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Interesting paper. I work in fluid dynamics and tend to agree that computational power often is wasted. To elaborate on a point not discussed in the paper: Man
by btrettel 8y ago
Interesting paper. I work in fluid dynamics and tend to agree that computational power often is wasted.
To elaborate on a point not discussed in the paper: Many methods used in practical engineering won't necessarily converge to the empirically correct answer as the computational cost increases, say, by decreasing the time step, increasing the number of computational particles, and/or refining a grid. This is true even if the uncertainty is quantified. Without this guarantee, additional computational expenditure could simply be wasted. The problem isn't the lack of computing power, it's the model.
One reason is that "reliable" physics like, for example, the Navier-Stokes equations for fluid flows, are not being used as the model because the computational cost for solving those equations directly is far too high. Instead lower cost models are being used which may be loosely based on the reliable model, but ultimately is not the same thing. A numerical implementation of the lower cost model might converge to something (which is not necessarily right) as the computational cost increases, but converging to the right answer is less likely than the reliable model. It depends on the reliability and generalizability of the lower cost model and how well it has been tuned.
(To those familiar with computational fluid dynamics, convergence is one reason to prefer LES to RANS, though it's worth pointing out that not all LES methods are convergent.)
A hypothesis I have is that computing resources have caused many scientists and engineers' theoretical abilities to atrophy. Perhaps they were never that good on average to begin with, but in my academic field I work in theory partly because I've found so much low hanging fruit. I don't see much of a point in making expensive computational models if applying basic regression techniques with a simple theory-based model is similarly or even more accurate. These simple models are much more efficient.
Also: I wasn't aware of attempts to bound the results. I attended an uncertainty quantification seminar before and asked one of the lecturers about this more generally and they seemed skeptical of its utility.
- nestorD 8y agoRegarding the bounding of numerical results, we currently have precise method that cannot be used on real size applications and methods that can be used at scale but tend to tell you that your result is between minus infinity and plus infinity. I have seen it used sucessfully in small math kernel and embedded systems but genuine application usually require deep modifications of the algorithms to get any useful result.
- snaky 8y agoAs Vladimir Arnold, who used to know a thing or two about fluid dynamics in particular, once said > At this point a special technique has been developed in mathematics. This technique, when applied to the real world, is sometimes useful, but can sometimes also lead to self-deception. This technique is called modelling. When constructing a model, the following idealisation is made: certain facts which are only known with a certain degree of probability or with a certain degree of accuracy, are considered to be "absolutely" correct and are accepted as "axioms". The sense of this "absoluteness" lies precisely in the fact that we allow ourselves to use these "facts" according to the rules of formal logic, in the process declaring as "theorems" all that we can derive from them. > It is obvious that in any real-life activity it is impossible to wholly rely on such deductions. The reason is at least that the parameters of the studied phenomena are never known absolutely exactly and a small change in parameters (for example, the initial conditions of a process) can totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible, no matter how much we develop computers and devices which record initial conditions. https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
- narimiran 8y ago>> a small change in parameters (for example, the initial conditions of a process) __can__ totally change the result. Say, for this reason a reliable long-term weather forecast is impossible and will remain impossible (emphasis mine) A small change in initial conditions can totally change the result in chaotic systems such as weather forecast, or a much simpler example — double pendulum. There are systems where a small change changes the results only slightly. E.g. a single pendulum.
- jeffreygoesto 8y agoSpeaking of bound results, interval arithmetic comes to my mind and the work of Ulrich Kulisch [0]. I've used it together with automatic differentiation with some remarkable results, but the problems were comparably small (approx. 40 dimensional optimization). [0] http://www.math.kit.edu/ianm2/~kulisch/de http://www.math.kit.edu/ianm2/~kulisch/de