Math 467/667

467/667 NUMERICAL METHODS II (3+0) 3 credits

Instructor  Course                             Time            Room
----------------------------------------------------------------------
Eric Olson  Math 467/667 Numerical Methods II  TR 3:00-4:15pm  Remote

Course Information

Instructor:

Eric Olson

email:

ejolson at unr dot edu

Office:

Through Zoom by appointment.

Homepage:

http://fractal.math.unr.edu/~ejolson/467/

Required Texts and Equipment

1. Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Second Edition, 2008. Note that an online copy of this book is available from the UNR library.
2. Home computer running Windows, Linux or MacOS and a suitable web camera. Note that it is possible to provision certain mobile phones as web cameras for use on a computer.

Supplemental Texts on Numerical Methods

3. Anthony Ralston and Philip Rabinowitz, A First Course in Numerical Analysis, Second Edition, Dover, 1978.
4. Justin Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning and Graphics, CRC Press, 2015.
5. David Kincaid and Ward Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd Revised Edition, Pure and Applied Undergraduate Texts, American Mathematical Society, 2002.
6. R.J. Hosking, S. Joe, D.C. Joyce, J.C. Turner, First Steps in Numerical Analysis, 2nd Edition, Hodder Education Publishers, 1998.
7. R.W. Hamming, Numerical Methods for Scientists and Engineers, Second Edition.
8. Kendall Atkinson, An Introduction to Numerical Analysis, Second Edition, Wiley, 1989.
9. Eugene Isaacson, Analysis of Numerical Methods, Revised Edition, Dover Books on Mathematics, 1993.

Information about Software

• The Julia Programming Language Download and Documentation.
• The Julia Project, The Julia 1.5 Language, official documentation.
• Thomas Breloff, Plots--Powerful Convenience for Visualization in Julia, official documentation.
• Jupyter Lab, A Comprehensive List of Links for the Juptyer Project, official documentation.
• The UNR DataWorks Remote Desktop.
• Mathematica, Wolfram Language and System Documentation Center, official documentation.

Student Learning Outcomes

1. Use Runge-Kutta and Multistep methods to solve IVP's for ODE's.
2. Use energy-conserving IRK methods to solve conservative ODE's.
3. Use techniques to solve elliptic, parabolic and hyperbolic PDE's.

Lecture Notes

• Convergence of Euler's Method
• Jan 26 -- Introduction and Derivation of Euler's Method
• Jan 28 -- Testing Euler's Method in Julia
  • euler.jl
• Feb 02 -- Convergence of Euler's Method
• Feb 04 -- Trapezoid Method and Truncation Error
  • traptrunc.wl
• Feb 09 -- Theta Method and Adams Bashforth
  • thetatrunc.wl
• Feb 11 -- Multistep Methods
• Gauss Quadrature
• Feb 16 -- Algebraic Approach to Multistep Methods
  • ab3sigma.wl, ab4sigma.wl, newcubic.wl
  • dotprod.wl
• Feb 18 -- Gaussian Quadrature Example
  • opolynom.wl, gaussquad.jl
• Feb 23 -- Accuracy of Gauss Quadrature
  • gqtest.jl, gausscheck.wl (pdf)
• Mar 02 -- RK Methods
  • rk2trunc.wl
• Mar 04 -- The RK Tableaux
  • rkgctrunc.wl
• Mar 11 -- Midterm Review
  • rk3trunc.wl
• Mar 16 -- Practical Example with RK2
  • rossler.jl
• Mar 18 -- Implicit Methods
• Mar 23 -- Linear Stability
  • eulerstab.wl (pdf), rk3stab.wl (pdf)
• Mar 25 -- A-Stability
  • rk3stab2.wl, ab2stab.wl
• Mar 30 -- The Cover of the Book
  • irkgauss.wl (pdf)
• Apr 01 -- Geometric Integrators
  • iserles.jl
• Apr 06 -- Conservative RK Methods
• Apr 08 -- Practical IRK Methods
  • irk.jl, irkdg.wl (pdf)
• Apr 13 -- Energy Conserving IRK Example
  • DG.wl, conserve.jl
• Apr 15 -- Finite Differences
• Apr 20 -- Approximation of Derivatives
  • loge.wl (pdf)
• Apr 22 -- The Poisson Equation
• Apr 27 -- Practical Finite Differences
  • poisson.jl, poisson-new.jl
• May 04 -- Review and Project 2

Announcements

[15-May-2021] Project 2 Solutions

[12-May-2021] Final Exam

In preparing for the final exam please consider the following topics, techniques and questions:

1. There were five video assignments and one reading assignment. Of these six participation activites, which one did you find the most meaningful. Explain why, what inspired you and something you learned.
2. Explain in a mathematical way how the logarithm of the shift operator is connected with the derivative? Provide an example based on the finite-difference method of how this connection can be used to construct a numerical approximation.
3. Compare and contrast the advantages and disadvantages of explicit schemes versus implicit schemes for approximating the solutions to ordinary differential equations.
4. Define the terms truncation error, convergence and stability. Explain the significance of each concept. List one other property a numerical scheme might possess and explain the importance of that property.
5. What's the difference difference between a Runge-Kutta method and a multistep method? Discuss how these two kinds of methods are used and when one might be preferred over the other.
6. Given a particular numerical scheme be able to find the truncation error and linear stability domain.
7. Given a multistep method be able to use the root condition to determine whether the method is convergent.
8. Be able to convert an RK tableau into computer code that approximates a given initial value problem.

[13-May-2021] Programming Project 2 Due

[01-May-2021] Solutions to Homework 2

[29-Apr-2021] In-Class Exam (written part)

1. The proof that Euler's method converges as in the lecture notes or Theorem 1.1 in the textbook.
2. The proof that the n-point Gaussian Quadrature formula is exact for polynomials of degree 2n-1 as in the lecture notes or Theorem 3.3 in the textbook.
3. The definition of linear stability domain, A-stability and how these notions effect the maximum step-size that can be used when approximating the solution to a differential equation.
4. What it means intuitively for a differential equation to be stiff.
5. What it means for a differential equation y'=f(t,y) to preserve a quadratic invariant of the form y(t)·Sy(t) where S is a symmetric matrix.
6. [Extra Credit and for Math 667] The definition of the famous matrix M and why M=0 implies the corresponding RK method exactly preserves all quadratic invariants up to rounding error as in Theorem 5.4.

[07-May-2021] Video Participation 6 Due

1. Who created the Connection Machine?
2. What is a data parallel computer?
3. Are GPUs also data parallel? Explain.
4. Give an example of a computational task that can be sped up using data parallel techniques?

[18-Apr-2021] Project 1 Solutions

[16-Apr-2021] Homework 2

• Iserles 3.1abcd, 3.6, 3.7

• Complete the multi-part question about Gaussian quadrature:
  1. Make the change of variables y=tan(z) so that
     ∫₀^π/2 1/sqrt(1+tan(z)) dz = ∫₀^∞ g(y) dy.

     Write down an explicit formula for g(y).

  2. Show the further change of variables x=2y/(1+y)-1 transforms the integral above into the form
     ∫₀^∞ g(y) dy = ∫_-1¹ h(x) sqrt(1-x) dx.

     Write down an explicit formula for h(x).

  3. Define the weighted inner product and norm as
     (α,β) = ∫_-1¹ α(x) β(x) sqrt(1-x) dx   and   ||α||=sqrt(α,α).

     Use a computer algebra system (or pencil and paper if you prefer) to find the orthonormal polynomials p_n of degree n with respect to this inner product for n = 0, 1, ..., 6.

  4. Find the six roots x_k of p₆(x) and the corresponding weights w_k for k = 1, 2, ..., 6 of such that
     ∫_-1¹ x^j sqrt(1-x) dx = ∑_k w_k x_k^j   for   j = 0, 1, ..., 11.

  5. Use the weighted six-point Gauss quadrature method and the change of variables developed above to approximate the integral
     ∫₀^π/2 1/sqrt(1+tan(z)) dz ≈ ∑_k w_k h(x_k).

     What is the error in the approximation? Hint: if it's way off, please check all of your work and fix the mistake.

• [Extra Credit] Iserles 4.2abcde

• Iserles 4.4, 4.5, 4.6

[09-Apr-2021] Video Participation 5 Due

1. When Leo Kouwenhoven went to university, what subject was his first choice of study?
2. List some possible applications for quantum computing.
3. What is a qubit? In theory how much faster is a 32-qubit quantum computer compared to a 16-qubit quantum computer?
4. Predict when, if ever, a quantum computer will be available for student use at UNR. Explain your reasoning.

[30-Mar-2021] Solutions to Homework 1

[26-Mar-2021] Reading Participation 4 Due

with more science and engineering codes in need of a refresh to make AI/ML a possibility, starting with a cleaner slate with something [Julia] that provides the best of C or C++ and the usability of Python, for example, is becoming more attractive

Early in our journey some VCs said it was easy to monetize by holding back and selling the last 2X performance. We didn't want to do that. We developed it [Julia] to give people the tooling to solve really hard problems and it didn't make sense to take that away to make money.

Please read the article linked above and answer the following questions:

• What do you like most about Julia?
• What do you like least about Julia?

[12-Mar-2021] Midterm Exam (computer part)

In preparing for the midterm please know the following topics and techniques:

1. Given a numerical scheme for approximating an ODE, be able to compute the truncation error of the scheme.
2. Under the assumption that the method converges, explain how the order of the truncation error is related to the expected order of the resulting ODE solver.
3. Convert between the explicit form of a multistep method and the polynomials rho(w) and sigma(w).
4. Use the root condition to prove a multistep method is convergent.
5. Given a polynomial rho(w) the satisfies the root condition be able to solve for sigma(w) to find a multistep method with a specified order.
6. [Extra Credit] If one is given sigma(w) is it possible to find a function rho(w) that constructs a multistep method? If having a high order of convergence is not necessary is it always possible to find such a rho(w) that satisfies the root condition?
7. Be able to rescale the [-1,1] interval of a Gaussian quadrature method to approximate an integral in an arbitrary interval [a,b].
8. Be able to translate an RK tableaux into the actual equations used to compute a numerical scheme and vice versa.
9. Be familiar enough with Julia and Mathematica to perform some calculations using Euler's method and/or Gaussian quadrature.

[12-Mar-2021] Homework 1

• Iserles 1.1 with the following modifications: Skip everything to do with the theta method and consider only the implicit midpoint rule. Prove that the implicit midpoint rule is second order and then apply the method of proof of Theorems 1.1 and 1.2 to prove it is convergent.

• Iserles 1.4, 1.5 and 1.8.

• Iserles 2.3abcd, 2.4

[19-Mar-2021] Programming Project 1 Due

[05-Mar-2021] Video Participation 3 Due

1. Where is the Citadel Campus located?
2. What three energy sources are used to generate the electricity used at the Citadel Campus?
3. A Tier 4 datacenter guarantees 99.995% availability. How many minutes could a datacenter be unavailable per year and still meet these requirements?
4. What is the advertised network latency from Reno to Las Vegas? How does this compare to the theoretical limits based on the speed of light found in Video Participation 2?

[19-Feb-2021] Video Participation 2 Due

1. Who is Grace Hopper?
2. How many nanoseconds does it take to travel at the speed of light from Reno to Las Vegas?
3. How many picoseconds does each clock cycle of a CPU running at 2.8 Ghz take?
4. Research other sources of information about Grace Hopper and relate something you found interesting which doesn't appear in the video.

[05-Feb-2021] Video Participation 1 Due

1. Where is Argonne National Laboratory?
2. List some planned applications for the Aurora Supercomputer.
3. What is an exascale computer and what does exascale mean?
4. What are the possible benefits of mixing numeric simulation with artifical intelligence?

[04-Feb-2021] Mathematica

[26-Jan-2021] First Day of Class

[21-Jan-2021] Mathematics and Statistics Book Club

The first book we will read and discuss is "Mathematics for Human Flourishing" by Francis Su. The e-book is available for free at the UNR Knowledge Center. Here is a brief introduction where the author explains his view of human flourishing.

We will meet via Zoom on Tuesdays from 1:30 to 2:30pm starting in Spring 2021. The schedule is

• Feb. 2 (Chapters 1 - 3)
• March 2 (Chapters 4 - 7)
• April 6 (Chapters 8 - 11)
• May 4 (Chapters 12 - 13, Epilogue)

I'll also put a link on our WebCampus page for your convenience. Note that this is not a mandatory assignment but just an announcement of an activity you might find fun while sheltering at home to escape the epidemic.

[20-Jan-2021] Zoom for Students

[20-Jan-2021] Zoom

[20-Jan-2021] WebCampus

[20-Jan-2021] Julia

[20-Jan-2021] Alternative Remote

Grading

     Midterm (written part)    50 points
     Midterm (computer part)   50 points
     2 Homework Assignments    40 points each
     2 Programming Projects    50 points each
     Attendence/Participation  20 points
     Final Exam               100 points
    ------------------------------------------
                              400 points total

    Grade        Minimum Percentage
      A                 90 %
      B                 80 %
      C                 70 %
      D                 60 %

Course Schedule

Jan 25-Jan 29   Week 1: 1.1-1.2 Euler's Method
Feb 01-Feb 05   Week 2: 1.3-1.4 Theta Method
Feb 08-Feb 12   Week 3: 2.1-2.2 Adams Bashforth

*** President's Day Monday Feb 15

Feb 16-Feb 19   Week 4: 2.3 Backwards Difference Formulas
Feb 22-Feb 24   Week 5: 3.1 Gaussian Quadrature

*** Reading Day Thursday Feb 25

Mar 01-Mar 05   Week 6: 3.2-3.3 Runge Kutta
Mar 11-Mar 12   Week 7: Review, Midterm

*** Reading Day Tuesday Mar 9
*** Reading Day Wednesday Mar 10

Mar 15-Mar 19   Week 8: 3.4-4.1 Implicit RK and Stability

*** No-Instruction Day Wednesday Mar 24

Mar 22-Mar 26   Week 9: 4.2-4.3 Linear and A Stability

*** Spring Break Cancelled

Mar 29-Apr 02   Week 10: Chapter 5 or 6 or other
Apr 05-Apr 09   Week 11: optional topics
Apr 12-Apr 16   Week 12: 8.1-8.2
Apr 19-Apr 23   Week 13: 8.3

*** Reading Day Wednesday Apr 21

Apr 26-Apr 30   Week 14: 9.1-9.2
May 03-May 04   Week 15: Review

*** Prep Day May 5
*** Final exam Wednesday, May 12 from 2:30 to 4:30am

Course Policies

Communications Policy

Late Policy

Plagiarism

Netiquette

Diversity

COVID-19 Policies

Statement on COVID-19 Training Policies

Statement on COVID-19 Face Coverings

Statement on COVID-19 Social Distancing

Statement on COVID-19 Disinfecting Your Learning Space

Contact with Someone Testing Positive for COVID-19

Local, State and Federal COVID-19 Information

• Return of the Pack
• Reno/Sparks/Washoe Regional Information Center
• State of Nevada Heath Response
• CDC: Coronavirus

Statement on Academic Success Services

Equal Opportunity Statement

Statement on Audio and Video Recording

Academic Conduct

Final Exam