Category Archives: Analysis

Navier-Stokes Fluid Computers


Smart fluid

Terry Tao posted a very intriguing post on the Navier-Stokes equation, based on a recently uploaded paper Finite time blowup for an averaged three-dimensional Navier-Stokes equation.

The paper proved a remarkable negative answer for the regularity conjecture for a certain variants of the NS equations, namely (or, perhaps, more precisely) the main theorem demonstrates finite time blowup for an averaged Navier-Stokes equation. (This already suffices to show that certain approaches for a positive answer to the real problem are not viable.) The introduction ends with the following words.

“This suggests an ambitious (but not obviously impossible) program (in both senses of
the word) to achieve the same e ffect for the true Navier-Stokes equations, thus obtaining a negative answer to Conjecture 1.1 (the regularity conjecture for 3D NS equation)… Somewhat analogously to how a quantum computer can be constructed from the laws of quantum mechanics [Here Tao links to Benio ff’s 1982 paper: “Quantum mechanical Hamiltonian models of Turing machines,”], or a Turing machine can be constructed from cellular automata such as “Conway’s Game of Life” , one could hope to design logic gates entirely out of ideal fluid (perhaps by using suitably shaped vortex sheets to simulate the various types of physical materials one would use in a mechanical computer). If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid that, when it runs, behaves qualitatively like the blowup solution used to establish Theorem 1.4.[The paper’s main theorem] Note that such replicators, as well as the related concept of a universal constructor, have been built within cellular automata such as the “Game of Life.”

Once enough logic gates of ideal fluid are constructed, it seems that the main difficulties in executing the above program are of a “software engineering” nature, and would be in principle achievable, even if the details could be extremely complicated in practice. The main mathematical difficulty in executing this “fluid computing” program would thus be to arrive at (and rigorously certify) a design for logical gates of inviscid fluid that has some good noise tolerance properties. In this regard, ideas from quantum computing (which faces a unitarity constraint somewhat analogous to the energy conservation constraint for ideal fluids, albeit with the key di fference of having a linear evolution rather than a nonlinear one) may prove to be useful. (Emphasis mine.)

Interesting idea!

And what Tao does go well beyond an idea, he essentially implement this program for a close relative of the NS equation!  I am not sure if universal computing is established for these systems but the proofs of the finite-time blow up theorem certainly uses some computational-looking gadget, and also as Terry explains some form of fault-tolerance.

Somewhat related ideas (unsupported by any results, of course,) appeared in the seventh post “Quantum repetition” of my debate with Aram Harrow on quantum computing.  (See, e.g., this remark, and this one, and this one.) The thread also contains interesting links, e.g. to Andy Yao’s paper “Classical physics and the Curch-Turing Thesis.”  In addition to the interesting question:

Does the NS-equation in three-dimension supports universal (classical) computation,

we can also ask what about two-dimensions?

Can NS-equations in two dimension be approximated in any scale by bounded depth circuits?

One general question suggested there was the following: “It can be of interest (and perhaps harder compared to the quantum case) to try to describe classical evolutions that do not enable/hide fault tolerance and (long) computation.”

Another interesting comment by Arie Israel is: “I was surprised to learn that experimental fluid mechanics people had thought of this analogy before. Apparently the key name is ‘Fluidics’ and those ideas date back at least to the sixties.”


Update: Here is the first paragraph from a nice article by  Erica Klarreich entitled A Fluid New Path in Grand Math Challenge on this development in Quanta Magazine:

In Dr. Seuss’s book “The Cat in the Hat Comes Back,” the Cat makes a stain he can’t clean up, so he calls upon the help of Little Cat A, a smaller, perfect replica of the Cat who has been hiding under the Cat’s hat. Little Cat A then calls forth Little Cat B, an even smaller replica hidden under Little Cat A’s hat. Each cat in turn lifts his hat to reveal a smaller cat who possesses all the energy and good cheer of the original Cat, just crammed into a tinier package. Finally, Little Cat Z, who is too small to see, unleashes a VOOM like a giant explosion of energy, and the stain disappears.

And here is a follow up post on Tao’s blog (and a few more II, III), and a post on Shtetl Optimized.

The flip side

Update (June 14): It is worth noting that while the purpose of Tao’s program is to show finite-time blow up of the 3D Navier Stokes equations (as is often the case) these lines of ideas can potentially be useful also toward a positive solution of the regularity conjectures. Specifically, one can try to show that 3D Navier-Stokes equations do not support universal classical computation and even more specifically do not support classical fault-tolerance and error correction. Also here some analogy with quantum computation can be useful: It is expected that for adiabatic processes computation requires “spectral gap” and that gapped evolutions with local Hamiltonians support only bounded depth computation. Something analogous may apply to NS equations in bounded dimensions.

There are many caveats, of course,  the quantum results are not proved for D>1, NS equations are non-linear which weakens the analogy, and showing that the evolution does not support computation does not imply, as far as we know, regularity.

Three more remarks: 1) On the technical level an important relevant technical tool for the results on gapped systems with local Hamiltonians is the Lieb-Robinson inequality. (See, e.g. this review paper.)  2) for classical evolutions a repetition mechanism (or the “majority function”) seems crucial for robust computation and it will be interesting specifically to test of 3D Navier-stokes support it; 3) If computation is not possible beyond bounded depth this fact may lead to additional conserved quantities for NS, beyond the classical ones. (One more, June 28): It looks to me that the crucial question is if NS equations only support bounded computation or not. So this distinction captures places where circuit complexity gives clear mathematical distinctions.

Real Analysis Introductory Mini-courses at Simons Institute

The Real Analysis ‘Boot Camp’ included three excellent mini-courses.

Inapproximability of Constraint Satisfaction Problems (5 lectures)
Johan Håstad (KTH Royal Institute of Technology)

(Lecture I, Lecture II, Lecture III, Lecture IV, Lecture V)


Unlike more traditional ‘boot camps’ Johan rewarded answers and questions by chocolates (those are unavailable for audience of the video).

Starting from the PCP-theorem (which we will take as given) we show how to design and analyze efficient PCPs for NP-problems. We describe how an efficient PCP using small amounts of randomness can be turned into an inapproximability result for a maximum constraint satisfaction problem where each constraint corresponds to the acceptance criterion of the PCP. We then discuss how to design efficient PCPs with perfect completeness in some interesting cases like proving the hardness of approximating satisfiable instances of 3-Sat.

We go on to discuss gadget construction and how to obtain optimal reductions between approximation problems. We present Chan’s result on how to take products of PCPs to get hardness for very sparse CSPs and give some preliminary new results using these predicates as a basis for a gadget reduction.

Finally we discuss approximation in a different measure, and in particular the following problem. Given a (2k+1)-CNF formula which admits an assignment that satisfies k literal in each clause, is it possible to efficiently find a standard satisfying assignment?

Analytic Methods for Supervised Learning​ (4 lectures)
Adam Klivans (University of Texas, Austin)

(Lecture I, Lecture II, Lecture III, Lecture IV) additional related lecture by Adam on Moment matching polynomials.

In this mini-course we will show how to use tools from analysis and probability (e.g., contraction, surface area and limit theorems) to develop efficient algorithms for supervised learning problems with respect to well-studied probability distributions (e.g., Gaussians). One area of focus will be understanding the minimal assumptions needed for convex relaxations of certain learning problems (thought to be hard in the worst-case) to become tractable.

Introduction to Analysis on the Discrete Cube (4 lectures)
Krzysztof Oleszkiewicz (University of Warsaw)

(Lecture I, Lecture II, Lecture III, Lecture IV) Here are the slides for the lecture which contain material for 1-2 additional lectures.

The basic notions and ideas of analysis on the discrete cube will be discussed, in an elementary and mostly self-contained exposition. These include the Walsh-Fourier expansion, random walk and its connection to the heat semigroup, hypercontractivity and related functional inequalities, influences, the invariance principle and its application to the Majority is Stablest problem. The mini-course will also contain some other applications and examples, as well as several open questions.

The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastava

…while we keep discussing why mathematics is possible…

The news

Adam Marcus, Dan Spielman, and Nikhil Srivastava posted a paper entitled “Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem,” where they prove the 1959 Kadison-Singer conjecture.

(We discussed part I of “interlacing families” in this post about new Ramanujan graphs.  Looks like a nice series.)

The Kadison-Singer Conjecture

The Kadison-Singer conjecture refers to a positive answer to a question posed by Kadison and Singer: “They asked ‘whether or not each pure state of \cal B is the extension of some pure state of some maximal abelian algebra’ (where \cal B is the collection of bounded linear transformations on a Hilbert space.”) I heard about this question in a different formulation known as the “Bourgain-Tzafriri conjecture” (I will state it below) and the paper addresses a related well known discrepancy formulation by Weaver. (See also Weaver’s comment on the appropriate “quantum” formulation of the conjecture.)

Updates: A very nice post on the relation of the Kadison-Singer Conjecture  to foundation of quantum mechanics is in this post in  Bryan Roberts‘ blog Soul Physics. Here is a very nice post on the mathematics of the conjecture with ten interesting comments on the proof by Orr Shalit, and another nice post on Yemon Choi’s blog and how could I miss the very nice post on James Lee’s blog.. Nov 4, 2013: A new post with essentially the whole proof appeared on Terry tao’s blog, Real stable polynomials and the Kadison Singer Problem.

Update: A very nice blog post on the new result was written by  Nikhil Srivastava on “Windows on theory.” It emphasizes the discrapancy-theoretic nature of the new result, and explains the application for partitioning graphs into expanders.

The Bourgain-Tzafriri theorem and conjecture

Let me tell again (see this post about Lior, Michael, and Aryeh where I told it first) about a theorem of Bourgain and Tzafriri related to the Kadison-Singer conjecture.

Jean Bourgain and Lior Tzafriri considered the following scenario: Let a > 0 be a real number. Let A be a n \times n matrix with norm 1 and with zeroes on the diagonal. An s by s principal minor M is “good” if the norm of M is less than a.

Consider the following hypergraph F:

The vertices correspond to indices {}[n]=\{1,2,\dots,n\}. A set S \subset [n] belongs to F if the S \times S sub-matrix of M is good.

Bourgain and Tzafriri showed that for every a > 0 there is C(a) > 0 so that for every matrix A we can find S \in F so that |S| \ge C(a)n.

Moreover, they showed that for every nonnegative weights w_1,w_2,\dots w_n there is S \in F so that the sum of the weights in S is at least C(a) times the total weight. In other words, (by LP duality,) the vertices of the hypergraph can be fractionally covered by C(a) edges.

The “big question” is if there a real number C'(a) > 0 so that for every matrix M, {}[n] can be covered by C'(a) good sets. Or, in other words, if the vertices of F can be covered by C'(a) edges. This question is known to be equivalent to an old conjecture by Kadison and Singer (it is also known as the “paving conjecture”). In view of what was already proved by Bourgain and Tzafriri what is needed is to show that the covering number is bounded from above by a function of the fractional covering number. So if you wish, the Kadison-Singer conjecture had become a statement about bounded integrality gap. Before proving the full result, Marcus, Spielman and Srivastava gave a new proof of the Bourgain-Tzafriti theorem.

Additional references:

KADISON-SINGER MEETS BOURGAIN-TZAFRIRI by PETER G. CASAZZA, ROMAN VERSHYNIN,  The Kadison-Singer Problem in Mathematics and Engineering: A Detailed Account pdf, and many other recent publications by Pete Casazza.

A Really Nice Talk About PDE, Numerics (and Pyramids)

My previous post recommended a really nice talk by Jonathan Israel about human rights. Here is the link again. It is very recommended. Actually, I was in the audience and after the lecture, at the reception,  I came to the lecturer to compliment him for the talk, except that I approached a different, rather similar looking, person.

I am not going to make it a habit on this blog to recommend good videotaped lectures available on the Internet (unless this new Genre will lead to a sky-rocketing rating), but, just one more time, let me recommend you a really really nice lecture. It is a lecture with a scary title High order FEM-approximation on pyramids, by Nilima Nigam. Click on the picture to go directly to the lecture or click here. (And if you want to see the picture enlarged, click here.)

The first part is about “What is a PDE,” and it is an eye opener for people with little familiarity of PDEs (like me) and perhaps also an eye opener for PDE professionals on how to introduce them. And then it goes on to explain what finite element methods are, and about the recent finite element exterior calculus.