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18
votes
2answers
4k views

Discontinuous Galerkin: Nodal vs Modal advantages and disadvantages

There are two general approaches to representing solutions in the discontinuous galerkin method: nodal and modal. Modal: Solutions are represented by sums of modal coefficients multiplied by a set of ...
18
votes
2answers
5k views

Unstructured quad mesh-generation?

What is the best (scalability and efficiency) algorithms for generating unstructured quad meshes in 2D? Where can I find a good unstructured quad mesh-generator? (open-source preferred)
18
votes
3answers
4k views

Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD (symmetric-positive-definite) because it requires less ...
18
votes
2answers
314 views

Is there an efficient algorithm for matrix-valued continued fractions?

Suppose I have a matrix equation recursively defined as A[n] = inverse([1 - b[n]A[n+1]]) * a[n] Then the equation for A[1] looks similar to a continued fraction,...
18
votes
1answer
409 views

Catastrophic cancellation in logsum

I'm trying to implement the following function in double-precision floating point with low relative error: $$\mathrm{logsum}(x,y) = \log(\exp(x) + \exp(y))$$ This is used extensively in statistical ...
18
votes
3answers
5k views

Solving unconstrained nonlinear optimization problems on GPU

I am trying to solve some unconstrained nonlinear optimization problems on GPU (CUDA). The objective function is a smooth nonlinear function, and its gradient is relatively cheap to compute ...
18
votes
1answer
5k views

Alternatives to hdf5

I've been using HDF5 for years, but as the size of the dataset grows I'm starting to experience the same problems listed here http://cyrille.rossant.net/moving-away-hdf5/ Can you point me to a ...
18
votes
1answer
645 views

Why are Octrees used for Multipole space decomposition?

In most (all?) implementations of the Fast Multipole Method (FMM), octrees are used to decompose the relevant domain. Theoretically, octrees provide a simple volumetric bound, which is useful for ...
17
votes
8answers
1k views

Is there any open-source or easy-to-access software that can simplify algebraic expressions like $x^{2}+2x+3, x=\sqrt{2}t-1$?

I always calculate things by hand, but now my comrades are getting nasty and making a lot of repetitive exercises involving just plugging things in like the expression above. I am particularly ...
17
votes
6answers
3k views

To what extent is generic and meta-programming using C++ templates useful in computational science?

The C++ language provides generic programming and metaprogramming through templates. These techniques have found their way into many large-scale scientific computing packages (e.g., MPQC, LAMMPS, CGAL,...
17
votes
4answers
20k views

What's the fastest software(open source) to solve mixed integer programming problem

I have a mixed integer programming problem. And I am current using GLPK as my solver. But I found that GLPK is good for Linear Programming problem, but for Mixed Integer programming, it requires much ...
17
votes
5answers
7k views

What is the best way to determine the number of non zeros in sparse matrix multiplication?

I was wondering whether there is a fast and efficient method to find the number of non zeros in advance for sparse matrix multiplication operation assuming both matrices are in CSC or CSR format. I ...
17
votes
5answers
3k views

What is the advantage of multigrid over domain decomposition preconditioners, and vice versa?

This is mostly aimed for elliptic PDEs over convex domains, so that I can get a good overview of the two methods.
17
votes
5answers
961 views

How to address numerical non-associativity for parallel reduction?

A parallel reduction assumes that the corresponding operation is associative. This assumption is violated for addition of floating point numbers. You might ask why I care about this. Well, it makes ...
17
votes
6answers
2k views

How to reorder variables to produce a banded matrix of minimum bandwidth?

I'm trying to solve a 2D Poisson equation by finite differences. In the process, I obtain a sparse matrix with only $5$ variables in each equation. For example, if the variables were $U$, then the ...
17
votes
2answers
10k views

Which version of Fortran should I learn?

I'm a Mechanical Engineering student interested in the field of aerospace engineering where, I'm told, Fortran is still commonly used. Which version of Fortran should I invest my time to learn?
17
votes
4answers
680 views

What are some applications which require interval arithmetic?

I have a very basic notion about interval arithmetic (IA), but it seems to be a very interesting branch of computational science both theoretically and practically. It is clear that the obvious ...
17
votes
3answers
2k views

Are BLAS implementations guaranteed to give the exact same result?

Given two different BLAS implementations, can we expect that they make the exact same floating point computations and return the same results? Or can it happen, for instance, that one computes a ...
17
votes
4answers
11k views

Do currently available GPUs support double precision floating point arithmetic?

I have run the molecular dynamics (MD) code GROMACS on a Ubuntu Linux cluster consisting of nodes containing 24 Intel Xeon CPUs. My particular point of interest turns out to be somewhat sensitive to ...
17
votes
3answers
352 views

What programming strategies can I take for easily modifying algorithm parameters?

Developing scientific algorithms is a highly iterative process often involving changing lots of parameters that I will want to vary either as part of my experimental design or as part of tweaking ...
17
votes
1answer
6k views

What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?

I have learnt about Finite Element Method (also a little on other numerical methods) but I don't know what are exactly definition of these two errors and differences between them?
17
votes
8answers
2k views

Parsing protein structure data in C

My background is in genomics, but I have recently been working with problems related to protein structure. I wrote a few relevant programs in C, building my own PDB file parser from scratch in the ...
17
votes
2answers
2k views

Practical example of why it is not good to invert a matrix

I am aware about that inverting a matrix to solve a linear system is not a good idea, since it is not as accurate and as efficient as directly solving the system or using LU, Cholesky or QR ...
17
votes
5answers
2k views

Finding a global minimum of a smooth, bounded, non-convex 2D function that is costly to evaluate

I have a bounded non-convex 2-D function which I'd like to find the minimum of. The function is quite smooth. Evaluating it is costly. An acceptable error is about 3% of the function's domain in each ...
17
votes
5answers
536 views

Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?

Given an evolution PDE $$u_t = Au + Bu$$ where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving $$u_t = Au$$ ...
17
votes
7answers
1k views

Scripted Mesh Generation Software

I'm looking for a mesh generation software that is free and open source, provides a sane scripting interface for domain specification, works for complex geometries, can generate 2D and 3D meshes, ...
17
votes
2answers
4k views

Disadvantages of common discretization schemes for CFD simulations

The other day, my computational fluid dynamics instructor was absent and he sent in his PhD candidate to substitute for him. In the lecture he gave, he seemed to indicate several disadvantages ...
17
votes
2answers
4k views

Stopping criteria for iterative linear solvers applied to nearly singular systems

Consider $Ax=b$ with $A$ nearly singular which means there is an eigenvalue $\lambda_0$ of $A$ that is very small. The usual stop criterion of an iterative method is based on the residual $r_n:=b-Ax_n$...
17
votes
2answers
4k views

Options for solving ODE systems on GPUs?

I would like to farm out solving systems of ODEs onto GPUs, in a 'trivially parallelisable' setting. For example, doing a sensitivity analysis with 512 different parameter sets. Ideally I want to do ...
17
votes
1answer
5k views

When is Newton-Krylov not an appropriate solver?

Recently I have been comparing different non-linear solvers from scipy and was particularly impressed with the Newton-Krylov example in the Scipy Cookbook in which they solve a second order ...
17
votes
3answers
337 views

Desktop software with HPC resources for back end number crunching

Our workgroup produces a desktop application that simulates building energy performance. It is a .NET application and when the user is running a lot of simulations, they can be quite time consuming. ...
17
votes
5answers
377 views

Databases of results for numerical codes

In the numerical methods literature, many research papers consist of a description of a new algorithmic variation, followed by a few test problems comparing the new method with one or two existing ...
17
votes
2answers
1k views

Which libraries have good high-level support for multigrid?

I'm planning to use multigrid to calulate some eigenvalues and vectors, and I noticed PETSc has high-level support for multigrid. The PETSc documentation says that this part of PETSc should not be ...
17
votes
5answers
3k views

Is there a good, easy-to-use, high quality open source CFD solver out there?

My thesis is on developing numerical methods for model reduction in combustion. I run my methods purely on the chemistry part of combustion simulations, and I have plenty of case studies for 0-D ...
17
votes
3answers
8k views

Efficient computation of the matrix square root inverse

A common problem in statistics is computing the square root inverse of a symmetric positive definite matrix. What would be the most efficient way of computing this? I came across some literature (...
17
votes
2answers
4k views

Drawbacks of Newton-Raphson approximation with approximate numerical derivative

Suppose I have some function $f$ and I want to find $x$ such that $f(x)\approx 0$. I might use the Newton-Raphson method. But this requires that I know the derivative function $f'(x)$. An analytic ...
17
votes
4answers
5k views

Portable multicore/NUMA memory allocation/initialization best practices

When memory bandwidth limited computations are performed in shared memory environments (e.g. threaded via OpenMP, Pthreads, or TBB), there is a dilemma of how to ensure that the memory is correctly ...
17
votes
1answer
991 views

How do you debug numerical code, what could be source of this oscillatory error?

Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
17
votes
1answer
2k views

How to Run MPI-3.0 in shared memory mode like OpenMP

I am parallelizing code to numerically solve a 5 Dimensional population balance model. Currently I have a very good MPICH2 parallelized code in FORTRAN but as we increase parameter values the arrays ...
16
votes
4answers
4k views

Should I rent computing resources, or buy my own computers

Since this question is related to computation, I decided to post here. Hopefully it will be seen as appropriate. I've just started running atmospheric and oceanic models, and I realize that I need ...
16
votes
6answers
2k views

Example of a continuous function that is difficult to approximate with polynomials

For teaching purposes I'd need a continuous function of a single variable that is "difficult" to approximate with polynomials, i.e. one would need very high powers in a power series to "fit" this ...
16
votes
4answers
6k views

uniform vs. non-uniform grid

It is probably a student level question but I can't exactly make it cleat to myself. Why is it more accurate to use non-uniform grids in the numerical methods? I am thinking in the context of some ...
16
votes
4answers
6k views

Row major versus Column major layout of matrices

In programming dense matrix computations, is there any reason to choose a row-major layout of the over the column-major layout? I know that depending on the layout of the matrix chosen, we need to ...
16
votes
7answers
1k views

Does Computational Science involve programming?

I read about computational science on Wikipedia, but my understanding is not very clear. Does computational science involve programming? How different is computational science from computational ...
16
votes
4answers
3k views

Why can't Householder reflections diagonalize a matrix?

When computing the QR factorization in practice, one uses Householder reflections to zero out the lower portion of a matrix. I know that for computing eigenvalues of symmetric matrices, the best you ...
16
votes
2answers
2k views

(how to) write simulations that run faster?

I have started using python as the programming language for doing all my assignments in CFD. I have a very little experience in programming. I am from mechanical engineering background and am pursuing ...
16
votes
1answer
3k views

How can I avoid catastrophic cancellation?

I have the following formula that I need to rewrite in order to avoid catastrophic cancellation. $$y =\sqrt{\frac{1}{2}\left(1-\sqrt{1-x^{2}}\right)}$$ As $x$ becomes smaller, $\sqrt{1-x^{2}}$ ...
16
votes
3answers
5k views

Finding which triangles points are in

Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
16
votes
4answers
8k views

Profiling CFD code with Callgrind

I'm using Valgrind + Callgrind to profile a solver I have written. As the Valgrind user manual states, I've compiled my code with the debugging options for the compiler: "Without debugging info, ...
16
votes
2answers
820 views

Preconditioning a Krylov method with another Krylov method

In methods like gmres or bicgstab it could be attractive to use another Krylov method as a preconditioner. After all they are easy to implement in a matrix-free way and in a parallel environment. For ...

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