I believe Julia is worth learning. I have used it to produce a few research finite element codes, and produce them very quickly. I have been over all very pleased with my experience. Julia has ...

This is a really interesting question, and there are a lot of possible explanations. If we are attempting to use a polynomial interpolation, then note that polynomial satisfy the following annoying ...

In general, all Krylov methods essentially seek a polynomial that is small when evaluated on the spectrum of the matrix. In particular, the $n$th residual of a Krylov method (with zero initial guess) ...

Problems which have a high arithmetic intensity and regular memory access patterns are typically easy(ier) to implement on GPUs, and perform well on them. The basic difficulty in having high ...

Nothing stops you from doing that technically, but when you integrate by parts you get more flexibility with the solution space in that they need not have $H^2$ regularity (required for the non I.B.P ...

This area has been fairly well researched, you may check e.g. Ketcheson's review of such methods: https://doi.org/10.1016/j.jcp.2009.11.006 which does contain some low-storage Runge-Kutta methods ...

I have maybe a unique perspective on this because I am a HPC practitioner with a scientific computation background as well as a functional programming language user. I don't want to equate HPC with ...

Yes you can do this, and it will converge in one iteration regardless of the starting value. This is because each step of Newton's method involves solving a linear system with the Jacobian of the ...

One thing that CG has in its favor is that it's not minimizing the discrete $l^2$ norm for its residual polynomials (what GMRES does). It's minimizing a matrix-induced norm instead, and very often ...

Generally there isn't a way to compute the inverse of a sum of Kronecker products. However, suppose there is a factor in common, let's say $I_T$ here and your sum is  A = K \otimes I_T + I_T \...

There aren't many results in this direction because it is more difficult than just fixing accuracy, since stability considerations can often require you to pick time-steps that are smaller than you ...

Estimates for GMRES convergence based on eigenvalue distribution often implicitly assume that the matrix is normal. Sometimes the convergence rate is still provable in an asymptotic sense in the non-...

This algorithm is most useful for two situations, which are related to each other in practice: You don't know the matrix entries explicitly, but instead can only compute matrix-vector products with ...

A related task to this is to find a subset of column vectors that are maximally linearly independent. Linear independence isn't exactly the same thing as asking for a large determinant, but if we can ...

This really depends on the operations you are including in your question. If you took the sparse equivalent of any level 1 BLAS or level 2 BLAS algorithm, then yes they are memory bound (not compute ...

Unfortunately the 1D reasoning will not directly carry into 2D and beyond. Depending on the domain for your integral, you will have considerable latitude in the node choice - but of course the ...

In the advection dominated case the problem can develop physics which is invisible to a computational mesh that is too coarse (say by having elements which contain many wavelengths). This leads to ...

Power iteration is the simplest, but as mentioned above it would likely converge very slowly if the matrix is very non-normal. You get a "hump" phenomenon where the sequence appears to diverge for ...

The difficulty is relative to something, in this case it is relative to diffusion dominated problems. Diffusion dominated aren't "easy" either, they have their own set of problems. I'll start with ...

I don't recommend it, but not because of performance issues. It will be a little less performant than a traditional matrix, which are usually allocated as a big chunk of contiguous data that is ...

I'm not an expert on this topic, but I have read a few papers concerning this in the hopes of finding a useful preconditioner for my problem. I think though that understanding of this topic is rather ...

You might be familiar with the following paper already: http://link.springer.com/chapter/10.1007%2F978-3-642-22061-6_10 Problems which are highly indefinite and oscillatory are very difficult to ...

The meanings of those terms depend on context. Superconvergence is usually used to mean you are converging faster than the "optimal" rate, and occasionally this sort of weirdly fast convergence can ...

In addition to Jed's great recommendations (I can personally vouch for Brenner+Scott as a great intro finite elements book), an excellent book for the numerical solution of ODEs is Butcher: http://...

Yes you absolutely can, I did. Much of the power out of MATLAB however does come with some of its hugely easy to use toolboxes and builtins which may or may not have equivalents in Octave. Also be ...

In general explicit Runge-Kutta methods of order $N$ require at least $N$ function evaluations, and there is absolutely no way to avoid this. Past $N=4$ they require more than $N$ function evaluations....

In a word: No. The reason MKL and other optimized libraries are so good is they are optimized to balance calculation versus memory use (i.e. your matrix is fully stored in RAM explicitly). If your ...

I think diagonal dominance + symmetry gives positive semidefinite (definite if the dominance is strict). Since you know it's invertible that gives positive definite. Edit: The diagonal entries must ...