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I have a huge set of coupled nonlinear integro-partial differential equations. After a long while trying to simplify the equations and solve them at least semi-analytically I have come to conclude there has been left no way for me but an efficient numerical method. Finite element seems most amenable as is based on Galerkin method which gives a weak form solution, so a great hope that it might finally solve the equations. But at the same time I am so new to this field to write the codes all from the scratch:
Is there any Python library already available that can efficiently do a Finite Element analysis?
Also I was interested if softwares like FEniCS/Dolphin might also solve integro-differential equations?
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I am looking for predefined functions that implement in Python the Gauss and the Jacobi methods for solving linear equations. Are there any or I must write them myself?
I don't think so. At least not in python cores predefined functions. There are some functions in NumPy to calculate the eigenvalue using LAPACK. But I doubt anyone uses Gaussian and Jacobian methods these days. They are not particularly general and there are some cases in which the algorithm breaks down.
On the other hand, there are various implementations on the internet that you can use if you want.
Here are some examples:
Link 1: Gausian Implementation
Link 2: Jacobian Implementation No.1
Link 3: Jacobian Implementation No.2
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I currently have a MIP model formulated in Gurobi's python API, but recently I've been looking into tools such as PuLP and OR-Tools that allow me to build a model and feed it to multiple different optimizers. One feature of Gurobi used extensively in my model is the ability to have constraints that use functions such as and, or, min, max, and abs. However it seems as if PuLP and OR-Tools do not support these. Are there any alternatives that do support these? Or would I have to reformulate my model if I want to use something like this?
For or-tools, we only provide the minimal API for the linear solver.
If your problem is more structured (scheduling, routing, CP-like constraints), you can have a look at the CP-SAT interface:
https://developers.google.com/optimization/
https://github.com/google/or-tools/blob/master/ortools/sat/doc/index.md
Python examples are here:
https://github.com/google/or-tools/tree/master/examples/python
You might also want to have a look at Pyomo. It supports a variety of modeling tools and can call different solvers.
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I am trying to perform test summarize using self organizing map (SOM) as the clustering model. Do we have any libraries for performing SOM in python.
There is one here, but in general SOM implementations are not part of the main machine learning libraries. There are two reasons
SOM's, although nice to look at, don't really perform well in real problems.
It is too easy to construct one by yourself.
I would suggest to make it yourself. It is very easy and a great way to introduce yourself to python. The main code of the SOM itself is about 3 lines (a loop and one update). The remaing of the code would be for loading the data and plotting them, but you won't avoid that part of the code by using an external library
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I am trying to implement the newton method for maximization in higher dimensions and I was wondering if there exists any solvers for this in Python? In Scipy there is a solver for the 1-dimensional case, but I do not see one for the multi-dimensional case. I suppose that it is possible to implement it using the Hessian and Gradient solvers in Numdifftools
EDIT:
It looks like scipy.optimize.minimize does this. I was looking under the multi-dimensional heading and it wasn't there, that's why I missed it. It was under the general-purpose heading
scipy.optimize.minimize does this.
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Is there any library available to have inverse of a function? To be more specific, given a function y=f(x) and domain, is there any library which can output x=f(y)? Sadly I cannot use matlab/mathematics in my application, looking for C/Python library..
I am a bit late, but for future readers of the post, I just published a python package that does this precisely. https://pypi.python.org/pypi/pynverse
There is a detailed description of how to use it and how it does it in the description!
As has already been mentioned, not all functions are invertible. In some cases imposing additional constraints helps: think about the inverse of sin(x).
Once you are sure your function has a unique inverse, solve the equation f(x) = y. The solution gives you the inverse, y(x).
In python, look for nonlinear solvers from scipy.optimize.