22M:270
Theoretical Numerical Analysis
Fall 2011
You can see me outside the office hours provided it is mutually
convenient.
22M:270 is the first half on the theoretical side of numerical
analysis.
As such we will be dealing with Banach and Hilbert spaces as a way
of formulating questions and methods in numerical analysis. As always
in numerical analysis, we can only store a finite amount of data,
and so we are dealing with finite-dimensional approximations to the
objects we wish to compute. The main questions are: Will our
approximations
converge to the desired solution? How quickly do the computed objects
converge to the desired solution(s)?
A central issue is the problem of solving partial differential
equations,
where the solution necessarily belongs to an infinite dimensional
space. Related questions such as multidimensional approximation theory
(such as approximating functions on triangles, spheres, cubes, etc.)
become very important.
Theoretical Numerical Analysis by K. Atkinson and
W. Han,
J. Wiley and Sons, 3rd Edition (2010).
- Spaces in which to do numerical analysis
- Banach spaces and Hilbert spaces
- \(L^p(\Omega)\) spaces
- Sobolev spaces: \(H^s(\Omega)\) and \(W^{k,p}(\Omega)\)
spaces
- Linear operators and functionals
- Dual spaces and adjoint operators
- Strong and weak (and weak*) convergence
- Convex functions and optimization
- Approximation theory
- Stone-Weierstrass approximation theorem
- Interpolation in one dimension (polynomial and trigonometric)
- Best approximation
- Lebesgue constants
- Jackson theorems
- \(L^2(\Omega)\) approximation
- Fourier analysis and wavelets
- Trigonometric approximation
- Gibb's phenomenon
- Fourier transform and tempered distributions
- The discrete Fourier transform and the FFT
- Wavelets
- Solving equations in Banach spaces
- Contraction mapping principle (Banach fixed point theorem)
- Differential equations in Banach spaces
- Calculus in Banach spaces
- Newton's method in Banach spaces
- Conjugate gradient method in Hilbert spaces
- Finite difference methods
- Finite difference approximations
- Lax equivalence theorem (stability + consistency implies
convergence)
There will be a midterm exam (30%), a final exam (30%), and homework
(40%).
The final exam will be held in the regular classroom.
- Course plan: The course
plan may be modified during the semester. Such modifications will
be announced in advance during class periods; the student has
responsibility
for keeping up with such changes. You should also make a habit of
reviewing the ICON web page for this course, which is accessible
via: ICON http://icon.uiowa.edu/
This page will have homework details
and other information posted to it as the class progresses.
- Administration: The administrative home of this
course
is The Department of Mathematics in the College of Liberal Arts and
Sciences: offices are in 14 McLean Hall (MLH). You can contact the
chair of the department through the Departmental Secretary at 14 MLH
or by calling 335-0714. Since the administrative home of this course
is the College of Liberal Arts and Sciences, which governs academic
matters relating to the course such as add/drop deadlines,
second-grade-only
option, issues concerning academic fraud or academic probation, and
how credits are applied for various CLAS requirements. Please keep
in mind that different colleges might have different policies. If
you have questions about these or other CLAS policies, visit your
academic advisor or 120 Schaeffer Hall and speak with the staff. The
CLAS Academic Handbook is another useful source of information on
CLAS academic policy: http://www.clas.uiowa.edu/students/handbook/
- Electronic Communication: University policy
specifies
that students are responsible for all official correspondences sent
to their University of Iowa e-mail address (@uiowa.edu). Faculty and
students should use this account for correspondences. (Operations
Manual, III.15.2. Scroll down to k.11.)
- Accommodations for Disabilities: A student seeking
academic accommodations should first register with Student Disability
Services and then meet privately with the course instructor to make
particular arrangements. See www.uiowa.edu/~sds/ for more
information.
- Academic Fraud: Plagiarism and any other activities
when students present work that is not their own are academic fraud.
Academic fraud is a serious matter and is reported to the departmental
DEO and to the Associate Dean for Undergraduate Programs and
Curriculum.
Instructors and DEOs decide on appropriate consequences at the
departmental
level while the Associate Dean enforces additional consequences at
the collegiate level. See the CLAS Academic Fraud section of the
Student
Academic Handbook.
- CLAS Final Examination Policies: Final exams may be
offered only during finals week. No exams of any kind are allowed
during the last week of classes. Students should not ask their
instructor
to reschedule a final exam since the College does not permit
rescheduling
of a final exam once the semester has begun. Questions should be
addressed
to the Associate Dean for Undergraduate Programs and Curriculum.
- Making a Suggestion or a Complaint: Students with
a suggestion or complaint should first visit the instructor, then
the course supervisor, and then the departmental DEO. Complaints must
be made within six months of the incident. See the CLAS Student
Academic
Handbook (see above).
- Understanding Sexual Harassment: Sexual harassment
subverts the mission of the University and threatens the well-being
of students, faculty, and staff. All members of the UI community have
a responsibility to uphold this mission and to contribute to a safe
environment that enhances learning. Incidents of sexual harassment
should be reported immediately. See the UI Comprehensive Guide on
Sexual Harassment http://www.uiowa.edu/~eod/policies/sexual-harassment-guide/index.htmlfor
assistance,
definitions, and the full University policy.
- Reacting Safely to Severe Weather: In severe
weather,
class members should seek appropriate shelter immediately, leaving
the classroom if necessary. The class will continue if possible when
the event is over. For more information on Hawk Alert and the siren
warning system, visit the Public Safety web site http://www.uiowa.edu/~pubsfty/intlinks.htm.
*These CLAS policy and procedural statements have been summarized
from the web pages of the College of Liberal Arts and Sciences and
The University of Iowa Operations Manual.
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