This time I encourage you to use MATLAB from a remote
computing platform, i.e., from the app.science.oregonstate.edu .
For this, you must have X-Windows under Cygwin working for you.
- I suggest you use Start X Win icon to start X Windows under CYGWIN.
Open a few terminal windows so you will have somehting to work with.
For example,
type xterm -bg red -fg black &
type xterm -fg yellow -bg black &
so you will have an additional "black" and "red" windows.
- in the black terminal, type
ssh -Y you@app.science.oregonstate.edu and input password
- You can start MATLAB either in the usual graphic mode matlab
- ... or, without the graphics mode
matlab -nodesktop
- Plan how you will copy all the files
to the application server: use scp.
Assuming you have the file myfile in your current folder ...
in the red terminal ...
ls myfile
scp myfile you@app.science.oregonstate.edu:
Turn in solutions to project 1, or 2 (advanced). In both, please
present plots of solutions, and of convergence of Newton (i.e., the
size of residual as iterations progress).
Project 1.
In the algorithm semilinear.m from Lab4, Pbm2, you solved a
semilinear problem using Newton's method.
Now apply the inexact Newton-Krylov method. (Use Chapters 3
and 6 in Kelley's book for background.)
- (i) Change the call to linear solver in the code to
GMRES. Look-up the properties of GMRES implementation in MATLAB. Start
with the demos with various matrices such as the one obtained
with gallery('wilk',n) so you can become familiar how to
use the various parameters controlling GMRES depending on the
dimension n of the problem you are solving.
- (ii) Use a fairly high tolerance for GMRES like tau_gmres=1e-1 at
first. Experiment how large tau_gmres affects the number of
iterations of Newton solver. Then adjust the tolerance to be tighter
when your Newton solver gets closer to the solution.
- (iii) Since GMRES expects to work with a sparse matrix, change the set-up of
matrices A to sparse.
- (iv) Solve the largest problem (change n) you can on the local computer and on
the applications server.
- (v) Compare with other Krylov-type solvers pcg, bicgstab .
Project 2.
Proceed as in Project 1 however for a problem with (i) widely varying
coefficients and (ii) disparate scales as suggested in 4)iv).
- (i) Set the
coefficients k to have values from a log-normal distribution, e.g.,
randn('seed',316);
data = exp(randn(n+1,1));
- (ii) Set the coefficient k to be equal to 1 in the first and
third part of the domain, and to 10^b in the middle part for a large
range of $b=2-6$. This choice works well with the nonlinearity. Now
use $-b=2-6$: this choice works againts the nonlinearity.
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