I am currently doing a course called "Modeling of dynamic systems" and have been given the task of modeling a warm water tank in modelica with a distributed temperature description.
Most of the tasks have gone well, and my group is left with the task of introducing the heat flux due to buoyancy effects into the model. Here is where we get stuck.
the equation given is this:
Given PDE
But how do we discretize this into something we can use in modelica?
The discretized version we ended up with was this:
(Qd_pp_b[k+1] - Qd_pp_b[k]) / h_dz = -K_b *(T[k+1] - 2 * T[k] + T[k-1]) / h_dz^2
where Qd_pp_b is the left-hand side variable, ie the heat flux, k is the current slice of the tank and T is the temperature in the slices.
Are we on the right path? or completely wrong?
This doesn't seem to be a differential equation (as is) so this does not make sense without surrounding problem. For the second derivative you should always create auxiliary variables and for each partial derivative a separate equation. I added dummy values for parameters and dummy equations for T[k]. This can be simulated, is this about what you expected?
model test
constant Integer n = 10;
Real[n] Qd_pp_b;
Real[n] dT;
Real[n] T;
parameter Real K_b = 1;
equation
for k in 1:n loop
der(Qd_pp_b[k]) = -K_b *der(dT[k]);
der(T[k]) = dT[k];
T[k] = sin(time+k);
end for;
end test;
Related
I am trying to evaluate points in a large piecewise polynomial, which is obtained from a cubic-spline. This takes a long time to do and I would like to speed it up.
As such, I would like to evaluate a points on a piecewise polynomial with parallel processes, rather than sequentially.
Code:
z = zeros(1e6, 1) ; % preallocate some memory for speed
Y = rand(11220,161) ; %some data, rand for generating a working example
X = 0 : 0.0125 : 2 ; % vector of data sites
pp = spline(X, Y) ; % get the piecewise polynomial form of the cubic spline.
The resulting structure is large.
for t = 1 : 1e6 % big number
hcurrent = ppval(pp,t); %evaluate the piecewise polynomial at t
z(t) = sum(x(t:t+M-1).*hcurrent,1) ; % do some operation of the interpolated value. Most likely not relevant to this question.
end
Unfortunately, with matrix form and using:
hcurrent = flipud(ppval(pp, 1: 1e6 ))
requires too much memory to process, so cannot be done. Is there a way that I can batch process this code to speed it up?
For scalar second arguments, as in your example, you're dealing with two issues. First, there's a good amount of function call overhead and redundant computation (e.g., unmkpp(pp) is called every loop iteration). Second, ppval is written to be general so it's not fully vectorized and does a lot of things that aren't necessary in your case.
Below is vectorized code code that take advantage of some of the structure of your problem (e.g., t is an integer greater than 0), avoids function call overhead, move some calculations outside of your main for loop (at the cost of a bit of extra memory), and gets rid of a for loop inside of ppval:
n = 1e6;
z = zeros(n,1);
X = 0:0.0125:2;
Y = rand(11220,numel(X));
pp = spline(X,Y);
[b,c,l,k,dd] = unmkpp(pp);
T = 1:n;
idx = discretize(T,[-Inf b(2:l) Inf]); % Or: [~,idx] = histc(T,[-Inf b(2:l) Inf]);
x = bsxfun(#power,T-b(idx),(k-1:-1:0).').';
idx = dd*idx;
d = 1-dd:0;
for t = T
hcurrent = sum(bsxfun(#times,c(idx(t)+d,:),x(t,:)),2);
z(t) = ...;
end
The resultant code takes ~34% of the time of your example for n=1e6. Note that because of the vectorization, calculations are performed in a different order. This will result in slight differences between outputs from ppval and my optimized version due to the nature of floating point math. Any differences should be on the order of a few times eps(hcurrent). You can still try using parfor to further speed up the calculation (with four already running workers, my system took just 12% of your code's original time).
I consider the above a proof of concept. I may have over-optmized the code above if your example doesn't correspond well to your actual code and data. In that case, I suggest creating your own optimized version. You can start by looking at the code for ppval by typing edit ppval in your Command Window. You may be able to implement further optimizations by looking at the structure of your problem and what you ultimately want in your z vector.
Internally, ppval still uses histc, which has been deprecated. My code above uses discretize to perform the same task, as suggested by the documentation.
Use parfor command for parallel loops. see here, also precompute z vector as z(j) = x(j:j+M-1) and hcurrent in parfor for speed up.
The Spline Parameters estimation can be written in Matrix form.
Once you write it in Matrix form and solve it you can use the Model Matrix to evaluate the Spline on all data point using Matrix Multiplication which is probably the most tuned operation in MATLAB.
I am new to Modelica, and I am wondering if it is possible to write a kind of dynamic programming equation. Assume time is discretized by an integer i, and in my specific application x is boolean and f is a boolean function of x.
x(t_i) = f(x(t_{i+d}))
Where d can be a positive or negative integer. Of course, I would initialize x accordingly, either true or false.
Any help or references would be greatly appreciated!
It is possible. In Modelica the discretization in time is usually carried on by the compiler, you have to take care of the equations (continous dynamics). Otherwise, if you want to generate events at discrete time points, you can do it using when statements.
I suggest you to take a look at Introduction to Object-Oriented Modeling and Simulation with OpenModelica (PDF format, 6.6 MB) - a more recent tutorial (2012) by Peter Fritzson. There is a section that on Discrete Events and Hybrid Systems, that should clarify how to implement your equations in Modelica.
Below you can find an example from that tutorial about the model of a bouncing ball, as you can see discretization in time is not considered when you write your dynamic equations. So the continous model of the ball v=der(s), a=der(v) and than the discrete part inside the when clause that handles the contact with the ground:
model BouncingBall "the bouncing ball model"
parameter Real g=9.81; //gravitational acc.
parameter Real c=0.90; //elasticity constant
Real height(start=10),velocity(start=0);
equation
der(height) = velocity;
der(velocity)=-g;
when height<0 then
reinit(velocity, -c*velocity);
end when;
end BouncingBall;
Hope this helps,
Marco
If I understand your question, you want to use the last n evaluations of x to determine the next value of x. If so, this code shows how to do this:
model BooleanHistory
parameter Integer n=10 "How many points to keep";
parameter Modelica.SIunits.Time dt=1e-3;
protected
Boolean x[n];
function f
input Integer n;
input Boolean past[n-1];
output Boolean next;
algorithm
next :=not past[1]; // Example
end f;
initial equation
x = {false for i in 1:n};
equation
when sample(0,dt) then
x[2:n] = pre(x[1:(n-1)]);
x[1] = f(n, x[2:n]);
end when;
end BooleanHistory;
I am looking for a tool similar to graphviz that can render graphs, but that will allow me to constrain just the x coordinate of each node. Then, the tool will automatically choose y coordinates to make the graph look neat.
Basically, I want to make a timeline.
Language / platform / rendering medium are not very important.
If you want a neat-looking graph a force-directed algorithm is going to be your best bet. One of the best ones is SFDP (developed by AT&T, included in graphviz) though I can't seem to find pseudocode or an easy implementation. I don't think there are any algorithms this specialized. Thankfully, it's easy to code your own. I'll present some pseudocode mostly lifted form Wikipedia, but with suitably one-dimensional modifications. I'll assume you have n vertices and the vector of x-positions is x, subscripted by x.i.
set all vertex velocities to (0,0)
set all vertex positions to (x.i, random)
while (KE > epsilon)
KE = 0
for each vertex v
force = (0,0)
for each vertex u != v
force = force + (0, coulomb(u, v).y)
if u is incident to v
force = force + (0, hooke(u, v).y)
v.velocity = (v.velocity + timestep * force) * damping
v.position = v.position + timestep * v.velocity
KE = KE + |v.velocity| ^ 2
here the .y denotes getting the y-component of the force. This ensures that the x-components of the positions of the vertices never change from what you set them to be. The epsilon parameter is to be set by you, and should be something small compared to what you expect KE (the kinetic energy) to be. Also, |v| denotes the magnitude of the vector v (all computations are of 2-vectors in the above, except the KE). Note I set the mass of all the nodes to be 1, but you can change that if you want.
The Hooke and Coulomb functions calculate the respective forces between nodes; the first is linear in distance between vertices, the second is quadratic, so there is a guaranteed equilibrium. These functions look something like
def hooke(u, v)
return -k * |u.position - v.position|
def coulomb(u, v)
return C * |u.position - v.position|
where again most computations are in vector form. C and k have real values but experiment to get the graph you want. This isn't usually necessary because the scaling factors will, in two dimensions, pretty much expand or contract the whole graph, but here the x-distances are set so to get a good-looking graph you will have to change the values a bit.
I'm trying to render the "mount" scene from Eric Haines' Standard Procedural Database (SPD), but the refraction part just doesn't want to co-operate. I've tried everything I can think of to fix it.
This one is my render (with Watt's formula):
(source: philosoraptor.co.za)
This is my render using the "normal" formula:
(source: philosoraptor.co.za)
And this one is the correct render:
(source: philosoraptor.co.za)
As you can see, there are only a couple of errors, mostly around the poles of the spheres. This makes me think that refraction, or some precision error is to blame.
Please note that there are actually 4 spheres in the scene, their NFF definitions (s x_coord y_coord z_coord radius) are:
s -0.8 0.8 1.20821 0.17
s -0.661196 0.661196 0.930598 0.17
s -0.749194 0.98961 0.930598 0.17
s -0.98961 0.749194 0.930598 0.17
That is, there is a fourth sphere behind the more obvious three in the foreground. It can be seen in the gap left between these three spheres.
Here is a picture of that fourth sphere alone:
(source: philosoraptor.co.za)
And here is a picture of the first sphere alone:
(source: philosoraptor.co.za)
You'll notice that many of the oddities present in both my version and the correct version is missing. We can conclude that these effects are the result of interactions between the spheres, the question is which interactions?
What am I doing wrong? Below are some of the potential errors I've already considered:
Refraction vector formula.
As far as I can tell, this is correct. It's the same formula used by several websites and I verified the derivation personally. Here's how I calculate it:
double sinI2 = eta * eta * (1.0f - cosI * cosI);
Vector transmit = (v * eta) + (n * (eta * cosI - sqrt(1.0f - sinI2)));
transmit = transmit.normalise();
I found an alternate formula in 3D Computer Graphics, 3rd Ed by Alan Watt. It gives a closer approximation to the correct image:
double etaSq = eta * eta;
double sinI2 = etaSq * (1.0f - cosI * cosI);
Vector transmit = (v * eta) + (n * (eta * cosI - (sqrt(1.0f - sinI2) / etaSq)));
transmit = transmit.normalise();
The only difference is that I'm dividing by eta^2 at the end.
Total internal reflection.
I tested for this, using the following conditional before the rest of my intersection code:
if (sinI2 <= 1)
Calculation of eta.
I use a stack-like approach for this problem:
/* Entering object. */
if (r.normal.dot(r.dir) < 0)
{
double eta1 = r.iorStack.back();
double eta2 = m.ior;
eta = eta1 / eta2;
r.iorStack.push_back(eta2);
}
/* Exiting object. */
else
{
double eta1 = r.iorStack.back();
r.iorStack.pop_back();
double eta2 = r.iorStack.back();
eta = eta1 / eta2;
}
As you can see, this stores the previous objects that contained this ray in a stack. When exiting the code pops the current IOR off the stack and uses that, along with the IOR under it to compute eta. As far as I know this is the most correct way to do it.
This works for nested transmitting objects. However, it breaks down for intersecting transmitting objects. The problem here is that you need to define the IOR for the intersection independently, which the NFF file format does not do. It's unclear then, what the "correct" course of action is.
Moving the new ray's origin.
The new ray's origin has to be moved slightly along the transmitted path so that it doesn't intersect at the same point as the previous one.
p = r.intersection + transmit * 0.0001f;
p += transmit * 0.01f;
I've tried making this value smaller (0.001f) and (0.0001f) but that makes the spheres appear solid. I guess these values don't move the rays far enough away from the previous intersection point.
EDIT: The problem here was that the reflection code was doing the same thing. So when an object is reflective as well as refractive then the origin of the ray ends up in completely the wrong place.
Amount of ray bounces.
I've artificially limited the amount of ray bounces to 4. I tested raising this limit to 10, but that didn't fix the problem.
Normals.
I'm pretty sure I'm calculating the normals of the spheres correctly. I take the intersection point, subtract the centre of the sphere and divide by the radius.
Just a guess based on doing a image diff (and without reading the rest of your question). The problem looks to me to be the refraction on the back side of the sphere. You might be:
doing it backwards: e.g. reversing (or not reversing) the indexes of refraction.
missing it entirely?
One way to check for this would be to look at the mount through a cube that is almost facing the camera. If the refraction is correct, the picture should be offset slightly but otherwise un-altered. If it's not right, then the picture will seem slightly tilted.
So after more than I year, I finally figured out what was going on here. Clear minds and all that. I was completely off track with the formula. I'm instead using a formula by Heckbert now, which I am sure is correct because I proved it myself using geometry and discrete math.
Here's the correct vector calculation:
double c1 = v.dot(n) * -1;
double c1Sq = pow(c1, 2);
/* Heckbert's formula requires eta to be eta2 / eta1, so I have to flip it here. */
eta = 1 / eta;
double etaSq = pow(eta, 2);
if (etaSq + c1Sq >= 1)
{
Vector transmit = (v / eta) + (n / eta) * (c1 - sqrt(etaSq - 1 + c1Sq));
transmit = transmit.normalise();
...
}
else
{
/* Total internal reflection. */
}
In the code above, eta is eta1 (the IOR of the surface from which the ray is coming) over eta2 (the IOR of the destination surface), v is the incident ray and n is the normal.
There was another problem, which confused the problem some more. I had to flip the normal when exiting an object (which is obvious - I missed it because the other errors were obscuring it).
Lastly, my line of sight algorithm (to determine whether a surface is illuminated by a point light source) was not properly passing through transparent surfaces.
So now my images line up properly :)
I would like to do an algebraic curve fit of 2D data points, but for various reasons - it isn't really possible to have much of the sample data in memory at once, and iterating through all of it is an expensive process.
(The reason for this is that actually I need to fit thousands of curves simultaneously based on gigabytes of data which I'm reading off disk, and which is therefore sloooooow).
Note that the number of polynomial coefficients will be limited (perhaps 5-10), so an exact fit will be extremely unlikely, but this is ok as I'm trying to find an underlying pattern in data with a lot of random noise.
I understand how one can use a genetic algorithm to fit a curve to a dataset, but this requires many passes through the sample data, and thus isn't practical for my application.
Is there a way to fit a curve with a single pass of the data, where the state that must be maintained from sample to sample is minimal?
I should add that the nature of the data is that the points may lie anywhere on the X axis between 0.0 and 1.0, but the Y values will always be either 1.0 or 0.0.
So, in Java, I'm looking for a class with the following interface:
public interface CurveFit {
public void addData(double x, double y);
public List<Double> getBestFit(); // Returns the polynomial coefficients
}
The class that implements this must not need to keep much data in its instance fields, no more than a kilobyte even for millions of data points. This means that you can't just store the data as you get it to do multiple passes through it later.
edit: Some have suggested that finding an optimal curve in a single pass may be impossible, however an optimal fit is not required, just as close as we can get it in a single pass.
The bare bones of an approach might be if we have a way to start with a curve, and then a way to modify it to get it slightly closer to new data points as they come in - effectively a form of gradient descent. It is hoped that with sufficient data (and the data will be plentiful), we get a pretty good curve. Perhaps this inspires someone to a solution.
Yes, it is a projection. For
y = X beta + error
where lowercased terms are vectors, and X is a matrix, you have the solution vector
\hat{beta} = inverse(X'X) X' y
as per the OLS page. You almost never want to compute this directly but rather use LR, QR or SVD decompositions. References are plentiful in the statistics literature.
If your problem has only one parameter (and x is hence a vector as well) then this reduces to just summation of cross-products between y and x.
If you don't mind that you'll get a straight line "curve", then you only need six variables for any amount of data. Here's the source code that's going into my upcoming book; I'm sure that you can figure out how the DataPoint class works:
Interpolation.h:
#ifndef __INTERPOLATION_H
#define __INTERPOLATION_H
#include "DataPoint.h"
class Interpolation
{
private:
int m_count;
double m_sumX;
double m_sumXX; /* sum of X*X */
double m_sumXY; /* sum of X*Y */
double m_sumY;
double m_sumYY; /* sum of Y*Y */
public:
Interpolation();
void addData(const DataPoint& dp);
double slope() const;
double intercept() const;
double interpolate(double x) const;
double correlate() const;
};
#endif // __INTERPOLATION_H
Interpolation.cpp:
#include <cmath>
#include "Interpolation.h"
Interpolation::Interpolation()
{
m_count = 0;
m_sumX = 0.0;
m_sumXX = 0.0;
m_sumXY = 0.0;
m_sumY = 0.0;
m_sumYY = 0.0;
}
void Interpolation::addData(const DataPoint& dp)
{
m_count++;
m_sumX += dp.getX();
m_sumXX += dp.getX() * dp.getX();
m_sumXY += dp.getX() * dp.getY();
m_sumY += dp.getY();
m_sumYY += dp.getY() * dp.getY();
}
double Interpolation::slope() const
{
return (m_sumXY - (m_sumX * m_sumY / m_count)) /
(m_sumXX - (m_sumX * m_sumX / m_count));
}
double Interpolation::intercept() const
{
return (m_sumY / m_count) - slope() * (m_sumX / m_count);
}
double Interpolation::interpolate(double X) const
{
return intercept() + slope() * X;
}
double Interpolation::correlate() const
{
return m_sumXY / sqrt(m_sumXX * m_sumYY);
}
Why not use a ring buffer of some fixed size (say, the last 1000 points) and do a standard QR decomposition-based least squares fit to the buffered data? Once the buffer fills, each time you get a new point you replace the oldest and re-fit. That way you have a bounded working set that still has some data locality, without all the challenges of live stream (memoryless) processing.
Are you limiting the number of polynomial coefficients (i.e. fitting to a max power of x in your polynomial)?
If not, then you don't need a "best fit" algorithm - you can always fit N data points EXACTLY to a polynomial of N coefficients.
Just use matrices to solve N simultaneous equations for N unknowns (the N coefficients of the polynomial).
If you are limiting to a max number of coefficients, what is your max?
Following your comments and edit:
What you want is a low-pass filter to filter out noise, not fit a polynomial to the noise.
Given the nature of your data:
the points may lie anywhere on the X axis between 0.0 and 1.0, but the Y values will always be either 1.0 or 0.0.
Then you don't need even a single pass, as these two lines will pass exactly through every point:
X = [0.0 ... 1.0], Y = 0.0
X = [0.0 ... 1.0], Y = 1.0
Two short line segments, unit length, and every point falls on one line or the other.
Admittedly, an algorithm to find a good curve fit for arbitrary points in a single pass is interesting, but (based on your question), that's not what you need.
Assuming that you don't know which point should belong to which curve, something like a Hough Transform might provide what you need.
The Hough Transform is a technique that allows you to identify structure within a data set. One use is for computer vision, where it allows easy identification of lines and borders within the field of sight.
Advantages for this situation:
Each point need be considered only once
You don't need to keep a data structure for each candidate line, just one (complex, multi-dimensional) structure
Processing of each line is simple
You can stop at any point and output a set of good matches
You never discard any data, so it's not reliant on any accidental locality of references
You can trade off between accuracy and memory requirements
Isn't limited to exact matches, but will highlight partial matches too.
An approach
To find cubic fits, you'd construct a 4-dimensional Hough space, into which you'd project each of your data-points. Hotspots within Hough space would give you the parameters for the cubic through those points.
You need the solution to an overdetermined linear system. The popular methods are Normal Equations (not usually recommended), QR factorization, and singular value decomposition (SVD). Wikipedia has decent explanations, Trefethen and Bau is very good. Your options:
Out-of-core implementation via the normal equations. This requires the product A'A where A has many more rows than columns (so the result is very small). The matrix A is completely defined by the sample locations so you don't have to store it, thus computing A'A is reasonably cheap (very cheap if you don't need to hit memory for the node locations). Once A'A is computed, you get the solution in one pass through your input data, but the method can be unstable.
Implement an out-of-core QR factorization. Classical Gram-Schmidt will be fastest, but you have to be careful about stability.
Do it in-core with distributed memory (if you have the hardware available). Libraries like PLAPACK and SCALAPACK can do this, the performance should be much better than 1. The parallel scalability is not fantastic, but will be fine if it's a problem size that you would even think about doing in serial.
Use iterative methods to compute an SVD. Depending on the spectral properties of your system (maybe after preconditioning) this could converge very fast and does not require storage for the matrix (which in your case has 5-10 columns each of which are the size of your input data. A good library for this is SLEPc, you only have to find a the product of the Vandermonde matrix with a vector (so you only need to store the sample locations). This is very scalable in parallel.
I believe I found the answer to my own question based on a modified version of this code. For those interested, my Java code is here.