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In a metric space, a set of point sets of roughly the same size and an integer $k\geq 1$ are given as the input and the goal of data-distributed $k$-center is to find a subset of size $k$ of the input points as the set of centers to minimize the maximum distance from the input points to their closest centers. Metric $k$-center is known to be NP-hard which carries to the data-distributed setting. We give a $2$-approximation algorithm of $k$-center for sublinear $k$ in the data-distributed setting, which is tight. This algorithm works in several models, including the massively parallel computation model (MPC).

Elastic geophysical properties (such as P- and S-wave velocities) are of great importance to various subsurface applications like CO$_2$ sequestration and energy exploration (e.g., hydrogen and geothermal). Elastic full waveform inversion (FWI) is widely applied for characterizing reservoir properties. In this paper, we introduce $\mathbf{\mathbb{E}^{FWI}}$, a comprehensive benchmark dataset that is specifically designed for elastic FWI. $\mathbf{\mathbb{E}^{FWI}}$ encompasses 8 distinct datasets that cover diverse subsurface geologic structures (flat, curve, faults, etc). The benchmark results produced by three different deep learning methods are provided. In contrast to our previously presented dataset (pressure recordings) for acoustic FWI (referred to as OpenFWI), the seismic dataset in $\mathbf{\mathbb{E}^{FWI}}$ has both vertical and horizontal components. Moreover, the velocity maps in $\mathbf{\mathbb{E}^{FWI}}$ incorporate both P- and S-wave velocities. While the multicomponent data and the added S-wave velocity make the data more realistic, more challenges are introduced regarding the convergence and computational cost of the inversion. We conduct comprehensive numerical experiments to explore the relationship between P-wave and S-wave velocities in seismic data. The relation between P- and S-wave velocities provides crucial insights into the subsurface properties such as lithology, porosity, fluid content, etc. We anticipate that $\mathbf{\mathbb{E}^{FWI}}$ will facilitate future research on multiparameter inversions and stimulate endeavors in several critical research topics of carbon-zero and new energy exploration. All datasets, codes and relevant information can be accessed through our website at //efwi-lanl.github.io/

We consider the problem of computing a function of $n$ variables using noisy queries, where each query is incorrect with some fixed and known probability $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$ function of $n$ bits (where queries correspond to noisy readings of the bits) and the $\mathsf{MAX}$ function of $n$ real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of \[ (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} \] is both sufficient and necessary to compute both functions with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Compared to previous work, our results tighten the dependence on $p$ in both the upper and lower bounds for the two functions.

Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.

Let $\{\Lambda_n=\{\lambda_{1,n},\ldots,\lambda_{d_n,n}\}\}_n$ be a sequence of finite multisets of real numbers such that $d_n\to\infty$ as $n\to\infty$, and let $f:\Omega\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function defined on a domain $\Omega$ with $0<\mu_d(\Omega)<\infty$, where $\mu_d$ is the Lebesgue measure in $\mathbb R^d$. We say that $\{\Lambda_n\}_n$ has an asymptotic distribution described by $f$, and we write $\{\Lambda_n\}_n\sim f$, if \[ \lim_{n\to\infty}\frac1{d_n}\sum_{i=1}^{d_n}F(\lambda_{i,n})=\frac1{\mu_d(\Omega)}\int_\Omega F(f({\boldsymbol x})){\rm d}{\boldsymbol x}\qquad\qquad(*) \] for every continuous function $F$ with bounded support. If $\Lambda_n$ is the spectrum of a matrix $A_n$, we say that $\{A_n\}_n$ has an asymptotic spectral distribution described by $f$ and we write $\{A_n\}_n\sim_\lambda f$. In the case where $d=1$, $\Omega$~is a bounded interval, $\Lambda_n\subseteq f(\Omega)$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, B\"ottcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[\lambda_{1,n},\ldots,\lambda_{d_n,n}]$ and the vector of samples $[f(x_{1,n}),\ldots,f(x_{d_n,n})]$, i.e., \[ \lim_{n\to\infty}\,\max_{i=1,\ldots,d_n}|f(x_{i,n})-\lambda_{\tau_n(i),n}|=0, \qquad\qquad(**) \] where $x_{1,n},\ldots,x_{d_n,n}$ is a uniform grid in $\Omega$ and $\tau_n$ is the sorting permutation. We extend this result to the case where $d\ge1$ and $\Omega$ is a Peano--Jordan measurable set (i.e., a bounded set with $\mu_d(\partial\Omega)=0$). See the rest of the abstract in the manuscript.

In 1-equation URANS models of turbulence the eddy viscosity is given by $\nu_{T}=0.55l(x,t)\sqrt{k(x,t)}$ . The length scale $l$ must be pre-specified and $k(x,t)$ is determined by solving a nonlinear partial differential equation. We show that in interesting cases the spacial mean of $k(x,t)$ satisfies a simple ordinary differential equation. Using its solution in $\nu_{T}$ results in a 1/2-equation model. This model has attractive analytic properties. Further, in comparative tests in 2d and 3d the velocity statistics produced by the 1/2-equation model are comparable to those of the full 1-equation model.

Given a continuous definable function $f: S \to \mathbb{R}$ on a definable set $S$, we study sublevel sets of the form $S^f_t = \{x \in S: f(x) \leq t\}$ for all $t \in \mathbb{R}$. Using o-minimal structures, we prove that the Euler characteristic of $S^f_t$ is right continuous with respect to $t$. Furthermore, when $S$ is compact, we show that $S^f_{t+\delta}$ deformation retracts to $S^f_t$ for all sufficiently small $\delta > 0$. Applying these results, we also characterize the relationship between the concepts of Euler characteristic transform and smooth Euler characteristic transform in topological data analysis.

A new generalization of multiquadric functions $\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$, where $x\in\mathbb{R}^n$, $c\in \mathbb{R}$, $d\in \mathbb{N}$, is presented to increase the accuracy of quasi-interpolation further. With the restriction to Euclidean spaces of odd dimensionality, the generalization can be used to generate a quasi-Lagrange operator that reproduces all polynomials of degree $2d-1$. In contrast to the classical multiquadric, the convergence rate of the quasi-interpolation operator can be significantly improved by a factor $h^{2d-n-1}$, where $h>0$ represents the grid spacing. Among other things, we compute the generalized Fourier transform of this new multiquadric function. Finally, an infinite regular grid is employed to analyse the properties of the aforementioned generalization in detail.

We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$

This paper deals with the problem of numerically computing the roots of polynomials $p_k(x)$, $k=1,2,\ldots$, of degree $n=2^k-1$ recursively defined by $p_1(x)=x+1$, $p_k(x)=xp_{k-1}(x)^2+1$. An algorithm based on the Ehrlich-Aberth simultaneous iterations complemented by the Fast Multi-pole Method and the fast search of near neighbors of a set of complex numbers is provided. The algorithm, which relies on a specific strategy of selecting initial approximations, costs $O(n\log n)$ arithmetic operations per step. A Fortran 95 implementation is given and numerical experiments are carried out. Experimentally, it turns out that the number of iterations needed to arrive at numerical convergence is $O(\log n)$. This allows us to compute the roots of $p_k(x)$ up to degree $n=2^{24}-1$ in about 16 minutes on a laptop with 16 GB RAM, and up to degree $n=2^{28}-1$ in about 69 minutes on a machine with 256 GB RAM. The case of degree $n=2^{30}-1$ would require higher memory and higher precision to separate the roots. With a suitable adaptation of FMM to the limit of 256 GB RAM and by performing the computation in extended precision (i.e. with 10-byte floating point representation) we were able to compute all the roots in about two weeks of CPU time for $n=2^{30}-1$. From the experimental analysis, explicit asymptotic expressions of the real roots of $p_k(x)$ and an explicit expression of $\min_{i\ne j}|\xi_i^{(k)}-\xi_j^{(k)}|$ for the roots $\xi_i^{(k)}$ of $p_k(x)$ are deduced. The approach is effectively applied to general classes of polynomials defined by a doubling recurrence.