This sounds to be something well explored, so I just need a reference.
I need to take a look (i.e. sketch the plot) to the solution of certain complex equations with several complex parameters and I would like to do that using Python/NumPy.
I have a certain number of complex functions of one complex variable depending on two parameters, and I need to add and compose them and finally to study the set of complex numbers such that the modulus of the above sum and composition is equal to a fixed positive real number.
For short: I have to plot the graph of the solutions of a certain complex equation, and see how does it move when parameters are touched.
Can somebody give a reference, please?
Thanks
Related
Lets assume we have a two dimensional function
def f(x,y): #some calculations here
return value
we know from observations the following
f(x,0.9y) = 10
f(x,0.8y)=15
f(x,0.8y) = 23
...
how can I find the right values for x,y to get the best fit to the observations? which procedure is recommended in case of such an optimization problem?
It seems you have a typo in your question. Anyhow, given that we only have two data points, optimization is not exactly doable, as a set of simultaneous equations gives you an exact result, for many different types of functions.
If you meant to give three data points, there is different approach:
Note that the x is constant. This means that whatever the function is, we cannot say anything about the x portion of the 2D function. So it's really a one dimensional problem: How does y behave?
Given that we know nothing about the data in question, and given the values you gave, I would by gut instinct probably go with an exponential/logarithmic function (assuming the final value is meant to read 0.7). It might also be a linear function, though once you fit a function that you produce, you would have to calculate the error (here is how you would do this). Beyond this, in my experience, there is not much that you can do.
I'm working on a panel method code at the moment. To keep us from being bogged down in the minutia, I won't show the code - this is a question about overall program structure.
Currently, I solve my system by:
Generating the corresponding rows of the A matrix and b vector in an explicit component for each boundary condition
Assembling the partial outputs into the full A, b.
Solving the linear system, Ax=b, using a LinearSystemComp.
Here's a (crude) diagram:
I would prefer to be able to do this by just writing one implicit component to represent each boundary condition, vectorising the inputs/outputs to represent multiple rows/cols in the matrix, then allowing openMDAO to solve for the x while driving the residual for each boundary condition to 0.
I've run into trouble trying to make this work, as each implicit component is underdetermined (more rows in the output vector x than the component output residuals; that is, A1.x - b1= R1, length(R1) < length(x). Essentially, I would like openMDAO to take each of these underdetermined implicit systems, and find the value of x that solves the determined full system - without needing to do all of the assembling stuff myself.
Something like this:
To try and make my goal clearer, I'll explain what I actually want from the perspective of my panel method. I'd like a component, let's say Influence, that computes the potential induced by a given panel at a given point in the panel's reference frame. I'd like to vectorise the input panels and points such that it can compute the influence coefficent of many panels on one point, of many points on one panel, or of many points on many panels.
I'd then like a system of implicit boundary conditions to find the correct value of mu to solve the system. These boundary conditions, again, should be able to be vectorised to compute the violation of the boundary condition at many points under the influence of many panels.
I get confused again at this part. Not every boundary condition will use the influence coefficient values - some, like the Kutta condition, are just enforced on the mu vector, e.g .
How would I implement this as an implicit component? It has no inputs, and doesn't output the full mu vector.
I appreciate that the question is rather long and rambling, but I'm pretty confused. To summarise:
How can I use openMDAO to solve multiple individually underdetermined (but combined, fully determined) implicit systems?
How can I use openMDAO to write an implicit component that takes no inputs and only uses a portion of the overall solution vector?
In the OpenMDAO docs there is a close analog to what you are trying to accomplish, with the node-voltage analysis tutorial. In that code, the balance comp is used to create an implicit relationship that is similar to what you're describing. Its singular on its own, but part of a larger group is a well defined system.
You'll need to find a way to build similar components for your model. Each "row" in your equation will be associated with one state variables (one entry in your x vector).
In the simplest case, each row (or set of rows) would have one input which is the associated row of the A matrix, and a second input which is ALL of the other values for x, and a final input which is the entry of the b vector (right hand side vector). Then you could evaluate the residual for that specific row, which would be the following
R['x_i'] = np.sum(A*x_full) - b
where x_full is the assembly of the full x-vector from the x_other input and the x_i state variable.
#########
Having proposed the above solution, I have to say that I don't think this is a particularly efficient way to build or solve this linear system. It is modular, and might give you some flexibility, but you're jumping through a lot of hoops to avoid doing some index-math, and shoving everything into a matrix.
Granted, the derivatives might be a bit easier in your design, because the matrix assembly is going to get handled "magically" by the connections you have to create between the various row-components. So maybe its worth the trade... but i would say you might be better of trying a more traditional coding approach and using JAX or some other AD code to make the derivatives easier.
I have been using the SLSQP algorithm to run some MDO problems with ExplicitComponents only. Each component has a runtime of around 10 seconds and 60-100 input variables. Most of the input variables are static input variables that will remain constant during the entire optimization. The static input variables originate from an IndepVarComp. The ExplicitComponents are black boxes, so no information is available on the partials.
I noticed that when the Jacobian is calculated in the compute_totals(), the components are linearized with respect to all their input values. In the compute_approximations() a finite difference is calculated over all the input values, including the static input values. So, my question is: why is a finite difference calculation performed over these static input variables? As the values remain constant, I’m not sure why this information would be useful?
Furthermore, if I understand it correctly, the components are linearized to get the sub-Jacobians, which are then used to calculate the total Jacobian. However, is it possible to directly calculate a finite-difference over the entire group instead of linearizing each component? With the runtimes of my components and amount of input variables, it will take a long time to perform the linearization of each component. However, the optimization problem has only 3 design variables. So, if I could perform three finite difference calculations over the entire MDA to calculate the total Jacobian, the total runtime will decrease significantly.
To answer your questions in reverse order:
1) Can you FD over the entire model instead of each individual component? Yes!
You can set up FD over any group in your model, including the top-level group. Then the FD is taken across that group rather than across each component in it.
We call that computing a semi-total derivative, because in general you can select a sub-group in your model, in which case the FD is approximating a total-derivative across that group but that total-derivative is still effectively a partial-derivative for the overall model. hence semi-total derivative.
2) Why is a finite difference calculation performed over these static input variables?
In theory, you're correct that you don't really need partial derivatives of the inputs that can't change. As of OpenMDAO 2.4, we don't handle that situation automatically though, and we don't have plans to add that in the near future. However, the framework is only taking FD across the partials you tell it to. It sounds like you are declaring your partials like this:
self.declare_partials(of=['*'], wrt=['*'], method='fd')
So you're specifically asking the framework to compute all those partials. Instead, you could specify in the wrt argument only the inputs you know are actually changing. Of course, this is mathematically incorrect because there is a derivative wrt to the static-inputs. If someone later on connects something to those inputs and tries and optimization, they would get a wrong answer. But as long as your careful, you can specifically ask for only the partials you wanted from any component and simple leave the non-changing inputs as effectively 0.
I am modeling electrical current through various structures with the help of FiPy. To do so, I solve Laplace's equation for the electrical potential. Then, I use Ohm's law to derive the field and with the help of the conductivity, I obtain the current density.
FiPy stores the potential as a cell-centered variable and its gradient as a face-centered variable which makes sense to me. I have two questions concerning face-centered variables:
If I have a two- or three-dimensional problem, FiPy computes the gradient in all directions (ddx, ddy, ddz). The gradient is a FaceVariable which is always defined on the face between two cell centers. For a structured (quadrilateral) grid, only one of the derivates should be greater than zero since for any face, the position of the two cell-centers involved should only differ in one coordinate. In my simulations however, it occurs frequently that more than one of the derivates (ddx, ddy, ddz) is greater than zero, even for a structured grid.
The manual gives the following explanation for the FaceGrad-Method:
Return gradient(phi) as a rank-1 FaceVariable using differencing for the normal direction(second-order gradient).
I do not see, how this differs from my understanding pointed out above.
What makes it even more problematic: Whenever "too many" derivates are included, current does not seem to be conserved, even in the simplest structures I model...
Is there a clever way to access the data stored in the face-centered variable? Let's assume I would want to compute the electrical current going through my modeled structure.
As of right now, I save the data stored in the FaceVariable as a tsv-file. This yields a table with (x,y,z)-positions and (ddx, ddy, ddz)-values. I read the file and save the data into arrays to use it in Python. This seems counter-intuitive and really inconvenient. It would be a lot better to be able to access the FaceVariable along certain planes or at certain points.
The documentation does not make it clear, but .faceGrad includes tangential components which account for more than just the neighboring cell center values.
Please see this Jupyter notebook for explicit expressions for the different types of gradients that FiPy can calculate (yes, this stuff should go into the documentation: #560).
The value is accessible with myFaceVar.value and the coordinates with myFaceVar.mesh.faceCenters. FiPy is designed around unstructured meshes and so taking arbitrary slices is not trivial. CellVariable objects support interpolation by calling myCellVar((xs, ys, zs)), but FaceVariable objects do not. See this discussion.
After using scipy.integrate for a while I am at the point where I need more functions like bifurcation analysis or parameter estimation. This is why im interested in using the PyDSTool, but from the documentation I can't figure out how to work with ModelSpec and if this is actually what will lead me to the solution.
Here is a toy example of what I am trying to do: I have a network with two nodes, both having the same (SIR) dynamic, described by two ODEs, but different initial conditions. The equations are coupled between nodes via the Epsilon (see formula below).
formulas as a picture for better read, the 'n' and 'm' are indices, not exponents ~>
http://image.noelshack.com/fichiers/2014/28/1404918182-odes.png
(could not use the upload on stack, sadly)
In the two node case my code (using PyDSTool) looks like this:
#multiple SIR metapopulations
#parameter and initial condition definition; a dict is a must
import PyDSTool as pdt
params={'alpha': 0.7, 'beta':0.1, 'epsilon1':0.5,'epsilon2':0.5}
ini={'s1':0.99,'s2':1,'i1':0.01,'i2':0.00}
DSargs=pdt.args(name='SIRtest_multi',
ics=ini,
pars=params,
tdata=[0,20],
#the for-macro generates formulas for s1,s2 and i1,i2;
#sum works similar but sums over the expressions in it
varspecs={'s[o]':'for(o,1,2,-alpha*s[o]*sum(k,1,2,epsilon[k]*i[k]))',
'i[l]':'for(l,1,2,alpha*s[l]*sum(m,1,2,epsilon[m]*i[m]))'})
#generator
DS = pdt.Generator.Vode_ODEsystem(DSargs)
#computation, a trajectory object is generated
trj=DS.compute('test')
#extraction of the points for plotting
pts=trj.sample()
#plotting; pylab is imported along with PyDSTool as plt
pdt.plt.plot(pts['t'],pts['s1'],label='s1')
pdt.plt.plot(pts['t'],pts['i1'],label='i1')
pdt.plt.plot(pts['t'],pts['s2'],label='s2')
pdt.plt.plot(pts['t'],pts['i2'],label='i2')
pdt.plt.legend()
pdt.plt.xlabel('t')
pdt.plt.show()
But in my original problem, there are more than 1000 nodes and 5 ODEs for each, every node is coupled to a different number of other nodes and the epsilon values are not equal for all the nodes. So tinkering with this syntax did not led me anywhere near the solution yet.
What I am actually thinking of is a way to construct separate sub-models/solver(?) for every node, having its own parameters (epsilons, since they are different for every node). Then link them to each other. And this is the point where I do not know wether it is possible in PyDSTool and if it is the way to handle this kind of problems.
I looked through the examples and the Docs of PyDSTool but could not figure out how to do it, so help is very appreciated! If the way I'm trying to do things is unorthodox or plain stupid, you are welcome to make suggestions how to do it more efficiently. (Which is actually more efficient/fast/better way to solve problems like this: subdivide it into many small (still not decoupled) models/solvers or one containing all the ODEs at once?)
(Im neither a mathematician nor a programmer, but willing to learn, so please be patient!)
The solution is definitely not to build separate simulation models. That won't work because so many variables will be continuously coupled between the sub-models. You absolutely must have all the ODEs in one place together.
It sounds like the solution you need is to use the ModelSpec object constructs. These let you hierarchically build the sub-model definitions out of symbolic pieces. They can have their own "epsilon" parameters, etc. You declare all the pieces when you're finished and let PyDSTool make the final strings containing the ODE definitions for you. I suggest you look at the tutorial example at:
http://www.ni.gsu.edu/~rclewley/PyDSTool/Tutorial/Tutorial_compneuro.html
and the provided examples: ModelSpec_test.py, MultiCompartments.py. But, remember that you still have to have a source for the parameters and coupling data (i.e., a big matrix or dictionary loaded from a file) to be able to automate the process of building the model, otherwise you'd still be writing it all out by hand.
You have to build some classes for the components that you want to have. You might also create a factory function (compare 'makeSoma' in the neuralcomp.py toolbox) that will take all your sub-components and create an ODE based on summing something up from each of the declared components. At the end, you can refer to the parameters by their position in the hierarchy. One might be 's1.epsilon' while another might be 'i4.epsilon'.
Unfortunately, to build models like this efficiently you will have to learn to do some more complex programming! So start by understanding all the steps in the tutorial. You can email me directly through the SourceForge support discussions or email once you've got started and have specific questions.