What are the fast solvers for FEM equations? I would prefer open source implementation, but if there is a commercial implementation, then I won't mind paying for it.
Code Aster is an open source FE code. code aster
The pre- and post-processing is usually done with Salome - both originate from EDF.
How about FEAP. It has full source code available when you purchase it. It is pretty big project, maybe its too much for your needs, but check it out.
FEAP is a general purpose finite
element analysis program which is
designed for research and educational
use. Source code of the full program
is available for compilation using
Windows (Compaq or Intel compiler),
LINUX or UNIX operating systems, and
Mac OS X based Apple systems.
It has also a Personal Edition called FEAPpv available for free, including source code. Differences between those versions are listed in this pdf.
"brad"? do you mean "broad"?
you don't say if your problem is linear or non-linear. that'll make a very big difference.
the solver depends on the type of equation and the size of your problem. for elliptical pdes you can choose standard linear algebra techniques like lu decomposition, iterative methods like successive over relaxation, or wavefront solvers that minimize memory consumption.
some people like solving non-linear steady-state problems as if they were dynamics problems. the idea is to create "fake" mass and damping matricies and use explicit time integration to converge to steady state.
lots of choices. standard linear algebra is a good starting point.
language? java?
Oops, that's kind of a brad question.
Solving differential equations usually starts with analyzing equation itself. Some equations are notoriously difficult to solve efficiently, e.g. indifinite boundary problems.
So if you have something else than an elliptic problem, you'll might better prepare for hard times ahead.
Next important and crutial part is transfering the contiouus problem into a discrete mesh. Typically the accuracy of your results will vary with different ways to generate this mesh. You'll need some sound experience here.
So I'd say there is nothing like the fast slover for FEM equations. Anyway, while Wikipedia gives a short overview of the topic, you might perhaps also have a look a the german Wikipedia page. It lists well-known FEM implementations.
OpenFoam and Elmer are two open source solvers. Not sure about Elmer, but I think OpenFoam might uses the control volume approach.
I used OpenFOAM for fluid dynamics research. You can do parallel processing with it with MPI. And if you have a Cray T3E it will be fast!
It's open source :D
http://www.opencfd.co.uk/openfoam/features.html#features
Please have look for Deal.II open source library:
http://www.dealii.org/
They provide also VirtualBox image which comes pre-installed libs.
Related
Here is a good linear solver named GotoBLAS. It is available for download and runs on most computing platforms. My question is, is there an easy way to link this solver with the Mathematica kernel, so that we can call it like LinearSolve? One thing most of you may agree on for sure is that if we have a very large Linear system then we better get it solved by some industry standard Linear solver. The inbuilt solver is not meant for really large problems.
Now that Mathematica 8 has come up with better compilation and library link capabilities we can expect to use some of those solvers from within Mathematica. The question is does that require little tuning of the source code, or you need to be an advanced wizard to do it. Here in this forum we may start linking some excellent open source programs like GotoBLAS with Mathematica and exchange our views. Less experienced people can get some insight from the pro users and at the end we get a much stronger Mathematica. It will be an open project for the ever increasing Mathematica community and a platform where these newly introduced capabilities of Mathematica 8 could be transparently documented for future users.
I hope some of you here will give solid ideas on how we can get GotoBLAS running from within Mathematica. As the newer compilation and library link capabilities are usually not very well documented, they are not used by the common users very often. This question can act as a toy example to document these new capabilities of Mathematica. Help in this direction by the experienced forum members will really lift the motivation of new users like me as well as it will teach us a very useful thing to extend Mathematica's number crunching arsenal.
The short answer, I think, is that this is not something you really want to do.
GotoBLAS, as I understand it, is a specific implementation of BLAS, which stands for Basic Linear Algebra Subroutines. "Basic" really means quite basic here - multiply a matrix times a vector, for example. Thus, BLAS is not a solver that a function like LinearSolve would call. LinearSolve would (depending on the exact form of the arguments) call a LAPACK command, which is a higher level package built on top of BLAS. Thus, to really link GotoBLAS (or any BLAS) into Mathematica, one would really need to recompile the whole kernel.
Of course, one could write a C/Fortran program that was compiled against GotoBLAS and then link that into Mathematica. The resulting program would only use GotoBLAS when running whatever specific commands you've linked into Mathematica, however, which rather misses the whole point of BLAS.
The Wolfram Kernel (Mathematica) is already linked to the highly-optimized Intel Math Kernel Library, and is distributed with Mathematica. The MKL is multithreaded and vectorized, so I'm not sure what GotoBLAS would improve upon.
I am planning to write a bunch of programs on computationally intensive algorithms. The programs would serve as an indicator of different compiler/hardware performance.
I would want to pick up some common set of algorithms which are used in different fields, like Bioinformatics, Gaming, Image Processing, et al. The reason I want to do this would be to learn the algorithms and have a personal mini benchmark suit that would be small | useful | easy to maintain.
Any advice on algorithm selection would be tremendously helpful.
Benchmarks are not worthy of your attention!
The very best guide is to processor performance is: http://www.agner.org/optimize/
And then someone will chuck it in a box with 3GB of RAM defeating dual-channel hopes and your beautifully tuned benchmark will give widely different results again.
If you have a piece of performance critical code and you are sure you've picked the winning algorithm you can't then go use a generic benchmark to determine the best compiler. You have to actually compile your specific piece of code with each compiler and benchmark them with that. And the results you gain, whilst useful for you, will not extrapolate to others.
Case in point: people who make compression software - like zip and 7zip and the high-end stuff like PPMs and context-mixing and things - are very careful about performance and benchmark their programs. They hang out on www.encode.ru
And the situation is this: for engineers developing the same basic algorithm - say LZ or entropy coding like arithmetic-coding and huffman - the engineers all find very different compilers are better.
That is to say, two engineers solving the same problem with the same high-level algorithm will each benchmark their implementation and have results recommending different compilers...
(I have seen the same thing repeat itself repeatedly in competition programming e.g. Al Zimmermann's Programming Contests which is an equally performance-attentive community.)
(The newer GCC 4.x series is very good all round, but that's just my data-point, others still favour ICC)
(Platform benchmarks for IO-related tasks is altogether another thing; people don't appreciate how differently Linux, Windows and FreeBSD (and the rest) perform when under stress. And benchmarks there - on the same workload, same machine, different machines or different core counts - would be very generally informative. There aren't enough benchmarks like that about sadly.)
There was some work done at Berkeley about this a few years ago. The identified 13 common application paterns for parallel programming, the "13 Dwarves". The include things like linear algebra, n-body models, FFTs etc
http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-183.html
See page 10 onwards.
There are some sample .NET implementations here:
http://paralleldwarfs.codeplex.com/
The typical one is Fast Fourier Transform, perhaps you can also do something like the Lucas–Lehmer primality test.
I remember a guy who tested computational performance of machines, compiler versions by inverting Hilbert matrices.
For image processing, median filtering (used for removing noise, bad pixels) is always too slow. It might make a good test, given a large enough image say 1000x1000.
We know there are like a thousand of classifiers, recently I was told that, some people say adaboost is like the out of the shell one.
Are There better algorithms (with
that voting idea)
What is the state of the art in
the classifiers.Do you have an example?
First, adaboost is a meta-algorithm which is used in conjunction with (on top of) your favorite classifier. Second, classifiers which work well in one problem domain often don't work well in another. See the No Free Lunch wikipedia page. So, there is not going to be AN answer to your question. Still, it might be interesting to know what people are using in practice.
Weka and Mahout aren't algorithms... they're machine learning libraries. They include implementations of a wide range of algorithms. So, your best bet is to pick a library and try a few different algorithms to see which one works best for your particular problem (where "works best" is going to be a function of training cost, classification cost, and classification accuracy).
If it were me, I'd start with naive Bayes, k-nearest neighbors, and support vector machines. They represent well-established, well-understood methods with very different tradeoffs. Naive Bayes is cheap, but not especially accurate. K-NN is cheap during training but (can be) expensive during classification, and while it's usually very accurate it can be susceptible to overtraining. SVMs are expensive to train and have lots of meta-parameters to tweak, but they are cheap to apply and generally at least as accurate as k-NN.
If you tell us more about the problem you're trying to solve, we may be able to give more focused advice. But if you're just looking for the One True Algorithm, there isn't one -- the No Free Lunch theorem guarantees that.
Apache Mahout (open source, java) seems to pick up a lot of steam.
Weka is a very popular and stable Machine Learning library. It has been around for quite a while and written in Java.
Hastie et al. (2013, The Elements of Statistical Learning) conclude that the Gradient Boosting Machine is the best "off-the-shelf" Method. Independent of the Problem you have.
Definition (see page 352):
An “off-the-shelf” method is one that
can be directly applied to the data without requiring a great deal of timeconsuming data preprocessing or careful tuning of the learning procedure.
And a bit older meaning:
In fact, Breiman (NIPS Workshop, 1996) referred to AdaBoost with trees as the “best off-the-shelf classifier in the world” (see also Breiman (1998)).
I remember solving a lot of indefinite integration problems. There are certain standard methods of solving them, but nevertheless there are problems which take a combination of approaches to arrive at a solution.
But how can we achieve the solution programatically.
For instance look at the online integrator app of Mathematica. So how do we approach to write such a program which accepts a function as an argument and returns the indefinite integral of the function.
PS. The input function can be assumed to be continuous(i.e. is not for instance sin(x)/x).
You have Risch's algorithm which is subtly undecidable (since you must decide whether two expressions are equal, akin to the ubiquitous halting problem), and really long to implement.
If you're into complicated stuff, solving an ordinary differential equation is actually not harder (and computing an indefinite integral is equivalent to solving y' = f(x)). There exists a Galois differential theory which mimics Galois theory for polynomial equations (but with Lie groups of symmetries of solutions instead of finite groups of permutations of roots). Risch's algorithm is based on it.
The algorithm you are looking for is Risch' Algorithm:
http://en.wikipedia.org/wiki/Risch_algorithm
I believe it is a bit tricky to use. This book:
http://www.amazon.com/Algorithms-Computer-Algebra-Keith-Geddes/dp/0792392590
has description of it. A 100 page description.
You keep a set of basic forms you know the integrals of (polynomials, elementary trigonometric functions, etc.) and you use them on the form of the input. This is doable if you don't need much generality: it's very easy to write a program that integrates polynomials, for example.
If you want to do it in the most general case possible, you'll have to do much of the work that computer algebra systems do. It is a lifetime's work for some people, e.g. if you look at Risch's "algorithm" posted in other answers, or symbolic integration, you can see that there are entire multi-volume books ("Manuel Bronstein, Symbolic Integration Volume I: Springer") that have been written on the topic, and very few existing computer algebra systems implement it in maximum generality.
If you really want to code it yourself, you can look at the source code of Sage or the several projects listed among its components. Of course, it's easier to use one of these programs, or, if you're writing something bigger, use one of these as libraries.
These expert systems usually have a huge collection of techniques and simply try one after another.
I'm not sure about WolframMath, but in Maple there's a command that enables displaying all intermediate steps. If you do so, you get as output all the tried techniques.
Edit:
Transforming the input should not be the really tricky part - you need to write a parser and a lexer, that transforms the textual input into an internal representation.
Good luck. Mathematica is very complex piece of software, and symbolic manipulation is something that it does the best. If you are interested in the topic take a look at these books:
http://www.amazon.com/Computer-Algebra-Symbolic-Computation-Elementary/dp/1568811586/ref=sr_1_3?ie=UTF8&s=books&qid=1279039619&sr=8-3-spell
Also, going to the source wouldn't hurt either. These book actually explains the inner workings of mathematica
http://www.amazon.com/Mathematica-Book-Fourth-Stephen-Wolfram/dp/0521643147/ref=sr_1_7?ie=UTF8&s=books&qid=1279039687&sr=1-7
I am looking for an efficient eigensolver ( language not important, although I would be programming in C#), that utilizes the multi-core features found in modern CPU. Being able to work directly with pardiso solver is a major plus. My matrix are mostly sparse matrix, so an ideal solver should be able to take advantage of this fact and greatly enhance the memory usage and performance.
So far I have only found LAPACK and ARPACK. The LAPACK, as implemented in Intel MKL, is a good candidate, as it offers multi-core optimization. But it seems that the drivers inside the LAPACK don't work directly with pardiso solver, furthermore, it seems that they don't take advantage of sparse matrix ( but I am not sure on this point).
ARPACK, on the other hand, seems to be pretty hard to setup in Windows environment, and the parallel version, PARPACK, doesn't work so well. The bonus point is that it can work with pardiso solver.
The best would be Intel MKL + ARPACK with multi-core speedup. Not sure whether there is any existing implementations that already do what I want to do?
I'm working on a problem with needs very similar to the ones you state. I'm considering FEAST:
http://www.ecs.umass.edu/~polizzi/feast/index.htm
I'm trying to make it work right now, but it seems perfect. I'm interested in hearing what your experience with it is, if you use it.
cheers
Ned
Have a look at the Eigen2 library.
I've implemented it already, in C#.
The idea is that one must convert the matrix format in CSR format. Then, one can use MKL to compute linear equation solving algorithm ( using pardiso solver), the matrix-vector manipulation.