Numerical Analysis

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Large Sparse Matrix Problem

There is one point I just can’t understand. I want to solve large
sparse matrices problem
arising from discretization of pdes, like laplace or structural
mechanic with let say about 6 millions dofs.

Currently I have assembled a 600.000 dofs problems, the sparse matrix
itself is about 1Gbyte of memory (I have 2Gbytes so I can stay
incore), the assembly takes about 1minute and solving by PCG takes 75
seconds and 75 iterations. The problem scale linearly with problem
size (matrix sizes, assembly and solving).

My question is: How can I solve efficiently a 6Mdof problem ?

The main option I see is to not compute the whole matrix and compute
Ax products
element-by-element.  The problem would require 750 iterations to
converge, and the computation of small stiffness matrices would for
one iteration require 270 seconds, the product
make about 56hours….

In core, the solving process would take about 750 seconds, but this
huge matrix
doesn’t fit in core (10Gbytes).  I know that the Ca&SI solver can
handle such matrices with "only" 4Gbytes of memory, but I don’t how
they do (if anyone has a hint).

The last option I see is to do out-of-core computations, but I’am not
sure of what the performance would be with IO operations on disk
(depends of the disk of course).

Anyone has an idea to solve such a huge problem on a desktop
computer ?
Best Regards…
thomas laverne

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Dual Infinity Boundary Function?

I have two infinite sets. The first set has a maximum of three subsets
and including infinite, one of the two sub infinite sets has a
deterministic boundary solution. In combination the known boundary
function can form a discrete subset over the second infinite set.

The second infinite set has a sub infinite number of sets, including
an infinite set of subsets. How do I establish the boundary conditions
for the second infinite set, to determine the region of the other two
boundary functions over the first set?

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set of coupled nonlinear PDEs

Hi Everybody,

Recently I’ve run across with a problem.
I wanna solve a set of coupled nonlinear mass diffusion and heat conduction. Diffusivity and thermak conductivity are both functions of concentration and temperature and thinkness of material is a function of time.
Do you happen to have an idea about that?

I am looking forward to hearing from you soon.

Thank you very much in advance.

Noodles

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resampling data

Hi all,

I have a vector of independent multivariate data with additive
(Guassian) noise that is regularly sampled, but needs to be resampled
so that it has the same sampling grid as some of my other data.  Sorry
for the naive question, but will the noise change if I resample the
data?  As sampled, each datum (i.e. each element in the data vector)
is statistically independent of the others, but if I resample the
vector using splines or any other function then I expect that the
resulting resampled data will not necessarily be completely
independent anymore.  If this is indeed the case, I’m a little unsure
of how I can calculate the extent to which the data is correlated.
Can anyone recommend a paper or book that discusses this?

Many thanks.
Graham

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A(m x n)=P(m x n)H(n x n) P has orthonormal columns, H is s.p.d.

Factor A m x n into (P m x n with orthonormal columns) and (H n x n
symmetric positive definite).

A=PH
A’A = H’P'PH
diagonalize A with n x n orthogonal matrix V
VA’AV’ = VH’P'PHV’ = S_A’ S_A  , where S_A diag matrix of singualr
values.

So H = S_A  (its diag but that’s spd, correct?)

And P = A H^(-1) = A  S_A^(-1)

the inverse of diag matrix of singular values of A (diagonal, the
reciprocals of the singular values) S_A^(-1) is multiplied by A to get
a P with orthogonal columns.

All that happens to A is each column is multiplied by a singular
value.  THAT’S NOT ENOUGH TO MAKE IT ORTHONORMAL.

is this about left and right singular vectors?
what’s wrong with what i have so far?

  We may do whatever we like, providing we can pay the bill.

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