Lecture 29:
Wed 4 Nov |
9: Eigenvalues of M-1K
lecture29.pdf: Notes for Lecture 29
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Lecture 28:
Mon 2 Nov |
9: Forward Euler solution for the finite element approximation; instability
MATLAB: heat_fem_fe.m: heat eqn: finite elements + forward Euler
lecture28.pdf: Notes for Lecture 28
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Lecture 27:
Fri 30 Oct |
9: Matrix exponential solution for the finite element approximation
MATLAB: show_eig.m: display eigenvalues of M-1K
lecture27.pdf: Notes for Lecture 27
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Lecture 26:
Wed 28 Oct |
9: Solution of the heat equation via finite elements
Galerkin problem for heat equation; matrix exponential
MATLAB: heat_fem_expm.m: heat eqn: finite elements + matrix exponential
lecture26.pdf: Notes for Lecture 39
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Lecture 25:
Mon 26 Oct |
8: Solution of the heat equation via spectral methods
Periodic boundary conditions: "the" Fourier series
9: Solution of the heat equation via finite elements
Derivation of the weak form of the heat equation
lecture25.pdf: Notes for Lecture 25
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Lecture 24:
Fri 23 Oct |
8: Solution of the heat equation via spectral methods
Handling inhomogeneous boundary conditions
lecture24.pdf: Notes for Lecture 39
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Lecture 23:
Wed 21 Oct |
8: Solution of the heat equation via spectral methods
Inhomogeneous case: ut = uxx + f
MATLAB demo: heat_demo_inhom.m solve inhomegenous heat eqn (const in time)
lecture23.pdf: Notes for Lecture 23
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Lecture 22:
Mon 19 Oct |
8: Solution of the heat equation via spectral methods (Gockenbach, Chapter 6)
Homogeneous case: ut = uxx
MATLAB demo: heat_demo0.m solve heat eqn, simple initial condition
MATLAB demo: heat_demo1.m solve heat eqn, more complicated initial condition
MATLAB demo: heat_demo1jump.m solve heat eqn, discontinuous initial condition
lecture22.pdf: Notes for Lecture 39
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Lecture 21:
Wed 14 Oct |
7: Error analysis for the finite element method
lecture21.pdf: Notes for Lecture 21
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Lecture 20:
Mon 12 Oct |
Review of the first half of the semester
cmda4604notes.pdf: Notes from Lectures 1 - 19
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Lecture 19:
Fri 9 Oct |
7: Crash course in interpolation and quadrature
lecture19.pdf: Notes for the lecture
MATLAB demo: lagrange.m generates Lagrange basis functions in Chebfun
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Lecture 18:
Fri 9 Oct |
7: Neumann boundary conditions: finite elements
Weak form with "natural" boundary conditions
Singular stiffness matrix mirrors the zero eigenvalue in the spectral method
MATLAB demo: fem_neumann.m solves PDE w/Neumann b.c.
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Lecture 17:
Wed 7 Oct |
7: Neumann boundary conditions: spectral method
Zero eigenvalue implies either no solutions or infinitely many solutions
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Lecture 16:
Mon 5 Oct |
7: The finite element method (Galerkin with hat functions)
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Lecture 15:
Fri 25 Sep |
7: The weak form of Laplace's equation and Galerkin's method (Gockenbach, Sections 5.4-5.6)
MATLAB demo: fem1.m: Galerkin method with hat functions (hat.m)
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Lecture 14:
Fri 25 Sep |
6: The spectral method (Gockenbach, Sections 5.2-5.3), continued
Inhomogeneous Dirichlet b.c.'s (add a linear function to a homogeneous solution)
Mixed b.c.'s (need new eigenvalues/eigenfunctions)
General operators (key: computing eigenvalue/eigenfunctions)
MATLAB demo: eigendemo1.m: Use "chebops" to compute eigenvalues/functions of the Laplacian
MATLAB demo: eigendemo2.m: As above, but for L u = -u''(x) + v(x)
MATLAB demo: eigendemo3.m: As above, but for L u = -(k(x) u'(x))
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Lecture 13:
Wed 23 Sep |
6: The spectral method (Gockenbach, Sections 5.2-5.3), continued
Applying the spectral method to solve -u''(x) = f(x)
MATLAB demo: decay.m: decay rate of Fourier coefficients
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Lecture 12:
Mon 21 Sep |
6: The spectral method (Gockenbach, Sections 5.2-5.3)
Applying the spectral method to solve -u''(x) = f(x)
Worked examples for two f: examples.pdf.
MATLAB demo: approx1.m: approximating f(x)=1 from eigenfunctions
MATLAB demo: lapex1.m: solving Laplace's equation w/spectral method
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Lecture 11:
Fri 18 Sep |
5: Eigenvalues and eigenfunctions (Gockenbach, Sections 5.1-5.2)
Eigenvalues and eigenfunctions of the Laplacian
MATLAB demo: lapeig_dd.m: eigenfunctions of Laplacian with u(0)=u(1)=0
MATLAB demo: lapeig_dn.m: eigenfunctions of Laplacian with u(0)=u'(1)=0
6: The spectral method
Derivation of the spectral method as the best approximation from the span of eigenvectors
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Lecture 10:
Wed 16 Sep |
5: Eigenvalues and eigenfunctions (Gockenbach, Sections 3.5 and 5.1)
Definition of eigenvalues of a linear operator
Symmetric linear operators always have real eigenvalues.
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Lecture 9:
Mon 14 Sep |
5: Eigenvalues and eigenfunctions (Gockenbach, Sections 3.5 and 5.1)
Symmetric linear operators; L u = -u'' with Dirichlet boundary conditions is symmetric.
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Lecture 8:
Fri 11 Sep |
4: Best approximation (Gockenbach, Section 3.4)
Best approximation from a higher dimensional subspaces
The error is orthogonal to the approximating subspace.
An orthonormal basis makes this matrix equation very simple (diagonal).
Chebfun error demo from class: error_approx2.m
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Lecture 7:
Wed 9 Sep |
4: Best approximation (Gockenbach, Section 3.3 and 3.4)
Best approximation from a higher dimensional subspaces
Finding best approximations requires solution of a matrix equation.
An orthonormal basis makes this matrix equation very simple (diagonal).
Chebfun demo: best_approx2.m
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Lecture 6:
Fri 4 Sep |
4: Best approximation (Gockenbach, Section 3.3 and 3.4)
Best approximation from a one-dimensional subspace
The error is orthogonal to the approximating subspace.
Projectors are linear operators that give best approximations.
Linear independence, subspaces
Chebfun demo: best_approx1.m
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Lecture 5:
Wed 2 Sep |
3: Differential equations as linear operator equations (Gockenbach, Section 3.1 and 3.4)
General inner products; norm properties; Cauchy-Schwarz inequality.
Want to dive deeper? Learn about Lp spaces,
Sobolev spaces,
Besov spaces
To do: Download Chebfun for MATLAB (free)
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Lecture 4:
Mon 31 Aug |
3: Differential equations as linear operator equations (Gockenbach, Section 3.1 and 3.4)
Definition of linear operators; inner product on C[0,1].
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Lecture 3:
Fri 28 Aug |
Wrap up heat equation; heat equation at steady state
3: Differential equations as linear operator equations (Gockenbach, Section 3.1)
MATLAB: bar_bc_ss.m: demo of heat equation at steady state
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Lecture 2:
Wed 26 Aug |
Derivation of the heat equation
2: Derivation of the Heat Equation (Gockenbach, Section 2.1)
MATLAB: bar_bc.m: demo of heat equation with Dirichlet boundary conditions
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Lecture 1:
Mon 24 Aug |
Why PDEs matter and how they arise from physical laws
1: The Name of the Game: a survey of partial differential equations
Handout: Syllabus
Handout: Tentative schedule
MATLAB: demo of various solutions of diff eqs
MATLAB: demo of Burgers' equation (shock formation)
Getting Started with MATLAB from MathWorks
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Resources:
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Notes from CMDA 3606 (Mathematical Modeling II), Spring 2014
Carl Meyer, Applied Matrix Analysis and Linear Algebra
Gilbert Strang, Introduction to Applied Mathematics
Gilbert Strang, Differential Equations and Linear Algebra
Jorge Nocedal and Steven Wright, Numerical Optimization
Ralph Smith, Uncertainty Quantification
D. J. Higham & N. J. Higham, MATLAB Guide, 2nd ed.
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