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The connection between random walks, diffusion, and the heat equation is an amazing example of "the unreasonable effectiveness of mathematics." However, it's important to understand that this doesn't …

Unfortunately you can't. With any orthogonal factorization (e.g. QR, LQ, or SVD) you have the problem that because some of the columns of the orthogonal matrix have to span a particular subspace, and …

You're really asking the wrong question here... Let's back up a bit. You're attempting to estimate some parameters here, either by a maximum likelihood method or more likely by $\chi^2$ minimization …

It's worth pointing out that many inverse problems in the functional analytic setting go beyond ill-conditioning to ill-posedness. That is, a small change in the data (noise) can lead to an aribtrari …

The real issue here is the constraint $\sum_{i} x_{i}1_{x_{i}>a} < b $ whose left hand side has horrible discontinuities. Rather than using a solver designed for problems with continuous variable …

Linear systems of 10,000 equations in 10,000 unknowns can easily be solved in a few seconds using double precision floating point arithmetic on typical consumer grade PC's and even laptop computers. …

As Fumiyo Eda already mentioned, you can use an iterative method such as GMRES to resolve the system after the change to $A$. If you want to use direct LU factorization rather than an iterative meth …

Lagrange interpolation is the classic approach to this problem. In order for it to work, you'll need very precise function values, since the problem of numerical differentiation is ill-conditioned. …

You haven't said whether $K$, $\Sigma$, or $L_{1}$ are sparse or otherwise specially structured (e.g. Toeplitz.) This could be hugely important in finding the most efficient way to do this. You als …

Unfortunately, you can't do any better than your "no so good method." It's a standard result that this is the best (as measured by the Frobenius norm) rank $n$ approximation to $M$.

This is hopeless without further assumptions, because no numerical procedure with a finite number of function evaluations can ever rule out a lack of injectivity in parts of the domain where the funct …

I agree that this is a better question for scicomp.stackexchange.com. Maybe. It depends on the sparsity structure of your particular matrix and the actual numerical values of the nonzero elements …

Being able to get elements of the matrix isn't very useful (particularly if you don't know where the nonzero elements of the matrix are without checking.) Iterative methods can be useful if you hav …

Now that you've provided some more information, I think I can make some useful suggestions. First, a quick review of linear transformations of multivariate normal random vectors. If $z$ is an MVN …

Yes, this is more or less true. See for example Timothy Davis's book, "Direct Methods for Sparse Linear Systems." You can work out in advance the locations of all possible non-zero entries in $L$ …