21 votes
Accepted

What is the state of the art in solving stiff initial value problems?

So there is a ton to say about this, and we will actually be putting a paper out that tries to summarize it a bit, but let me narrow it down to something that can be put into a quick StackOverflow ...
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19 votes

Robust algorithm for $2 \times 2$ SVD

See https://math.stackexchange.com/questions/861674/decompose-a-2d-arbitrary-transform-into-only-scaling-and-rotation (sorry, I would have put that in a comment but I've registered just to post this ...
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17 votes

Looking for Runge-Kutta 8th order in C/C++

If you're doing celestial mechanics over long time scales, using a classical Runge-Kutta integrator will not preserve energy. In that case, using a symplectic integrator would probably be better. ...
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17 votes
Accepted

Is it well known that some optimization problems are equivalent to time-stepping?

As Jed Brown mentioned, the connection between gradient descent in nonlinear optimization and time stepping of dynamical systems is rediscovered with some frequency (understandably, since it's a very ...
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14 votes

Fast explicit solution for $\mathbf{A}\mathbf{x} = \mathbf{b}$, $ \mathbf{b} \in \mathbf{R}^3$, low condition number

You can't beat an explicit formula. You can write down the formulas for the solution $x=A^{-1}b$ on a piece of paper. Let the compiler optimize things for you. Any other method will almost inevitably ...
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13 votes

Is it well known that some optimization problems are equivalent to time-stepping?

While I haven't seen the exact formulation that you have written down here, I keep seeing talks in which people "rediscover" a connection to integrating some transient system, and proceed to write ...
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  • 25.2k
13 votes
Accepted

How to directly compute the inverse of an ill-conditioned dense matrix

Though it is a relatively rare situation when you actually have to calculate an inverse of the matrix, not all techniques were created equally. I would use the term badly-conditioned instead of ill-...
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  • 8,287
12 votes

Modern C++ in scientific computing?

Two examples of libraries that use modern C++ constructs: Both the eigen and armadillo libraries (linear algebra) use several modern C++ constructs. For instance, they use both expression templates ...
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  • 2,165
12 votes
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Benchmarks for Gröbner bases and polynomial system solution

I posted some benchmarks here: http://www.cecm.sfu.ca/~rpearcea/mgb.html (archived copy) These are for total degree orders. To solve systems you typically need to do more work. Timings are for a ...
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12 votes
Accepted

$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)

Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that $L^2$ is its own dual space to write the $L^2$-norm as an operator norm $$\|u\|_{L^2} = \sup_{\phi\in L^2\...
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11 votes
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How important is learning hardware/architecture for scientific computing?

I haven't worked in quantum chemistry specifically, but I've worked in other areas where high performance is a correctness requirement (along with scientific accuracy), so I think we're on the same ...
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  • 351
10 votes

Robust algorithm for $2 \times 2$ SVD

@Pedro Gimeno "I doubt it can be any more robust than that." Challenge accepted. I noticed the usual approach is to use trig functions like atan2. Intuitively, there shouldn't be a need to use trig ...
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10 votes

Modern C++ in scientific computing?

I would suggest taking a look at Deal.II. It uses the STL, it's own iterators, shared pointers, etc. The various linear solvers can use the various matrices because of how it was designed. I ...
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  • 1,000
10 votes
Accepted

Intro to DG Finite Element methods

For parabolic/elliptic PDE's, I highly recommend Beatrice Riviere's book: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. For hyperbolic PDE'...
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  • 11.8k
9 votes
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Looking for Runge-Kutta 8th order in C/C++

Both GNU Scientific Library (GSL) (C) and Boost Odeint (C++) feature 8th order Runge-Kutta methods. Both are opensource, and under linux and mac they should be directly available from the package ...
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  • 2,463
9 votes

Fast explicit solution for $\mathbf{A}\mathbf{x} = \mathbf{b}$, $ \mathbf{b} \in \mathbf{R}^3$, low condition number

Since the matrix is so close to the identity, the following Neumann series will converge very rapidly: $$A^{-1} = \sum_{k=0}^\infty (I-A)^k$$ Depending on the accuracy required it might even be good ...
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  • 3,003
9 votes

Benchmarks for Gröbner bases and polynomial system solution

Googling benchmark polynomial systems leads to a few hits, including the University of Mannheim's Computer Algebra Benchmark Initiative. Sadly, most of these are ...
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9 votes
Accepted

First appearance of the phrase "inverse crime"

The term inverse crime for a numerical test of a parameter identification method that uses data contained in the range of the discrete(!) forward operator used for the inversion (thus essentially ...
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9 votes
Accepted

Consumer hardware for scientific computing?

One issue that you should be aware of is that NVIDIA has a market segmentation strategy in which it sells relatively inexpensive GPU's to the gaming and graphics workstation markets (GeForce and ...
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9 votes
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4th order tensor rotation - sources to refer

There are two main ways to write stress/strain tensors as 6 components vectors: Voigt notation, that is the most common; and Mandel-Kelvin notation, that has the advantage of writing stress and ...
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  • 7,902
9 votes
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FEM for vector valued problems: reference request

Short answer: Just replicate the vector of interpolation functions into a block-diagonal matrix, as showed e.g. on page 5 in this lecture note. Detailed answer: Mathematically oriented texts typically ...
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8 votes
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Books on mathematical foundation of finite element methods

I have to disagree with Paul: Functional analysis is a beautiful, elegant topic with an enormous range of applications (but de gustibus... :)). But in any case, you don't need to know a lot of (pure)...
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8 votes

Robust algorithm for $2 \times 2$ SVD

I needed an algorithm that has little branching (hopefully CMOVs) no trigonometric function calls high numerical accuracy even with 32 bit floats We want to calculate $c_1, s_1, c_2, s_2, \sigma_1$ ...
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8 votes
Accepted

Finite elements on manifold

I think you start by looking at something like FEniCS. Marie Rognes has a presentation with code examples and a paper discussing the theory and implementation. libMesh is supposed to be able to do ...
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7 votes

Modern C++ in scientific computing?

The HPX library makes heavy use of a range of C++11 features such as move constructors and is also aiming to be a complete implementation of N4409 (Working Draft, Technical Specification for C++ ...
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  • 171
7 votes
Accepted

Term for the typical "linear in the larger dimension, quadratic in the smaller" cost for linear algebra

The book "Introduction to Applied Linear Algebra" by Boyd and Vandenberghe has an appendix about complexity of basic operations in linear algebra and they call this case big-times-small-...
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  • 1,728
7 votes
Accepted

The Formula of Explicit Runge-Kutta Fourteen order

The 14th order methods due to Feagin can be found in DifferentialEquations.jl. An example using them with 128-bit floating point arithmetic is as follows: ...
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6 votes
Accepted

Point inside curved finite element

I don't think you can do better in general than mapping to the reference element and testing there. If your mapping is somehow special, you might be able to develop a test that's more efficient than ...
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  • 10.8k
6 votes

Modern C++ in scientific computing?

I suggest taking a look at Scientific and Engineering C++: An Introduction with Advanced Techniques and Examples by Barton and Nackmann. The fact that this book was published in 1994 makes it appear ...
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  • 5,734
6 votes

Benchmarks for Gröbner bases and polynomial system solution

Always keep in mind that the results of any benchmark will depend, in addition to the problem's size, on the base field over which the polynomial ring is defined (rational numbers or integers modulo ...
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