This question has already confused me several days. While I referred to senior students, they also cannot give a reply.
We have ten ODEs, into which each a noise term should be added. The noise is defined as follows. since I always find that I cannot upload a picture, the formula below maybe not very clear. In order to understand, you can either read my explanation or go the this address: Plos one. You could find the description of the equations directly above the Support Information in this address
The white noise term epislon_i(t) is assumed with Gaussian distribution. epislon_i(t) means that for equation i, and at t timepoint, the value of the noise.
the auto-correlation of noise are given:
(EQ.1)
where delta(t) is the Dirac delta function and the diffusion matrix D is defined by
(EQ.2)
Our problem focuses on how to explain the Dirac delta function in the diffusion matrix. Since the property of Dirac delta function is delta(0) = Inf and delta(t) = 0 if t neq 0, we don't know how to calculate the epislonif we try to sqrt of 2D(x, t)delta(t-t'). So we simply assume that delta(0) = 1 and delta(t) = 0 if t neq 0; But we don't know whether or not this is right. Could you please tell me how to use Delta function of diffusion equation in MATLAB?
This question associates with the stochastic process in MATLAB. So we review different stochastic process to inspire our ideas. In MATLAB, the Wienner process is often defined as a = sqrt(dt) * rand(1, N). N is the number of steps, dt is the length of the steps. Correspondingly, the Brownian motion can be defined as: b = cumsum(a); All of these associate with stochastic process. However, they doesn't related to the white noise process which has a constraints on the matrix of auto-correlation, noted by D.
Then we consider that, we may simply use randn(1, 10) to generate a vector representing the noise. However, since the definition of the noise must satisfy the equation (2), this cannot enable noise term in different equation have the predefined partial correlation (D_ij). Then we try to use mvnrnd to generate a multiple variable normal distribution at each time step. Unfortunately, the function mvnrnd in MATLAB return a matrix. But we need to return a vector of length 10.
We are rather confused, so could you please give me just a light? Thanks so much!
NOTE: I see two hazy questions in here: 1) how to deal with a stochastic term in a DE and 2) how to deal with a delta function in a DE. Both of these are math related questions and http://www.math.stackexchange.com will be a better place for this. If you had a question pertaining to MATLAB, I haven't been able to pin it down, and you should perhaps add code examples to better illustrate your point. That said, I'll answer the two questions briefly, just to put you on the right track.
What you have here are not ODEs, but Stochastic differential equations (SDE). I'm not sure how you're using MATLAB to work with this, but routines like ode45 or ode23 will not be of any help. For SDEs, your usual mathematical tools of separation of variables/method of characteristics etc don't work and you'll need to use Itô calculus and Itô integrals to work with them. The solutions, as you might have guessed, will be stochastic. To learn more about SDEs and working with them, you can consider Stochastic Differential Equations: An Introduction with Applications by Bernt Øksendal and for numerical solutions, Numerical Solution of Stochastic Differential Equations by Peter E. Kloeden and Eckhard Platen.
Coming to the delta function part, you can easily deal with it by taking the Fourier transform of the ODE. Recall that the Fourier transform of a delta function is 1. This greatly simplifies the DE and you can take an inverse transform in the very end to return to the original domain.
Related
I have the following equation that I want to solve with respect to a:
x = (a-b-c+d)/log((a-b)/(c-d))
where x, b, c, and d are known. I used Wolfram Alpha to solve the equation, and the result is:
a = b-x*W(-((c-d)*exp(d/x-c/x))/x)
where W is the is the product log function (Lambert W function). It might be easier to see it at the Wolfram Alpha page.
I used the Matlab's built-in lambertW function to solve the equation. This is rather slow, and is the bottleneck in my script. Is there another, quicker, way to do this? It doesn't have to be accurate down to the 10th decimal place.
EDIT:
I had no idea that this equation is so hard to solve. Here is a picture illustrating my problem. The temperatures b-d plus LMTD varies in each time step, but are known. Heat is transferred from red line (CO2) to blue line (water). I need to find temperature "a". I didn't know that this was so hard to calculate! :P
Another option is based on the simpler Wright ω function:
a = b - x.*wrightOmega(log(-(c-d)./x) - (c-d)./x);
provided that d ~= c + x.*wrightOmega(log(-(c-d)./x) - (c-d)./x) (i.e., d ~= c+b-a, x is 0/0 in this case). This is equivalent to the principal branch of the Lambert W function, W0, which I think is the solution branch you want.
Just as with lambertW, there's a wrightOmega function in the Symbolic Math toolbox. Unfortunately, this will probably also be slow for a large number of inputs. However, you can use my wrightOmegaq on GitHub for complex-valued floating-point (double- or single-precison) inputs. The function is more accurate, fully-vectorized, and can be three to four orders of magnitude faster than using the built-in wrightOmega for floating-point inputs.
For those interested, wrightOmegaq is based on this excellent paper:
Piers W. Lawrence, Robert M. Corless, and David J. Jeffrey, "Algorithm 917: Complex Double-Precision Evaluation of the Wright omega Function," ACM Transactions on Mathematical Software, Vol. 38, No. 3, Article 20, pp. 1-17, Apr. 2012.
This algorithm goes beyond the cubic convergence of the Halley's method used in Cleve Moler's Lambert_W and uses a root-finding method with fourth-order convergence (Fritsch, Shafer, & Crowley, 1973) to converge in no more than two iterations.
Also, to further speed up Moler's Lambert_W using series expansions, see my answer at Math.StackExchange.
Two (combinable) options:
Is your script already vectorized? Evaluate the function for more than a single argument. Executing for i = 1:100, a(i)=lambertw(rhs(i)); end is slower than a=lambertw(rhs).
If you are dealing with the real valued branch of LambertW (i.e. your arguments are in the interval [-1/e, inf) ), you can use the implementation of Lambert_W submitted by Cleve Moler on the File Exchange.
Do you know the mass flow rates at both sides of the heat exchanger at each time-step?
If yes, temperature 'a' can be solved by the 'effectiveness-NTU' approach which does not need any iteration, rather than the LMTD approach. Reference: e.g. http://ceng.tu.edu.iq/ched/images/lectures/chem-lec/st3/c2/Lec23.pdf
I have a set of 4 PDEs:
du/dt + A(u) * du/dx = Q(u)
where,u is a matrix and contains:
u=[u1;u2;u3;u4]
and A is a 4*4 matrix. Q is 4*1. A and Q are function of u=[u1;u2;u3;u4].
But my questions are:
How can I solve above equation in MATLAB?
If I solved it by PDE functions of Matlab,can I convert it to a
simple function that is not used from ready functions of Matlab?
Is there any way that I calculate A and Q explicitly. I mean that in
every time step, I calculate A and Q from data of previous time step
and put new value in the equation that causes faster run of program?
PDEs require finite differences, finite elements, boundary elements, etc. You can also turn them into ODEs using transforms like Laplace, Fourier, etc. Solve those using ODE functions and then transform back. Neither one is trivial.
Your equation is a non-linear transient diffusion equation. It's a parabolic PDE.
The equation you posted has the additional difficulty of being non-linear, because both the A matrix and Q vector are functions of the independent variable q. You'll have to start by linearizing your equations. Solve for increments in u rather than u itself.
Once you've done that, discretize the du/dx term using finite differences, finite elements, or boundary elements. You should start with a weighted residual integral formulation.
You're almost done: Next to integrate w.r.t. time using the method of your choice.
It's not trivial.
Google found this: maybe it will help you.
http://www.mathworks.com/matlabcentral/fileexchange/3710-nonlinear-diffusion-toolbox
I'm not too familiar with MATLAB or computational mathematics so I was wondering how I might solve an equation involving the sum of squares, where each term involves two vectors- one known and one unknown. This formula is supposed to represent the error and I need to minimize the error. I think I'm supposed to use least squares but I don't know too much about it and I'm wondering what function is best for doing that and what arguments would represent my equation. My teacher also mentioned something about taking derivatives and he formed a matrix using derivatives which confused me even more- am I required to take derivatives?
The problem that you must be trying to solve is
Min u'u = min \sum_i u_i^2, u=y-Xbeta, where u is the error, y is the vector of dependent variables you are trying to explain, X is a matrix of independent variables and beta is the vector you want to estimate.
Since sum u_i^2 is diferentiable (and convex), you can evaluate the minimal of this expression calculating its derivative and making it equal to zero.
If you do that, you find that beta=inv(X'X)X'y. This maybe calculated using the matlab function regress http://www.mathworks.com/help/stats/regress.html or writing this formula in Matlab. However, you should be careful how to evaluate the inverse (X'X) see Most efficient matrix inversion in MATLAB
I would like to measure the goodness-of-fit to an exponential decay curve. I am using the lsqcurvefit MATLAB function. I have been suggested by someone to do a chi-square test.
I would like to use the MATLAB function chi2gof but I am not sure how I would tell it that the data is being fitted to an exponential curve
The chi2gof function tests the null hypothesis that a set of data, say X, is a random sample drawn from some specified distribution (such as the exponential distribution).
From your description in the question, it sounds like you want to see how well your data X fits an exponential decay function. I really must emphasize, this is completely different to testing whether X is a random sample drawn from the exponential distribution. If you use chi2gof for your stated purpose, you'll get meaningless results.
The usual approach for testing the goodness of fit for some data X to some function f is least squares, or some variant on least squares. Further, a least squares approach can be used to generate test statistics that test goodness-of-fit, many of which are distributed according to the chi-square distribution. I believe this is probably what your friend was referring to.
EDIT: I have a few spare minutes so here's something to get you started. DISCLAIMER: I've never worked specifically on this problem, so what follows may not be correct. I'm going to assume you have a set of data x_n, n = 1, ..., N, and the corresponding timestamps for the data, t_n, n = 1, ..., N. Now, the exponential decay function is y_n = y_0 * e^{-b * t_n}. Note that by taking the natural logarithm of both sides we get: ln(y_n) = ln(y_0) - b * t_n. Okay, so this suggests using OLS to estimate the linear model ln(x_n) = ln(x_0) - b * t_n + e_n. Nice! Because now we can test goodness-of-fit using the standard R^2 measure, which matlab will return in the stats structure if you use the regress function to perform OLS. Hope this helps. Again I emphasize, I came up with this off the top of my head in a couple of minutes, so there may be good reasons why what I've suggested is a bad idea. Also, if you know the initial value of the process (ie x_0), then you may want to look into constrained least squares where you bind the parameter ln(x_0) to its known value.
If i have to use a partial derivative in modelica, how can that be used. I am not sure if partial derivative can be solved in modelica but i would like to know, if it can be used then, how should it be implemented.
There are two different potential "partial derivatives" you might want. One is the partial derivative with respect to spatial variables (if you are interested in solving PDEs) or you might want the partial derivative of an expression with respect to a simulation variable.
But it doesn't matter, because you cannot express either of these in Modelica.
If your motivation is to solve PDEs, then I'm afraid you will simply have to process the spatial aspects in your models (using some kind of discretization, weak formulation, etc) so that the resulting equations are simple ODEs.
If you want to compute the derivative of expressions with respect to variables other than time, the question would be ... why? I'm hard pressed to think of an application where this is really necessary. But if you can explain your use case, I could comment further on how to handle it.
I've discretized PDE systems for solution in Modelica: heat equation, wave equation, PDEs from double-pipe heat exchangers, PDEs from water hammer to model pressure surges in pipelines etc
At a simple level, you can replace the spatial derivative with a central difference approximation, and then generate the entire set of ODEs with a for loop. For example. here's a Modelica code snippet for a simple discretization of the heat equation.
parameter Real L = 1 "Length";
parameter Integer n = 50 "Number of sections";
parameter Real alpha = 1;
Real dL = L/n "Section length";
Real[n] u(each start = 0);
equations
u[1] = 273; //boundary condition
u[n] =0; //boundary condition
for i in 2:n-1 loop
der(u[i]) = alpha * (u[i+1] - 2 * u[i] + u[i-1]) / dL^2;
end for;
This is just a simple example entered from the top of my head, so excuse any mistakes.
Do you have a specific example or application?