I have to solve the following nonlinear coupled partial differential system (link). All the functions are better explained here.
Even if I use the finite difference method (i.e. central difference in space and forward Crank-Nicholson in time in order to obtain an implicit scheme) I don't know how to linearize the problem (I have to solve a nonlinear system: I tried using Newton method, but sometimes I get a singular Jacobian; moreover, the function tau(lambda) is a non-convex energy, so I'm near a spinoidal point). Could you suggest some method?
Thanks in advance for all your help.
Petrus
Related
The N2 diagram for my full problem is below.
The N2 diagram for the coupled portion of the problem is below.
I have a DirectSolver handling the coupling between LLTForces and ImplicitLiftingLine, and an LNBGS solver handling the coupling between LiftingLineGroup and TestCL.
The gist for the problem is here: https://gist.github.com/eufren/31c0e569ed703b2aea3e2ef5360610f7
I have implemented guess_nonlinear() on ImplicitLiftingLine, which should use various outputs from LLTGeometry to give a good initial guess for the vortex strengths based on a linearised form of the governing equations.
def guess_nonlinear(self, inputs, outputs, resids):
freestream_unit_vector = inputs['freestream_unit_vector']
freestream_velocity = inputs['freestream_velocity']
n = inputs['normal_vectors']
A = inputs['surface_areas']
l = inputs['bound_vortices']
ic_tot = inputs['influence_coefficients_total']
v_inf = freestream_velocity
v_inf_vec = v_inf*freestream_unit_vector
lin_numerator = np.pi * v_inf * A * np.sum(n * v_inf_vec, axis=1)
lin_denominator = (np.linalg.norm(np.cross(v_inf_vec, l), axis=1) - np.pi * v_inf * A * np.sum(np.sum(n * ic_tot, axis=2), axis=1))
lin_vtx_str = lin_numerator / lin_denominator
outputs['vortex_strengths'] = lin_vtx_str
However, when the problem is run for the first time, any inputs not explicitly set with p.set_val() are all 1s. This causes guess_nonlinear() to give a bad output and so the system fails to converge:
As far as I can tell, the execution order for the LLT group is correct, and the geometry components should be being executed before the implicit component. I'm confused as to why this doesn't seem to actually be happening when the code is run, and instead these inputs are taking their default values.
What do I need to change to get this to work properly? Additionally, I've found difficulty in getting LNBGS to converge (hence adding guess_nonlinear()) during optimisation - only DirectSolver gets all the way through the optimisation without issues, but it's very slow for large numbers of LLT nodes). How can I improve the linear and nonlinear solver selection, and improve the reliability of the iterative solver?
Note: Thanks for providing a testable example. It made figuring out the answer to your question a lot simpler. Your problem was a bit subtle and I would not have been able to give a good answer without runnable code
Your first question: "Why are all the inputs 1"
"Short" Answer
You have put the nonlinear solver to high in the model hierarchy, which then included a key precurser component that computed your input values. By moving the solver down to a lower level of the model, I was able to ensure that the precurser component (LTTGeometry) ran and had valid outputs before you got to the guess_nonlinear of implicit component.
Here is what you had (Notice the implicit solver included LTTGeometry even though the data cycle does not require that component:
I moved both the nonlinear solver and the linear solver down into the LTTCycle group, which then allows the LTTGeometry component to execute before getting to the nonlinear solver and guess_nonlinear step:
My fix is only partially correct, since there is a secondary cycle from the TestCL component that also needs a solver and does not have one. However, that cycle still does not involve the LTTGeometry group. So the fully correct fix is to restructure you model top run geometry first, and then put the LTTCycle and TestCL groups together so you can run a solver over just them. That was a bit more hacking than I wanted to do on your test problem, but you can see the general idea from the adjusted N2 above.
Long Answer
The guess_nonlinear sequence in OpenMDAO does NOT run the compute method of explicit components or of groups. It follows the execution hierarchy, and calls any guess_nonlinear that it finds. So that means that any explicit components you have in your model will NOT get executed, their outputs will not get updated with computed values, and those computed values will not get passed to the inputs of downstream components.
Things get a little tricky when you have deep model hierarchies. The guess_nonlinear method is called as the first step in the nonlinear solver process. If you have a NonLinearRunOnce solver at the top level, it will follow the compute chain down the line calling compute or solve_nonlinear on each child and doing a data transfer after each one. If one of those children happens to be a group with a nonlinear solver, then that solver will call guess_nonlinear on its children (grandchildren of the top group with the NonLinearRunOnce solver) as the first step. So any outputs that were computed by the siblings of this group will be valid, but none of the outputs from the grandchild level will have been computed yet.
You may be wondering why not just have the guess_nonlinear method call the compute for any explicit components? There is a difficult to balance trade off here. If you assume that all explicit components are very cheap to run, then it might make sense to run the compute methods --- or it might not. A lot depends on the cyclic data structure. If some early component in the group needs guesses from the later one, then running its compute isn't going to help you much at all. Perhaps more importantly though, not all explicit components are cheap to run. You might have a very expensive computation, and calling compute as part of the guess process would be way too costly.
The compromise here, if you need some kind of top level guess process, is that you can implement guess_nonlinear at the group level. It's less common to do, but it gives you total control over what happens. You can call whatever you need to call in whatever sequence.
So the absolute key thing to remember is that the only data you have available to you when a guess_nonlinear is called is any data that was computed before your containing solver was executed. That means any thing that was computed before you got to the model scope of the containing solver (not the scope of the component with the guess_method itself).
Your second question: "How can I speed this up when the number of nodes gets large?"
This one not possible to give a generic answer to at all. I noticed that you have already specified sparse partial derivatives. That is a great start, but if its still not fast enough for you then it means you're reaching the limits of what you can do with a DirectSolver. You note that this solver is the only one that gets you through the optimization without issues, which I will take to mean that ScipyKryloventer link description here and PetscKrylov are not converging the linear system well for you --- at least not by themselves. Thats not surprising, as krylov solvers almost always require some kind of preconditioner... and this is why I can't offer a generic answer. Setting up efficient linear solvers for larger-scale compute is a tricky subject. If you look into the literature, you'll find some good suggestions. You can also study open source implementations like VSPAero for some tips.
effectively, you've reached the limit of what simple linear solvers can offer you. From this point forward, OpenMDAO can help a bit by making it easier to implement some preconditioning, but you'll have to suffer the math side yourself.
I don't understand this issue:
Issue: HIS Metriken - Cyclomatic (CR-MET4): [function_name] 13>10
It appears in Klocwork analysis while checking the issues of Code: METRICS.E.HIS_Metriken___Cyclomatic__CR_MET4_
Can anyone support?
Thanks
Do you see all those ifs, elses, loops in that function?
Those are the problem, you need to either design this function's logic more elegantly or split it into more functions with well-defined purposes.
By the way, I can only see that problematic function of yours because I am especially clairvoyant. For this kind of question you should normally show your code, just to be fair towards all those other users which cannot read your mind like I did.
Naaa, not really. The cyclomatic complexity is a measure for number of potential paths through your function. And that you have crossed the treshold of 10 by 3 means your function must be full of control structs, which create many paths.
I'm using caffe to do the object detection with SSD model, and recently work I adjust the loss type of "MultiBoxLoss".
In the multibox_loss_layer.cpp file, its loss has SOFTMAX as default and LOGISTIC option, I add the hingeloss(SVM) option into caffe code, and do the training but the result is bad.
Now the boss want me to use SVM to classify the feature map by python sklearn.
And a question come across to me, in the multibox_loss_layer.cpp file, there can use the softmax, logistic and hingeloss to calculate the loss. On this step, its data is just "one-dimension", but the feature map is high-dimension, and I internet the article, it seem softmax can't classify high-dimension data.
Ex: if there have three class: cat, dog and rabbit, then it's one-dimension data just have three value to represent cat, dog and rabbit(one value for each class), but the high-dimension data, it have many value(like feature map) for each class, and on the high-dimension case, softmax seems have no work for this.
so I wonder what's the different between softmax, logistic and SVM. Can anybody help? thank you!
Never seen applying SVM loss function into NN. However softmax is a loss function which should be used in order to optimize solution multiclass classifiaction problem. Softmax "transform" NN outputs into probability of each class occurance. Logistic function usually optimize each neuron output as a logistic problem, so it's not force output to be only one class. You should use this function if you want to solve multi labeling problem.
SVM is not a function, is a different classifier. There is no sense in comparing softmax with SVM, because first one is a loss function second one is a classifier.
I need to implement an intelligent agent to play Abalone game, for this kind of game the best way to proceed seems a min-max strategy with alpha beta pruning.
I have already implemented a naive search algorithm that use min-max with pruning,
my problem is...
How to generate the nodes of the tree where perform the search?
I have no idea of the right way to do this, and how assign the weigh to each node.
For generating the tree nodes: You need to implement a method that returns a collection of all possible legal moves given the current board position and the player whose turn it is. All these moves will become children of the node representing the current board position. Repeat until memory is exhausted to generate the game tree ;) or rather until you reach a reasonable tree depth.
For alpha-beta search you also need an evaluation function which calculates the weight for each board position/node. You can do some research or think about such a function yourself, maybe considering the number of stones still on the board. However a bad evaluation function can seriously screw up your results, so take care and run a lot of tests.
If you have trouble coming up with a reasonable evaluation function, I recommend you take a look into Monte-Carlo techniques such as UCT.
http://en.wikipedia.org/wiki/Monte_Carlo_tree_search
These tackle the problem using a probabilistic approach and have some nice advantages over alpha-beta. Also they don't require an evaluation function so you could skip this step.
Good luck!
I have published two libraries for move generation in Abalone. You didn't mention the programming language used for your search implementation, but you can easily port the functions.
For C++, https://sourceforge.net/projects/abnet/
For Python, https://gitlab.com/peer.sommerlund/haliotis
For an evaluation function, distance between all your marbles, or distance to their gravity center (same thing), works nicely. Tino Werner used this with a twist for his program that won ICGA 2003.
For understanding distance when using hex coordinates, I can recommend Amit Patel's page: https://www.redblobgames.com/grids/hexagons/
I'll be writing the online Google test tomorrow as a fresher. Apparently, they definitely ask one problem on Dynamic Programming?
Does anyone know of a good resource for collection of DP problems in C along with solutions? I know what DP is & have used it on an occasion or twice. However I feel to crack a DP problem in test, prior practice of typical problems will make it easier to approach.
Any good resources or problem sets with solutions in C will be highly appreciated. Thanks.
Okay, so I really hope this doesn't count as "shameless self-promotion," since all of these links are to code snippets I've posted on my personal site. If this is inappropriate, please let me know and I can take them down.
Here are a few fun DP problems that are pretty much classics:
Minimum edit distance: Given two strings A and B, find the shortest number of edits (insertions, deletions, or substitutions) necessary to convert A into B. This is called the Levenshtein distance. (My solution)
Optimal sequence alignment: Given two strings A and B, find the minimum number of gaps that must be inserted into the sequence to align A and B. This is called the Needleman-Wunsch algorithm. (My solution)
Single-source shortest paths: Given a directed graph G and a single node s, find the lengths of the shortest paths from s to each other node in the graph, assuming edges can be positive or negative but that no cycles exist. This is the Bellman-Ford algorithm. (My solution)
All-pairs shortest paths: Given a directed graph G, find the minimum distances between all pairs of nodes. This is the Floyd-Warshall algorithm. (My solution)
Hopefully this is somewhat useful, and best of luck tomorrow!
The Topcoder website is amazing. Not all of the problems use DP, but many do. Free full access to all problems from past competitions, which are at 3 different difficulty levels, as well as after-match explanations of every single problem from the problem author. Not only that, but you can quickly dig up the source code solution submitted by any coder in the competition.
Haven't been back there for a while, but they allow at least C++, Java, C# and I believe several other languages now.
I suggest u,collect a book "An Introduction to Bioinformatics Algorithms".This has a chapter fully on DP.As #templatetypedef mentioned Minimum edit distance,Optimal sequence alignment it has other problem with them.Though there is no implementation in it.You have to do it on your own.But you will find pretty interesting reading them.
To practice you can take one of the available problems at SPOJ. To recognize DP ones easily you can check at Problems Classifier (keyword: dp).