I have been trying to make a biologically accurate 2D spatial model of tissue layers, where different physiological processes happen. This includes mainly chemical reactions, diffusion and fluxes over boundaries.
I am making this model in COMSOL Multiphysics, a finite element software package that solves different physics like reaction-diffusion systems, although for my question this might not be really relevant.
In my geometry, I have really small regions between the cells of the tissue layers. These regions serve as openings where diffusion can take place between the cells (junctions). The quality of the mesh is not great here and if I want to improve the quality (mainly by introducing more elements and such), my simulation time increases drastically. The lesser quality mesh also causes convergence to take longer. I added a picture of the geometry to give an idea. I tried different meshes, all with different qualities of the elements and the number of elements ranging from 16000 to 50000.
My background in FEM is really limited and I wanted to know if I can tackle this problem in such a way that it
doesn't negatively affect the biology (keep the tissue domain sizes/problem etc as biologically accurate as possible),
doesn't increase the simulation time drastically,
give a better mesh quality.
So I really want to know what the best way to go is, since I have already thought of some things.
Can I go with the lesser quality mesh (which is not really bad, but not good either), so that I can keep the small regions for optimum biological accuracy and have a relatively small computation time (and hope I won't run into convergence errors).
But maybe there are possibilities that I am missing, for instance: is it possible to make the small domain bigger and then add some kind of factor to the diffusion rates. In other words, if I want to make the domain twice as large, do I factor the diffusion rate with half? Is that even accurate in chemical/physical laws :S.
Hopefully I made the problem a bit clear and thank you greatly in advance for the help.
Cheers,
Mesh of the tissue model
I know this thread was posted some months back but I am unsure if you found a solution.
In order to find the relationship between accuracy and computational time would be that you run a mesh analysis on your model and see how the mesh size directly affects the results you are expecting to obtain (pore pressure, fluid velocity, strain, etc.) This will allow you to determine the most appropriate mesh strategy for your specific problem.
Also, you might need to keep in mind that the diffusion rate of a material will depend on the pore size and the permeability (by means of Darcy's law) so depending on the assumptions you are making for the implementation of your constitutive law and your problem boundary conditions you might simplify/enlarge some of the smaller domains you have in your model so long they are within your previously made assumptions.
Related
I'm trying to determine the best DSP method for what I'm trying to accomplish, which is the following:
In real-time, detect the presence of a frequency from a set of different predefined frequencies (no more than 40 different frequencies all within a 1000Hz range). I need to be able to do this even when there are other frequencies (outside of this set or range) that are more dominant.
It is my understanding that FFT might not be the best method for this, because it tells you the most dominant frequency (magnitude) at any given time. This seems like it wouldn't work because if I'm trying to detect say a frequency at 1650Hz (which is present), but there's also a frequency at 500Hz which is stronger, then it's not going to tell me the current frequency is 1650Hz.
I've heard that maybe the Goertzel algorithm might be better for what I'm trying to do, which is to detect single frequencies or a set of frequencies in real-time, even within sounds that have more dominant frequencies than the ones trying to be detected .
Any guidance is greatly appreciated and please correct me if I'm wrong on these assumptions. Thanks!
In vague and somewhat inaccurate terms, the output of the FFT is the magnitude and phase of all[1] frequencies. That is, your statement, "[The FFT] tells you the most dominant frequency (magnitude) at any given time" is incorrect. The FFT is often used as a first step to determine the most dominant frequency, but that's not what it does. In fact, if you are interested in the most dominant frequency, you need to take extra steps over and beyond the FFT: you take the magnitude of all frequencies output by the FFT, and then find the maximum. The corresponding frequency is the dominant frequency.
For your application as I understand it, the FFT is the correct algorithm.
The Goertzel algorithm is closely related to the FFT. It allows for some optimization over the FFT if you are only interested in the magnitude and/or phase of a small subset of frequencies. It might be the right choice for your application depending on the number of frequencies in question, but only as an optimization -- other than performance, it won't solve any problems the FFT won't solve. Because there is more written about the FFT, I suggest you start there and use the Goertzel algorithm only if the FFT proves to not be fast enough and you can establish the Goertzel will be faster in your case.
[1] For practical purposes, what's most inaccurate about this statement is that the frequencies are grouped together in "bins". There's a limited resolution to the analysis which depends on a variety of factors.
I am leaving my other answer as-is because I think it stands on it's own.
Based on your comments and private email, the problem you are facing is most likely this: sounds, like speech, that are principally in one frequency range, have harmonics that stretch into higher frequency ranges. This problem is exacerbated by low quality microphones and electronics, but it is not caused by them and wouldn't go away even with perfect equipment. Once your signal is cluttered with noise in the same band, you can't really distinguish on from off in a simple and reliable way, because on could be caused by the noise. You could try to do some adaptive thresholding based on noise in other bands, and you'll probably get somewhere, but that's no way to build a robust system.
There are a number of ways to solve this problem, but they all involve modulating your signal and using error detection and correction. Basically, you are building a modem and/or radio. Ultimately, what I'm saying is this: you can't solve your problem on the detector alone. You need to build some redundancy into your signal, and you may need to think about other methods of detection. I know of three methods of sending complex signals:
Amplitude modulation, which is what it sounds like you are doing now.
Frequency modulation, which tends to be more robust in the face of ambient noise. (compare FM and AM radio)
Phase modulation, which is more subtle and tricky.
These methods can be combined and multiplexed in various ways. Read about them on wikipedia. Moreover, once your base signal is transmitted, you can add error correction and detection on top.
I am not an expert in this area, but off the top of my head, I am not sure you'll be able to use PM silently, and AM is simply too sensitive to noise, as you've discovered, although it might work with the right kind of redundancy. FM is probably your best bet.
I implemented a differential evolution algorithm for a side project I was doing. Because the crossover step seemed to involve a lot of parameter choices (e.g. crossover probabilities), I decided to skip it and just use mutation. The method seemed to work ok, but I am unsure whether I would get better performance if I introduced crossover.
Main Question: What is the motivation behind introducing crossover to differential evolution? Can you provide a toy example where introducing crossover out-performs pure mutation?
My intuition is that crossover will produce something like the following in 2-dimensions. Say
we have two parent vectors (red). Uniform crossover could produce a new trial vector at one of the blue points.
I am not sure why this kind of exploration would be expected to be beneficial. In fact, it seems like this could make performance worse if high-fitness solutions follow some linear trend. In the figure below, lets say the red points are the current population, and the optimal solution is towards the lower right corner. The population is traveling down a valley such that the upper right and lower left corners produce bad solutions. The upper left corner produces "okay" but suboptimal solutions. Notice how uniform crossover produces trials (in blue) that are orthogonal to the direction of improvement. I've used a cross-over probability of 1 and neglected mutation to illustrate my point (see code). I imagine this situation could arise quite frequently in optimization problems, but could be misunderstanding something.
Note: In the above example, I am implicitly assuming that the population was randomly initialized (uniformly) across this space, and has begun to converge to the correct solution down the central valley (top left to bottom right).
This toy example is convex, and thus differential evolution wouldn't even be the appropriate technique. However, if this motif was embedded in a multi-modal fitness landscape, it seems like crossover might be detrimental. While crossover does support exploration, which could be beneficial, I am not sure why one would choose to explore in this particular direction.
R code for the example above:
N = 50
x1 <- rnorm(N,mean=2,sd=0.5)
x2 <- -x1+4+rnorm(N,mean=0,sd=0.1)
plot(x1,x2,pch=21,col='red',bg='red',ylim=c(0,4),xlim=c(0,4))
x1_cx = list(rep(0, 50))
x2_cx = list(rep(0, 50))
for (i in 0:N) {
x1_cx[i] <- x1[i]
x2_cx[i] <- x2[sample(1:N,1)]
}
points(x1_cx,x2_cx,pch=4,col='blue',lwd=4)
Follow-up Question: If crossover is beneficial in certain situations, is there a sensible approach to a) determining if your specific problem would benefit from crossover, and b) how to tune the crossover parameters to optimize the algorithm?
A related stackoverflow question (I am looking for something more specific, with a toy example for instance): what is the importance of crossing over in Differential Evolution Algorithm?
A similar question, but not specific to differential evolution: Efficiency of crossover in genetic algorithms
I am not particularly familiar with the specifics of the DE algorithm but in general the point of crossover is that if you have two very different individuals with high fitness it will produce an offspring that is intermediate between them without being particularly similar to either. Mutation only explores the local neighbourhood of each individual without taking the rest of the population into account. If you think of genomes as points in some high dimensional vector space, then a mutation is shift in a random direction. Therefore mutation needs to take small steps since if your are starting from a significantly better than random position, a long step in a random direction is almost certain to make things worse because it is essentially just introducing entropy into an evolved genome. You can think of a cross over as a step from one parent towards the other. Since the other parent is also better than random, it is more promising to take a longer step in that direction. This allows for faster exploration of the promising parts of the fitness landscape.
In real biological organisms the genome is often organized in such a way that genes that depend on each other are close together on the same chromosome. This means that crossover is unlikely to break synergetic gene combinations. Real evolution actually moves genes around to achieve this (though this is much slower than the evolution of individual genes) and sometimes the higher order structure of the genome (the 3 dimensional shape of the DNA) evolves to prevent cross-overs in particularly sensitive areas. These mechanisms are rarely modeled in evolutionary algorithms, but you will get more out of crossovers if you order your genome in a way that puts genes that are likely to interact close to each other.
No. Crossover is not useful. There I said it. :P
I've never found a need for crossover. People seem to think it does some kind of magic. But it doesn't (and can't) do anything more useful than simple mutation. Large mutations can be used to explore the entire problem space and small mutations can be used to exploit niches.
And all the explanations I've read are (to put it mildly) unsatisfactory. Crossover only complicates your algorithms. Drop it asap. Your life will be simpler. .... IMHO.
As Daniel says, cross over is a way to take larger steps across the problem landscape, allowing you to escape local maxima that a single mutation would be unable to do so.
Whether it is appropriate or not will depend on the complexity of the problem space, how the genotype -> phenotype expression works (will related genes be close together), etc.
More formally this is the concept of 'Connectivity' in Local Search algorithms, providing strong enough operators that the local search neighbourhood is sufficentally large to escape local minima.
http://msl.cs.uiuc.edu/rrt/
Can anyone explain how rrt works with simple wording that is easy to understand?
I read the description in the site and in wikipedia.
What I would like to see, is a short implementation of a rrt or a thorough explanation of the following thing:
Why does the rrt grow outwards instead of just growing very dense around the center?
How is it different from a naive random tree?
How is the next new vertex that we attempt to reach picked?
I know there is an Motion Strategy Library I could download but I would much rather understand the idea before I delve into the code rather than the other way around.
The simplest possible RRT algorithm has been so successful because it is pretty easy to implement. Things tend to get complicated when you:
need to visualise planning concepts in more than two dimensions
are unfamiliar with the terminology associated with planning, and;
in the huge number of variants of RRT that are have been described in the literature.
Pseudo code
The basic algorithm looks something like this:
Start with an empty search tree
Add your initial location (configuration) to the search tree
while your search tree has not reached the goal (and you haven't run out of time)
3.1. Pick a location (configuration), q_r, (with some sampling strategy)
3.2. Find the vertex in the search tree closest to that random point, q_n
3.3. Try to add an edge (path) in the tree between q_n and q_r, if you can link them without a collision occurring.
Although that description is adequate, after a while working in this space, I really do prefer the pseudocode of figure 5.16 on RRT/RDT in Steven LaValle's book "Planning Algorithms".
Tree Structure
The reason that the tree ends up covering the entire search space (in most cases) is because of the combination of the sampling strategy, and always looking to connect from the nearest point in the tree. This effect is described as reducing the Voronoi bias.
Sampling Strategy
The choice of where to place the next vertex that you will attempt to connect to is the sampling problem. In simple cases, where search is low dimensional, uniform random placement (or uniform random placement biased toward the goal) works adequately. In high dimensional problems, or when motions are very complex (when joints have positions, velocities and accelerations), or configuration is difficult to control, sampling strategies for RRTs are still an open research area.
Libraries
The MSL library is a good starting point if you're really stuck on implementation, but it hasn't been actively maintained since 2003. A more up-to-date library is the Open Motion Planning Library (OMPL). You'll also need a good collision detection library.
Planning Terminology & Advice
From a terminology point of view, the hard bit is to realise that although lots of the diagrams you see in the (early years of) publications on RRT are in two dimensions (trees that link 2d points), that this is the absolute simplest case.
Typically, a mathematically rigorous way to describe complex physical situations is required. A good example of this is planning for a robot arm with n- linkages. Describing the end of such an arm requires a minimum of n joint angles. This set of minimum parameters to describe a position is a configuration (or some publications state). A single configuration is often denoted q
The combination of all possible configurations (or a subset thereof) that can be achieved make up a configuration space (or state space). This can be as simple as an unbounded 2d plane for a point in the plane, or incredibly complex combinations of ranges of other parameters.
I'm trying to design a nonlinear fitness function where I maximize variable A and minimize the variable B. The issue is that maximizing A is much more important at single digit values, almost logarithmic. B needs to be minimized and in contrast to A, it becomes less important when small (less than one) and more important when it's larger (>1), so exponential decay.
The main goal is to optimize A, so I guess an analog is A=profits, B=costs
Should I aim to keep everything positive so that the I can use a roulette wheel selection, or would it be better to use a rank/torunament kind of system? The purpose of my algorithm is shape optimization.
Thanks
When considering a multi-objective problem the goal is usually to identify all solutions that lie on the Pareto curve - the Pareto optimal set. Have a look here for a 2-dimensional visual example. When the algorithm completes you want a set of solutions that are not dominated by any other solution. You therefore need to define a pareto ranking mechanism to take into account both objectives - for a more in depth explanation, as well as links to even more reading, go here
With this in mind, in order to effectively explore all solutions along the pareto front you do not want an implementation that encourages premature convergence, otherwise your algorithm will only explore the search space in one specific area of the Pareto curve. I would implement a selection operator that keeps all members of each iteration's optimal set of solutions, that is all solutions which are not dominated by another + plus a parameter controlled percentage of other solutions. This way you encourage exploration all along the Pareto curve.
You also need to ensure your mutation and crossover operators are tuned correctly too. With any novel application of Evolutionary Algorithms, part of the problem is trying to identify an optimal parameter set for the problem domain... this is where it gets really interesting!!
The description is very vague, but assuming that you actually have an idea of what the function should look like and you're just wondering whether you need to modify it so that proportional selection can be used easily, then no. Regardless of fitness function, you should probably default to using something like tournament selection. Controlling selection pressure is one of the most important things you have to do in order to get consistently good results, and roulette wheel selection doesn't allow you that control. You typically get enormous pressure very early, which drives premature convergence. That might be preferable in a few cases, but it's not where I'd start my investigations.
I am designing a RPG game like final fantasy.
I have the programming part done but what I lack is the maths. I am ok at maths but I am having trouble incorporating the players stas into mu sums.
How can I make an action timer that is based on the players speed?
How can I use attack and defence so that it is not always exactly the same damage?
How can I add randomness into the equations?
Can anyone point me to some resources that I can read to learn this sort of stuff.
EDIT: Clarification Of what I am looking for
for the damage I have (player attack x move strength) / enemy defence.
This works and scales well but i got a look at the algorithms from final fantasy 4 a while a got and this sum alone was over 15 steps. mine has only 2.
I am looking for real game examples if possible but would settle for papers or books that have sections that explain how they get these complex sums and why they don't use simple ones.
I eventually intent to implement but am looking for more academic knowledge at the moment.
Not knowing Final fantasy at all, here are some thoughts.
Attack/Defence could either be a 'chance to hit/block' or 'damage done/mitigated' (or, possibly, a blend of both). If you decide to go for 'damage done/mitigated', you'll probably want to do one of:
Generate a random number in a suitable range, added/subtracted from the base attack/defence value.
Generate a number in the range 0-1, multiplied by the attack/defence
Generate a number (with a Gaussian or Poisson distribution and a suitable standard deviation) in the range 0-2 (or so, to account for the occasional crit), multiplied by the attack/defence
For attack timers, decide what "double speed" and "triple speed" should do for the number of attacks in a given time. That should give you a decent lead for how to implement it. I can, off-hand, think of three methods.
Use N/speed as a base for the timer (that means double/triple speed gives 2/3 times the number of attacks in a given interval).
Use Basetime - Speed as the timer (requires a cap on speed, may not be an issue, most probably has an unintuitive relation between speed stat and timer, not much difference at low levels, a lot of difference at high levels).
Use Basetime - Sqrt(Speed) as the timer.
I doubt you'll find academic work on this. Determining formulae for damage, say, is heuristic. People just make stuff up based on their experience with various functions and then tweak the result based on gameplay.
It's important to have a good feel for what the function looks like when plotted on a graph. The best advice I can give for this is to study a course on sketching graphs of functions. A Google search on "sketching functions" will get you started.
Take a look at printed role playing games like Dungeons & Dragons and how they handle these issues. They are the inspiration for computer RPGs. I don't know of academic work
Some thoughts: you don't have to have an actual "formula". It can be rules like "roll a 20 sided die, weapon does 2 points of damage if the roll is <12 and 3 points of damage if the roll is >=12".
You might want to simplify continuous variables down to small ranges of integers for testing. That way you can calculate out tables with all the possible permutations and see if the results look reasonable. Once you have something good, you can interpolate the formulas for continuous inputs.
Another key issue is play balance. There aren't necessarily formulas for telling you whether your game mechanics are balanced, you have to test.