


Lecture 41: Wed 9 Dec 
Recap for final exam
lecture41.pdf: Notes for Lecture 41



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



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



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



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



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



Lecture 35: Fri 20 Nov 
11: Implementing explicit and implicit timestepping 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



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



Lecture 33: Fri 13 Nov 
10: Wave equation as a firstorder system
lecture33.pdf: Notes for Lecture 33



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



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



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



Lecture 29: Wed 4 Nov 
9: Eigenvalues of M^{1}K
lecture29.pdf: Notes for Lecture 29



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



Lecture 27: Fri 30 Oct 
9: Matrix exponential solution for the finite element approximation
MATLAB: show_eig.m: display eigenvalues of M^{1}K
lecture27.pdf: Notes for Lecture 27



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



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



Lecture 24: Fri 23 Oct 
8: Solution of the heat equation via spectral methods
Handling inhomogeneous boundary conditions
lecture24.pdf: Notes for Lecture 39



Lecture 23: Wed 21 Oct 
8: Solution of the heat equation via spectral methods
Inhomogeneous case: u_{t} = u_{xx} + f
MATLAB demo: heat_demo_inhom.m solve inhomegenous heat eqn (const in time)
lecture23.pdf: Notes for Lecture 23



Lecture 22: Mon 19 Oct 
8: Solution of the heat equation via spectral methods (Gockenbach, Chapter 6)
Homogeneous case: u_{t} = u_{xx}
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



Lecture 21: Wed 14 Oct 
7: Error analysis for the finite element method
lecture21.pdf: Notes for Lecture 21



Lecture 20: Mon 12 Oct 
Review of the first half of the semester
cmda4604notes.pdf: Notes from Lectures 1  19



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



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.



Lecture 17: Wed 7 Oct 
7: Neumann boundary conditions: spectral method
Zero eigenvalue implies either no solutions or infinitely many solutions



Lecture 16: Mon 5 Oct 
7: The finite element method (Galerkin with hat functions)



Lecture 15: Fri 25 Sep 
7: The weak form of Laplace's equation and Galerkin's method (Gockenbach, Sections 5.45.6)
MATLAB demo: fem1.m: Galerkin method with hat functions (hat.m)



Lecture 14: Fri 25 Sep 
6: The spectral method (Gockenbach, Sections 5.25.3), 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))



Lecture 13: Wed 23 Sep 
6: The spectral method (Gockenbach, Sections 5.25.3), continued
Applying the spectral method to solve u''(x) = f(x)
MATLAB demo: decay.m: decay rate of Fourier coefficients



Lecture 12: Mon 21 Sep 
6: The spectral method (Gockenbach, Sections 5.25.3)
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



Lecture 11: Fri 18 Sep 
5: Eigenvalues and eigenfunctions (Gockenbach, Sections 5.15.2)
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



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.



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.



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



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



Lecture 6: Fri 4 Sep 
4: Best approximation (Gockenbach, Section 3.3 and 3.4)
Best approximation from a onedimensional 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



Lecture 5: Wed 2 Sep 
3: Differential equations as linear operator equations (Gockenbach, Section 3.1 and 3.4)
General inner products; norm properties; CauchySchwarz inequality.
Want to dive deeper? Learn about L^{p} spaces,
Sobolev spaces,
Besov spaces
To do: Download Chebfun for MATLAB (free)



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].



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



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



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



Resources: 
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.


