tag:blogger.com,1999:blog-5822805028291837738.post5211543931454957413..comments2019-08-17T12:19:14.454-04:00Comments on Various Consequences: Recurrence, Averaging and PredictabilityJoshua Stultshttp://www.blogger.com/profile/03506970399027046387noreply@blogger.comBlogger20125tag:blogger.com,1999:blog-5822805028291837738.post-35098661854914870582012-08-10T06:42:15.207-04:002012-08-10T06:42:15.207-04:00A Rigorous ODE solver and Smale's 14th Problem...<a href="http://www2.math.uu.se/~warwick/main/rodes.html" rel="nofollow">A Rigorous ODE solver and Smale's 14th Problem</a>jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-76588001809116420252011-10-15T10:56:06.064-04:002011-10-15T10:56:06.064-04:00Interesting sounding paper, from the Abstract: Non...Interesting sounding paper, from the Abstract: <i>Nonlinear systems driven by noise and periodic forces with more than one frequency exhibit the phenomenon of Ghost Stochastic Resonance (GSR) found in a wide and disparate variety of fields ranging from biology to geophysics. The common novel feature is the emergence of a "ghost" frequency in the system's output which it is absent in the input. As reviewed here, the uncovering of this phenomenon helped to understand a range of problems, from the perception of pitch in complex sounds or visual stimuli, to the explanation of climate cycles. Recent theoretical efforts show that a simple mechanism with two ingredients are at work in all these observations. The first one is the linear interference between the periodic inputs and the second a nonlinear detection of the largest constructive interferences, involving a noisy threshold. These notes are dedicated to review the main aspects of this phenomenon, as well as its different manifestations described on a bewildering variety of systems ranging from neurons, semiconductor lasers, electronic circuits to models of glacial climate cycles. </i><br /><a href="http://arxiv.org/abs/1110.0136" rel="nofollow">The Ghost of Stochastic Resonance: an Introductory Review</a>jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-88963493100889117822011-03-08T11:14:43.047-05:002011-03-08T11:14:43.047-05:00These neat little examples on parameter continuati...These neat little <a href="http://www.mathworks.com/help/techdoc/math/f1-713877.html#brfhdqp-1" rel="nofollow">examples on parameter continuation</a> were linked from the Maxima list in a completely unrelated context, but goes to the IVP / BVP distinction. My question to folks at Dr Curry's site about why GCM's aren't using all the standard BVP and convergence acceleration techniques (dropping the time-derivative term) if they are really solving a BVP has been met with limp-wristed hand-waving or silence. <br /><br />I guess they didn't know that Steven Schneider (certainly not on anyone's list of 'skeptics') thought climate was probably an IVP problem too: <br />"A common view of the climate system and ecosystem structure and function is that of path independence (no memory of previous conditions). However, the multiple stable equilibria for both THC and for atmosphere-biosphere interactions in West Africa suggest a more complex reality. In such systems, the equilibrium state reached is dependent on the initial conditions of the system."<br /><a href="http://www.oecd.org/dataoecd/9/59/2482280.pdf" rel="nofollow">Abrupt Non-Linear Climate Change, Irreversibility and Surprise</a>jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-76570570180500402672011-02-09T05:28:55.578-05:002011-02-09T05:28:55.578-05:00Yes interesting paper with many details .
They fin...Yes interesting paper with many details .<br />They find basically that :<br /><i>Increasing the<br />number of significant digits beyond the 420 digits used in this work increases the<br />interval on which the Lorenz solution is computable.</i><br />what is an elaborated way to say what I wrote above in a fast 0 order approximation :<br /><i>T=dt/A</i><br /><br />There is a fine point of critics that I would make and that is that T is an upper bound and not a constant of the system .<br />Depending on X0 , the numerical solution may become wrong (e.g an artefact) BEFORE T for a GIVEN hardware precision .<br />The real T cannot be independent of X0 .<br />When I played with Lorenz solvers at low precision and with the same dt , I noticed that the time at which the trajectories intersect (e.g T when the numerical solution becomes an artefact) depends on X0 .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-13298948146863492782011-02-08T13:17:30.109-05:002011-02-08T13:17:30.109-05:00Tom said: Focusing a moment on the "simple&qu...Tom said: <i>Focusing a moment on the "simple" interpretation .<br />A characteristics of a chaotic solution is , as I already wrote above , that :<br />f(X0+dX0,t) - f(X0,t) = g(X0).exp(L.t)<br />From there follows trivially that for times less than 1/L the trajectories don't diverge too wildly and regardless of the "time step" which is necessarily much smaller than 1/L , when the time step decreases you will get something that looks like "convergence" .</i><br />I think you'll appreciate the introductory error analysis in <a href="http://lorenzsystem.net/paper/" rel="nofollow">the paper</a> that goes with the site from reference [8] linked above.<br /><br />Tom also said: <i>Normally what you should see if you plot differences between trajectories with different time steps what is equivalent to take the differences between different initial conditions , e.g f(X0+dX0,t) - f(X0,t)(this should be proven if one wants to be rigorous), is a horizontal 0 line untill approximately T and then a more or less sudden explosion .</i><br />Yes, that's exactly what I found; see the <a href="http://www.variousconsequences.com/2010/01/lorenz-63-ensemble.html" rel="nofollow">plots in Figure 2 of this post</a>, differences between trajectories in an initial condition ensemble.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-91214301450419377262011-02-08T11:00:09.218-05:002011-02-08T11:00:09.218-05:00I just meant that I'm actually going to add a ...I just meant that I'm actually going to add a periodic forcing term rather than "force" the system by having a time-dependent parameter value; so it's a different system, but that's already coded up because that's what I needed for verifying the numerical implementation with MMS.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-35998934456763559452011-02-08T08:53:10.668-05:002011-02-08T08:53:10.668-05:00Argh I am muddled today .
The end should read N = ...Argh I am muddled today .<br />The end should read N = 1/A so as T =N.dt , T=dt/A .<br />Correct this time . And better for the dimensional consistency .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-1547216320046028682011-02-08T08:40:04.799-05:002011-02-08T08:40:04.799-05:00Of course there is a nonsense in the above that I ...Of course there is a nonsense in the above that I could not edit.<br />The time after which a numerical solution becomes an artefact can obviously NOT be greater than T because T is an upper bound .<br />It can only be smaller and what one cannot know is by how much smaller it is because that depends on X0 (initial condition) .<br /><br />T can be easily computed .<br />If one supposes that the solution is normalised so that it is a number between 0 and 1 , then if A is the accuracy of the hardware for [0,1] , and dt the step chosen , the maximum of steps you can compute is N = 1/(A.dt) so T = 1/A . Beyond this T surely , and possibly earlier , your solution becomes an artefact .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-88838279124677584482011-02-08T07:44:26.520-05:002011-02-08T07:44:26.520-05:00Joshua
My skills and experience with numerical me...Joshua<br /><br />My skills and experience with numerical methods are far below yours and Dan Hugh's .<br />I am a physicist who does 95% of his physics with a pencil and a sheet of paper so when I look at your scripts , my eyes glaze over .<br />Sure I learned some time ago things about Runge Kuttas and such but have never really do it in practice .<br />That's why your very target as in the link you gave above stays largely mysterious to me .<br />My certainly imperfect interpretation is that you want to know things about the trajectory divergence/convergence and then I cannot decide if the questions you ask are simple or on the contrary so complex that I did not understand them .<br /><br />Focusing a moment on the "simple" interpretation .<br />A characteristics of a chaotic solution is , as I already wrote above , that :<br />f(X0+dX0,t) - f(X0,t) = g(X0).exp(L.t)<br />From there follows trivially that for times less than 1/L the trajectories don't diverge too wildly and regardless of the "time step" which is necessarily much smaller than 1/L , when the time step decreases you will get something that looks like "convergence" .<br />One of course doesn't know exactly g(X0) but if it is small , it will contribute to keeping the trajectories near to each other for times below 1/L .<br /><br />On the other hand for times greater than 1/L (note that this is just a convention , an order of magnitude) , the term exp(L.t) will overwhelm everything and the increase of the length of convergence by taking smaller and smaller dt will become marginal .<br />In summary depending on the exact form of g(X0) and the value of L , there will be a time T beyond which no decrease of dt will improve the convergence significantly .<br /><br />Ultimately the smallest possible dt is given by the construction properties of the computer .<br />The corollary of the above is that once one uses the smalles dt allowed by the hardware , everything computed beyond T is just an artefact .<br /><br />For fun it is possible to mathematically prove that ANY numerical solution of a chaotic system is NECESSARILY an artefact LATEST beyond some finite T .<br />This doesn't however mean that it becomes an artefact exactly at T . It may become an artefact even before T (for grossly large dt) or a long time after T . You simply can't know .<br /><br />A numerical analysis of convergence will give hints about this T and eventually about the g(X0) .<br />Normally what you should see if you plot differences between trajectories with different time steps what is equivalent to take the differences between different initial conditions , e.g f(X0+dX0,t) - f(X0,t)(this should be proven if one wants to be rigorous), is a horizontal 0 line untill approximately T and then a more or less sudden explosion .<br /><br />Last but not least in order to be complete .<br />Of course it would be nice if things were as simple as that but obviously the divergence can't go forever because the exponetial goes to infinity .<br />And we know that chaotic systems having an attractor (like Lorenz one) must stay in a finite volume of the phase space because the attractor is always bounded .<br />From there follows that when f(X0+dX0,t) - f(X0,t) begins to reach the size of the attractor , the trajectories don't diverge exponentially anymore . They are then so different and in so diferent places of the attractor that they have nothing in common anymore .<br />That's why the Lyapounov coefficient can only be used and makes sense for some finite time interval [0,TL] and looses more and more its significance when one approaches TL .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-49184113141531792972011-02-07T15:18:31.041-05:002011-02-07T15:18:31.041-05:00Good paper Tom; thanks. I was going to approach i...Good paper Tom; thanks. I was going to approach it from <a href="http://www.variousconsequences.com/2010/01/chaotic-time-convergence-and-mms.html" rel="nofollow">a slightly different angle</a>, but I'm going to incorporate a bit of discussion of that one now too.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-57458090401104499122011-02-07T09:24:57.798-05:002011-02-07T09:24:57.798-05:00Joshua :
I think I'll get to a forced Lorenz ...Joshua :<br /><br /><i>I think I'll get to a forced Lorenz system (just periodic forcing), before I get to stochastic resonance, but all these ideas are related. </i><br /><br />Well thought .<br />But before going there , read that : http://arxiv.org/abs/chao-dyn/9405012<br /><br />Just to save your time :)TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-68337239252888565602011-02-01T01:09:00.813-05:002011-02-01T01:09:00.813-05:00Is that a moment closure technique that Tom/Tomas ...Is that a moment closure technique that Tom/Tomas is using to show that the average diverges?<br /><br />I was looking at this paper a few weeks ago in the context of Monte Carlo sims, which struck a chord:<br />http://www.mas.ncl.ac.uk/~ncsg3/talks/talks/cisban08.pdfWHThttps://www.blogger.com/profile/18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-33256729123634260952011-01-31T09:41:52.529-05:002011-01-31T09:41:52.529-05:00Thanks, That answers my question. So it is really ...Thanks, That answers my question. So it is really dependent on the size of the forcing; if the forcing is not big enough, a stochastic resonance mechanism can kick in.WHThttps://www.blogger.com/profile/18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-79225483884228511682011-01-31T08:05:32.184-05:002011-01-31T08:05:32.184-05:00WHT, you must be spying on me; I've been noodl...WHT, you must be spying on me; I've been noodling around a stochastic resonance post, but haven't really got a good "hook" for it yet. I don't think I'm understanding your question. Forcings that should be too small to drive the oscillations are able to because the stochastic noise pushes the system "over the top" (and back again). I think I'll get to a forced Lorenz system (just periodic forcing), before I get to stochastic resonance, but all these ideas are related.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-54204447093850853352011-01-30T23:20:43.932-05:002011-01-30T23:20:43.932-05:00I think this is very important work, essentially s...I think this is very important work, essentially showing how robust the formulations are to uncertainties. It seems like the complement to this is the introduction of order out of noise and disturbances, which is the basic idea behind stochastic resonance. If you could unify these two ideas, it would help clear things up in my mind. <br /><br />So in a particular situation, is the oscillation observed because it is always there and just immune to disturbances, or does it emerge from the noise, resonantly amplified as a kind of principle component from the nonlinear positive feedback of the system? Or could these two ideas be the same principle?WHThttps://www.blogger.com/profile/18297101284358849575noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-74853542171497176642011-01-26T12:46:07.834-05:002011-01-26T12:46:07.834-05:00Found it; it went to the spam bucket.
Here's ...Found it; it went to the spam bucket.<br /><br />Here's a <a href="http://amath.colorado.edu/faculty/juanga/Papers/PhysRevLett_96_254103.pdf" rel="nofollow">clickable link for the paper</a>.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-33898328041612454402011-01-26T09:35:35.831-05:002011-01-26T09:35:35.831-05:00The comment I wanted to add was that this result c...The comment I wanted to add was that this result can't be easily transported to the climate .<br />Indeed climate is spatio-temporal chaos and the result obtained here is only valid for chaotic solutions in temporal chaos .<br /><br />This difficulty is due to the fact that you correctly identified - temporal chaos deals with functions while climate science deals with functionals .<br />I know of only one person who could deal with functionals correctly and even did it so correctly that he got a Nobel for that - Feynman :)<br /><br />Once you said that you said almost everything .<br />A differential of a functional is not uniquely defined so you loose all the impressive and necessary weaponry of differential calculus once you start having only functionals .<br /><br />As a side note I go ROFL everytime when I see expressions like dTg/dF (which is supposed to define the climate sensitivity) and where Tg is an average global temperature and F some forcing .<br />Tg being a functional that makes dTg be .... what ?<br />A nonsense .<br />But nonsense won't of course stop climate "scientists" to come to far reaching conclusions .<br /><br />So it is a kind of damned if you do , damned if you don't .<br />If you don't use functionals , you face a full blown spatio temporal chaos with its untractable infinite dimensionality (of the phase space) .<br />If you use functionals , a miracle seems to happen because you obtain things that superficially look like functions of time only .<br />But it is only an illusion and you are strictly forbidden to plug that thing in the good old temporal chaos theory .<br /><br />So what is left ?<br />I have already written about it here .<br />If a full spatio-temporal chaos theory is not feasible in the next 100 years or so , then what is left is this : http://amath.colorado.edu/faculty/juanga/Papers/PhysRevLett_96_254103.pdf .<br />This is one of the ways to discretize space and thus transform the world in a <b>FINITE</b> spatial network of coupled oscillators chaotic or not .<br /><br />And of course hope that this discretization will produce dynamics that are approximations of the real system for a sufficiently long time .<br />As you are apparently expert in numerical treatements , you will intuitively see the difficulty at once - imagine that you want to study a convergence of an algorithm depending on the grid size but you don't know the function that has to be represented by the algorithm .<br /><br />Actually it is even worse :)<br />On every <b>node</b> of the grid you have an unknown and different function of time and you are supposed to combine them in some way to obtain a known result and this independently of the grid size .<br />And the amusing part is that if you divide the grid step by 2 , not only you need X times more functions but it can't be excluded that the functions that worked with the previous size must be changed too .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-400802615615264152011-01-25T11:33:41.979-05:002011-01-25T11:33:41.979-05:00Thanks again for your thoughtful comments Tom; I r...Thanks again for your thoughtful comments Tom; I really appreciate your contributions.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-46676863519529237842011-01-25T09:30:30.618-05:002011-01-25T09:30:30.618-05:00Very informed and excellent post !
I just found it...Very informed and excellent post !<br />I just found it following a kind mention by Dan Hughes and will perhaps comment more in detail later .<br />It resonates with a post I made here a year ago or so , pointing out that the ergodicity was the single most important issue in climate matters .<br />It is largely the same thing as what Lorenz said in the quotes above .<br /><br />You wrote<br /><i>"Lorenz would seem to agree, “most climatic elements, and certainly climatic means, are not predictable in the first sense at infinite range, since a non-periodic series cannot be made periodic through averaging [1].” We’re not going to just take his word on it. We’ll see if we can demonstrate this with our toy model. </i><br /><br />So just for fun a much faster demonstration in 3 lines :)<br /><br />A solution is chaotic if the trajectories diverge exponentially.<br />That is : f(X0+dX0,t) - f(X0,t) = g(X0).exp(L.t) where :<br />X0 are the initial conditions , f is the chaotic solution , g is some function of IC only and L>0 (Lyapounov coefficient) .<br />Let's define the time average TA(X0,t) = 1/T Integral t to T + t (f(X0,x)dx).<br /><br />T is some arbitrary averaging period<br />Then<br />TA(X0+dX0,t) - TA(X0,t) = 1/T Integral t to T + t [f(X0+dX0,x) - f(X0,x)]dx = 1/T Integral t to T + t [g(X0).exp(L.x)]dx = <br />Something(T,X0,L).exp(L.t)<br /><br />The trajectories of the average diverge exponentially for any arbitrary averaging period T .<br />Therefore if a solution is chaotic , all its time averages are chaotic too .<br />QED .TomVonknoreply@blogger.comtag:blogger.com,1999:blog-5822805028291837738.post-32601543502547594682011-01-24T16:24:01.107-05:002011-01-24T16:24:01.107-05:00Dr Pielke has a post up about recent news coverage...Dr Pielke has a <a href="http://pielkeclimatesci.wordpress.com/2011/01/24/confessions-in-the-news-on-the-predictability-of-the-climate-system/" rel="nofollow">post up</a> about recent news coverage on predictability. He was also kind enough to link this post on one of <a href="http://pielkeclimatesci.wordpress.com/2011/01/23/recommended-reading-on-whether-climate-is-an-initial-value-problem/" rel="nofollow">his previous entries</a>.jstultshttps://www.blogger.com/profile/03506970399027046387noreply@blogger.com