Reflections on the Millennium Problems
In this essay I reflect on the status of three of the Millennium Prize problems: the Riemann Hypothesis, P vs. NP, and the solvability of the Navier-Stokes equations.
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arXiv subjects
Lloyd N. Trefethen
Publications and source records attributed to Lloyd N. Trefethen.
$L^2$ and $L^\infty$ rational approximation
Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the $L^2$ best approximations, we employ a new formulation of TF-IRKA in barycentric form.
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Applications of AAA rational approximation
The AAA algorithm for rational approximation is employed to illustrate applications of rational functions all across numerical analysis.
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Numerical conformal mapping
Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $Ω$ in the complex plane ${\mathbb{C}}$ (assuming $Ω\ne {\mathbb{C}}$) can be mapped bijectively to the unit disk by an analytic function with nonvanishing derivative, as in Figure 1. If $Ω$ is doubly-connected, it can be mapped to a circular annulus $1<|z|<R$ for some $R$, called the conformal modulus, which is uniquely determined by $Ω$, as in Figure 2. If $Ω$ has connectivity higher than $2$, it can be mapped onto various canonical domains such as a disk with exclusions in the form of slits or smaller disks, as in Figure 3.
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Quadrature formulas from rational approximations
It is shown that quadrature formulas in many different applications can be derived from rational approximation of the Cauchy transform of a weight function. Since rational approximation is now a routine technology, this provides an easy new method to derive all kinds of quadrature formulas as well as fundamental insight into the mathematics of quadrature. Intervals or curves of quadrature nodes correspond to near-optimal branch cuts of the Cauchy transform.
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Computation of Zolotarev rational functions
An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^*$ of a given degree that are as small as possible on one set $E\subseteq\complex\cup\{\infty\}$ relative to their size on another set $F\subseteq\complex\cup\{\infty\}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).
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Numerical computation of the Schwarz function
An analytic function can be continued across an analytic arc $Γ$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in Γ$. We show how $S(z)$ can be computed with the AAA algorithm of rational approximation, an operation that is the basis of the AAALS method for solution of Laplace and related PDE problems in the plane. We discuss the challenge of computing $S(z)$ further away from from $Γ$, where it becomes multi-valued.
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Unbounded growth of band-limited functions
It is shown that a band-limited function bounded by 1 for negative x can grow arbitrarily fast for positive x.
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Rational approximation
Potential theory for rational approximation is reviewed by means of examples computed with the AAA algorithm.
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Evaluation of resonances: adaptivity and AAA rational approximation of randomly scalarized boundary integral resolvents
This paper presents a novel algorithm, based on use of rational approximants of a randomly scalarized boundary integral resolvent in conjunction with an adaptive search strategy and an exponentially convergent secant-method termination stage, for the evaluation of acoustic and electromagnetic resonances in open and closed cavities. The desired cavity resonances are obtained as the poles of associated rational approximants; both the approximants and their poles are obtained by means of the recently introduced AAA rational-approximation algorithm. In fact, the proposed resonance-search method applies to any nonlinear eigenvalue problem associated with a given function $F: U \to \mathbb{C}^{d\times d}$, wherein, denoting $F(k) = F_k$, a complex value $k$ is sought for which $F_kw = 0$ for some nonzero $w\in \mathbb{C}^d$. For the scattering problems considered in this paper, $F_k$ is taken to equal a spectrally discretized version of a Green function-based boundary integral operator at spatial frequency $k$. In all cases, the scalarized resolvent is given by an expression of the form $u^* F_k^{-1} v$, where $u,v \in \mathbb{C}^d$ are fixed random vectors. The proposed adaptive search strategy relies on use of a rectangular subdivision of the resonance search domain which is locally refined to ensure that all resonances in the domain are captured. The approach works equally well in the case in which the search domain is an interval of the real line, in which case the rectangles used degenerate into subintervals of the search domain. A variety of numerical results are presented, including comparisons with well-known methods based on complex contour integration, and a discussion of the asymptotics that result as open cavities approach closed cavities -- in all, demonstrating the accuracy provided by the method, for low- and high-frequency states alike.
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The first five years of the AAA algorithm
The AAA algorithm, introduced in 2018, computes best or near-best rational approximations to functions or data on subsets of the real line or the complex plane. It is much faster and more robust than previous algorithms for such problems and has been used in many applications since its appearance, including the numerical solution of Laplace, Poisson, and biharmonic PDE problems in irregular domains. AAA has also been extended in new directions and seems likely to be a tool of lasting importance in the future.
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Computation of 2D Stokes flows via lightning and AAA rational approximation
Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen (2022) recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the "LARS" algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.
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Polynomial and rational convergence rates for Laplace problems on planar domains
Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is nonconvex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates, and the theory of crowding of conformal maps.
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Resolution of singularities by rational functions
Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collaborators, based on exponentially clustering poles close to the singularities. Our results are obtained by augmenting the lightning approximation set with either a low-degree polynomial basis or poles clustering towards infinity, in order to obtain a robust approximation of the smooth behaviour of the function. This leads to a significant increase in the achievable accuracy as well as the convergence rate of the numerical scheme. For the approximation of $x^α$ on $[0,1]$, the optimal convergence rate as shown by Stahl in 1993 is now achieved simply by least-squares fitting.
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Lightning Helmholtz Solver
In this dissertation we have applied Trefethen and Gopal's Lightning Method to solve the Helmholtz equation in the exterior of two dimensional piecewise smooth domains. The background theory motivating the method is presented, and we explore the optimal method implementation for the unit square, which is subsequently used to give a guide on parameter selection for a general region. The behaviour of the computed solutions is verified to act in accordance with our intuition and current understanding of wave propagation, and we show that the wave decays to approximately 0 in the shadow region.
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AAA rational approximation on a continuum
AAA rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum AAA algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains; the latter two are equivalent by a Moebius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a AAA-Lawson option for improvement of a AAA approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain.
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Sigmoid functions and multiscale resolution of singularities
In this short, conceptual paper we observe that essentially the same mathematics applies in three contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions near singularities, and (3) $hp$ mesh refinement for solution of PDEs. The relationship of (1) and (2) is as simple as the change of variables $s = \log(x)$, and our informal mnemonic for this relationship is ``sigmoid = log(ratapprox).''
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Spectacularly large expansion coefficients in Müntz's theorem
Müntz's theorem asserts, for example, that the even powers $1, x^2, x^4,\dots$ are dense in $C([0,1])$. We show that the associated expansions are so inefficient as to have no conceivable relevance to any actual computation. For example, approximating $f(x)=x$ to accuracy $\varepsilon = 10^{-6}$ in this basis requires powers larger than $x^{280{,}000}$ and coefficients larger than $10^{107{,}000}$. We present a theorem establishing exponential growth of coefficients with respect to $1/\varepsilon$.
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