**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

- Instructor:
- Eric Olson
- email:
- ejolson at unr dot edu
- Office:
- Through Zoom by appointment.
- Homepage:
- http://fractal.math.unr.edu/~ejolson/467/

- 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.
- 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.

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

- 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.

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

- Convergence of Euler's Method
- Jan 26 -- Introduction and Derivation of Euler's Method
- Jan 28 -- Testing Euler's Method in Julia
- Feb 02 -- Convergence of Euler's Method
- Feb 04 -- Trapezoid Method and Truncation Error
- Feb 09 -- Theta Method and Adams Bashforth
- Feb 11 -- Multistep Methods
- Gauss Quadrature
- Feb 16 -- Algebraic Approach to Multistep Methods
- Feb 18 -- Gaussian Quadrature Example
- Feb 23 -- Accuracy of Gauss Quadrature
- Mar 02 -- RK Methods
- Mar 04 -- The RK Tableaux
- Mar 11 -- Midterm Review
- Mar 16 -- Practical Example with RK2
- Mar 18 -- Implicit Methods
- Mar 23 -- Linear Stability
- eulerstab.wl (pdf), rk3stab.wl (pdf)

- Mar 25 -- A-Stability
- Mar 30 -- The Cover of the Book
- Apr 01 -- Geometric Integrators
- Apr 06 -- Conservative RK Methods
- Apr 08 -- Practical IRK Methods
- Apr 13 -- Energy Conserving IRK Example
- Apr 15 -- Finite Differences
- Apr 20 -- Approximation of Derivatives
- Apr 22 -- The Poisson Equation
- Apr 27 -- Practical Finite Differences
- May 04 -- Review and Project 2

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

- 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.
- 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.
- Compare and contrast the advantages and disadvantages of explicit schemes versus implicit schemes for approximating the solutions to ordinary differential equations.
- 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.
- 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.
- Given a particular numerical scheme be able to find the truncation error and linear stability domain.
- Given a multistep method be able to use the root condition to determine whether the method is convergent.
- Be able to convert an RK tableau into computer code that approximates a given initial value problem.

- The proof that Euler's method converges as in the lecture notes or Theorem 1.1 in the textbook.
- 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.
- 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.
- What it means intuitively for a differential equation to be stiff.
- 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.
- [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.

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

- Iserles 3.1abcd, 3.6, 3.7
- Complete the multi-part question about Gaussian quadrature:
- 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).

- Show the further change of variables x=2y/(1+y)-1 transforms
the integral above into the form
∫ _{0}^{∞}g(y) dy = ∫_{-1}^{1}h(x) sqrt(1-x) dx.Write down an explicit formula for h(x).

- Define the weighted inner product and norm as
(α,β) = ∫ _{-1}^{1}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. - Find the six roots x
_{k}of p_{6}(x) and the corresponding weights w_{k}for k = 1, 2, ..., 6 of such that∫ _{-1}^{1}x^{j}sqrt(1-x) dx = ∑_{k}w_{k}x_{k}^{j}for j = 0, 1, ..., 11. - 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.

- Make the change of variables y=tan(z) so that
- [Extra Credit] Iserles 4.2abcde
- Iserles 4.4, 4.5, 4.6

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

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?

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

- Given a numerical scheme for approximating an ODE, be able to compute the truncation error of the scheme.
- 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.
- Convert between the explicit form of a multistep method and the polynomials rho(w) and sigma(w).
- Use the root condition to prove a multistep method is convergent.
- 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.
- [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?
- Be able to rescale the [-1,1] interval of a Gaussian quadrature method to approximate an integral in an arbitrary interval [a,b].
- Be able to translate an RK tableaux into the actual equations used to compute a numerical scheme and vice versa.
- Be familiar enough with Julia and Mathematica to perform some calculations using Euler's method and/or Gaussian quadrature.

- 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

- Where is the Citadel Campus located?
- What three energy sources are used to generate the electricity used at the Citadel Campus?
- A Tier 4 datacenter guarantees 99.995% availability. How many minutes could a datacenter be unavailable per year and still meet these requirements?
- 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?

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

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

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.

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 totalExams and quizzes will be interpreted according to the following grading scale:

Grade Minimum Percentage A 90 % B 80 % C 70 % D 60 %The instructor reserves the right to give plus or minus grades and higher grades than shown on the scale if he believes they are warranted.

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

- Return of the Pack
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- CDC: Coronavirus

Last Updated: Mon Jan 18 14:41:57 PST 2021