I studied nonlinear PDEs in grad school. My advisor, Ralph Showalter, would remind us occasionally what ‘nonlinear’ means.
“Nonlinear” is not a hypothesis but the lack of a hypothesis.
He meant a couple things by this. First, when people say “nonlinear,” they often mean “not necessarily linear.” That is, they use “nonlinear” as a generalization of linear. If a statement doesn’t hold for linear equations, it can’t hold more generally. So try the linear case first.
Second, and more importantly, you usually have to specify in what way an equation is nonlinear before you can say anything useful. If you’re not assuming linearity, what are you assuming? Maybe you need to assume a function is convex. Or maybe you need to assume an upper or lower bound on a function’s growth. In any case, focus on what you are assuming rather than what you are not assuming, and make your assumptions explicit.
Related post: Three views of differential equations
Here are a couple things I thought of after publishing this post. Analogs in geometry and computing.
In geometry “non-Euclidean” is analogous to “nonlinear.” Non-Euclidean geometry isn’t completely unlike Euclidean geometry. It differs in one particular way: it rejects Euclid’s fifth postulate. Geometry distinguishes between “not-necessarily” Euclidean geometry and non-Euclidean geometry. The former is called “absolute geometry” and simply does not use the fifth postulate. Truly non-Euclidean geometry doesn’t ignore the fifth postulate; it replaces it with a different postulate. You can’t really study “non-Euclidean geometry” in the abstract. You have to specify what you’re assuming in place of Euclid’s fifth postulate and hence what flavor of non-Euclidean geometry you’re committing to such as hyperbolic or elliptic.
In computation you’ll hear of “ustructured data.” It can’t be completely unstructured or else it wouldn’t be data. By “unstructured” people usually mean “not stored in a relational database.” Or not necessarily stored in a relational database; software libraries for working with unstructured data usually accept data from relational databases but also from other sources.
There are varying degrees of “unstructuredness.” Some unstructured data is highly structured, but not in a relational format. Or the structure might be fuzzy. Instead of this always means that, the rules may say that this probably means that. But you have to assume some kind of structure. Software vendors are often vague on just what data their products can analyze. “Any kind” is not an honest answer.
Good point, and it’s a confusing use of terms – when people say “not X” to mean “not necessarily X”.
I often see the term “nonclassical” or “nonstandard” which is similarly inspecific.
When someone says “highly nonlinear,” it’s not clear what they mean. They may mean a function has a large departure from linearity in the sense of, for example, having a large second derivative.
But they may mean a large departure from the simplicity of a linear expression, i.e. they may use “highly nonlinear” to mean “messy” even though the function is in fact nearly linear as measured by some norm.
John: I’ve heard that said about fluid dynamics, I think on one of LANL’s public-facing papers on turbulence. (I guess as you change the Rayleigh number and perhaps one other parameter, “the nonlinearity increases” in some sense.)
Maybe the fairest / most universally understandable reason to say a (ℝ→ℝ) function ƒ is “highly” versus “sort of” nonlinear would be to measure the L&sub2; norm of the difference between ƒ and some affine function that reasonably fits at least part of the data.
Does a big fourth derivative make something “more nonlinear” than a big third derivative? Should we count
sin(x)as “highly nonlinear” more thanexp(-x)sin(x)? I think people talk past each other when they use the word “linear” as well (think how people describe art/music, or strategy, or …normal stuff… versus linear ODE’s)What do you think?
I think it’s safe to say something is “highly nonlinear” if the nonlinearity results in qualitatively different behavior than the corresponding linear system. Turbulence is a good example of such behavior.
How you quantify degree of nonlinearity depends on context. A sine wave might be considered highly nonlinear in some contexts because it oscillates. In other contexts it may be considered practically constant because it’s bounded. It depends on what aspect of linearity is important to you.
As far as what derivative you should look at, it depends on what you’re trying to do. There are tons of PDE theorems relating norms of inputs and outputs. The more you want in your output, the more you need to require of your input. For example, simply proving existence of a solution may only require than an input be integrable, but proving additional regularity of a solution may require additional regularity assumptions on the inputs. The way this is done is by Sobolev norms. You bound a Sobolev norm on your solution by another Sobolev norm on your input.
Sobolev norms consider all derivatives up to a certain order. So you’d seldom look at just the fourth derivative, for example. You’d look at all derivatives up to and including the fourth order.
Interesting. Is there an orderly progression of “increasingly turbulent” flows on the same system, ordered by Sobolev norms?
Thanks, interesting points, particularly re Euclid’s 5th (had to look that up and think about it). I don’t get to (have to?) deal with nonlinearity much in my work but it’s interesting to consider.
John, thanks for the thought provoking post. I was wondering if you might explain more your claim: “If a statement doesn’t hold for linear equations, it can’t hold more generally.”
Is this a universal mathematical truth? That would seem to make theorizing about complex systems a bit too easy. How does it apply to statistical reasoning, especially with regard to models linear in variables?
John
This is great. We live in a nonlinear world and make do with liner approximations c.f. Lorenz
What’s your take on the other emerging non linear hypothesis that in regression http://www.sciencemag.org/content/334/6062/1518.abstract would love to see a blog post on this
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Well the first objection is definitely a reason it is not a hypothesis, although even with the second objection it can be used as a null hypothesis. There are situations in statistics where you would in fact test for linearity and confirm the null hypothesis that it is not linear, but in that case you are using “not linear” which I suppose is not the meaning of “nonlinear.” Would never hold in English grammar but is perfectly fine in mathematics I suppose.
Ran across this from Dan North: My maths lecturer at college used to say the term “non-linear algebra” was as useful as saying “non-elephant zoology.”
“Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.” – Stanislaw Ulam
There’s another aspect to the nonlinear versus linear discussion, I think, and that is continuity. If a function is continuous then no matter how ill-behaved it is, you can shrink a neighborhood small enough and get a linear approximation via Taylor. There are, of course, such beasts as functions which are discontinuous everywhere, like the Dirichlet function or Thomae’s function. Presumably, the characterizations and predictions of physical or social phenomena which act like these need another kind of descriptive mechanism, perhaps stochastic. There is a view which suggests doing that is a way of imposing a structure on them so you can take some kind of meaningful gradient, and have a plane or hyperplane.
Re: al castro; There could be practical reasons for such (input) restrictions such as ‘Physiology of animals I can carry in my truck’ or ‘Non-black hole physics’** or ‘Psychology of available patients not in my family’**.
** Of course you risk (perhaps imperceptible) fouling of the results if there is interaction between the excluded inputs and the remaining field of investigation.
Fun stuff!