arXiv subjects

Elizabeth Hawkins

Publications and source records attributed to Elizabeth Hawkins.

7 recordsLinked to original sources

arXiv2026

The sum-product problem for small sets II

We establish that every set of $k=10$ natural numbers determines at least $30$ distinct pairwise sums or at least $30$ distinct pairwise products, as well as the analogous result for $k=11$ and at least $34$ sums/products, with sharpness uniquely (up to scaling) exhibited by $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18\}$ and $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24\}$, respectively. This extends previous work of the fifth author with Clevenger, Havard, Heard, Lott, and Wilson, which established the corresponding thresholds for $k\leq 9$. Included is a classification result for sets of $10$ real numbers (resp. positive real numbers) determining at most $29$ pairwise sums (resp. pairwise products) that do not contain $8$ elements of any single arithmetic progression (resp. geometric progression), as well as some observations controlling additive quadruples in small subsets of two-dimensional generalized geometric progressions.

math.CO↗

arXiv2025

Notes and computations on forbidden differences

We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that differ by an element of $X$? The set of forbidden differences, $X$, is called \textit{intersective} if $D(X,N)=o(N)$, with the most well-studied examples being $X=S=\{n^2: n\in \mathbb{N}\}$ and $X=\mathcal{P}-1=\{p-1: p\text{ prime}\}$. In addition to some new results, including exact formulas and estimates for $D(X,N)$ in some non-intersective cases like $X=\mathcal{P}$ and $X=S+k$, $k\in \mathbb{N}$, we also provide a comprehensive survey of known bounds and extensive computational data. In particular, we utilize an existing algorithm for finding maximum cliques in graphs to determine $D(S,N)$ for $N\leq 300$ and $D(\mathcal{P}-1,N)$ for $N\leq 500$. None of these exact values appear previously in the literature.

math.NT↗

arXiv2025

On the choice of optimization norm for Anderson acceleration of the Picard iteration for Navier-Stokes equations

While the most recent Anderson acceleration (AA) convergence theory [Pollock et al, {\it IMA Num. An.}, 2021] requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations the $\ell^2$ norm is perhaps the most common choice. Unfortunately, so far there is little research done regarding this discrepancy which might reveal when it is fine to use $\ell^2$. To address this issue, we consider AA applied to the Picard iteration for the Navier-Stokes equations (NSE) with varying choices of the AA optimization norm. We first prove a sharpened and generalized convergence estimate for depth $m$ AA-Picard for the NSE with the $H^1_0$ AA optimization norm by using a problem-specific analysis, utilizing a sharper treatment of the nonlinear terms than previous AA-Picard convergence studies, removing a small data assumption, and developing new AA term identities in the NSE nonlinear term estimates. Next, we prove a convergence result for when $L^2$ is used as the AA optimization norm, and this estimate is found to be very similar to that of the $H^1_0$ case. While no analogous theory seems possible for the $\ell^2$ norm, several numerical tests were run to compare AA-Picard convergence with varying choices of AA optimization norm. These tests revealed that convergence behavior was always similar for $L^2$ and $H^1_0$ and {\it usually but not always} similar for $\ell^2$: on a test problem for channel flow past a cylinder with coarser meshes, convergence of AA-Picard using $\ell^2$ performs significantly worse than using $L^2$ and $H^1_0$.

math.NA↗

arXiv2025

Accelerating convergence of a natural convection solver by continuous data assimilation

The Picard iteration for the Boussinesq model of natural convection can be an attractive solver because it stably decouples the fluid equations from the temperature equation (for contrast, the Newton iteration does not stably decouple). However, the convergence of Picard for this system is only linear and slows as the Rayleigh number increases, eventually failing for even moderately sized Rayleigh numbers. We consider this solver in the setting where sparse solution data is available, e.g. from data measurements or solution observables, and enhance Picard by incorporating the data into the iteration using a continuous data assimilation (CDA) approach. We prove that our approach scales the linear convergence rate by $H^{1/2}$, where $H$ is the characteristic spacing of the measurement locations. This implies that when Picard is converging, CDA will accelerate convergence, and when Picard is not converging, CDA (with enough data) will enable convergence. In the case of noisy data, we prove that the linear convergence rate of the nonlinear residual is similarly scaled by $H^{1/2}$ but the accuracy is limited by the accuracy of the data. Several numerical tests illustrate the effectiveness of the proposed method, including when the data is noisy. These tests show that CDA style nudging adapted to an iteration (instead of a time stepping scheme) enables convergence at much higher $Ra$.

math.NA↗

arXiv2024

Anderson-Picard based nonlinear preconditioning of the Newton iteration for non-isothermal flow simulations

We propose, analyze, and test a nonlinear preconditioning technique to improve the Newton iteration for non-isothermal flow simulations. We prove that by first applying an Anderson accelerated Picard step, Newton becomes unconditionally stable (under a uniqueness condition on the data) and its quadratic convergence is retained but has less restrictive sufficient conditions on the Rayleigh number and initial condition's accuracy. Since the Anderson-Picard step decouples the equations in the system, this nonlinear preconditioning adds relatively little extra cost to the Newton iteration (which does not decouple the equations). Our numerical tests illustrate this quadratic convergence and stability on multiple benchmark problems. Furthermore, the tests show convergence for significantly higher Rayleigh number than both Picard and Newton, which illustrates the larger convergence basin of Anderson-Picard based nonlinear preconditioned Newton that the theory predicts.

math.NA↗

arXiv2024

An optimization-based coupling of reduced order models with efficient reduced adjoint basis generation approach

Optimization-based coupling (OBC) is an attractive alternative to traditional Lagrange multiplier approaches in multiple modeling and simulation contexts. However, application of OBC to time-dependent problems has been hindered by the computational cost of finding the stationary points of the associated Lagrangian, which requires primal and adjoint solves. This issue can be mitigated by using OBC in conjunction with computationally efficient reduced order models (ROM). To demonstrate the potential of this combination, in this paper we develop an optimization-based ROM-ROM coupling for a transient advection-diffusion transmission problem. We pursue the ``optimize-then-reduce'' path towards solving the minimization problem at each timestep and solve reduced-space adjoint system of equations, where the main challenge in this formulation is the generation of adjoint snapshots and reduced bases for the adjoint systems required by the optimizer. One of the main contributions of the paper is a new technique for efficient adjoint snapshot collection for gradient-based optimizers in the context of optimization-based ROM-ROM couplings. We present numerical studies demonstrating the accuracy of the approach along with comparison between various approaches for selecting a reduced order basis for the adjoint systems, including decay of snapshot energy, average iteration counts, and timings.

cs.CE↗

arXiv2023

Removing splitting/modeling error in projection/penalty methods for Navier-Stokes simulations with continuous data assimilation

We study continuous data assimilation (CDA) applied to projection and penalty methods for the Navier-Stokes (NS) equations. Penalty and projection methods are more efficient than consistent NS discretizations, however are less accurate due to modeling error (penalty) and splitting error (projection). We show analytically and numerically that with measurement data and properly chosen parameters, CDA can effectively remove these splitting and modeling errors and provide long time optimally accurate solutions.

math.NA↗