arXiv subjects
Publications and source records attributed to David Lee.
We consider a shallow water model in a homogenization framework. For periodic topographies, Craig, Lannes and Sulem have established a consistency result under some non-resonance conditions. In the present contribution, we significantly relax the periodicity condition and treat general topographies under minimal assumptions.
We study the possibility of a gradual improvement as time progresses of the regularity of solutions to evolution problems of parabolic type driven by Lévy-type operators, not necessarily translation invariant. In the course of our analysis we study the equivalence between general smoothing effects and a family of logarithmic Sobolev inequalities. This equivalence allows us to identify a new type of regularization, strong hypercontractivity, characterized by the existence of a time at which solutions belong to every $L^p$ space with $p$ finite. It can also be used to prove logarithmic Sobolev inequalities in a context not previously seen in the literature. We then show that any purely nonlocal Lévy-type operator whose kernel is comparable to that of $\log(I-Δ)$ is strongly hypercontractive, but fails to be supercontractive and, consequently, also fails to be ultracontractive. Furthermore, in the translation-invariant case, we also prove that solutions get bounded eventually and start improving in differentiability right after doing so. Finally, we show that this behaviour only appears if the kernel defining the operator behaves as $|x-y|^{-N}$ for small interactions ($0^+$-order operators): more singular kernels yield instantaneous smoothing, while less singular ones do not produce any regularization.
In-context learning has established itself as an important learning paradigm for Large Language Models (LLMs). In this paper, we demonstrate that LLMs can learn encoding keys in-context and perform analysis directly on encoded representations. This finding enables lossless prompt compression via dictionary encoding without model fine-tuning: frequently occurring subsequences are replaced with compact meta-tokens, and when provided with the compression dictionary in the system prompt, LLMs correctly interpret these meta-tokens during analysis, producing outputs equivalent to those from uncompressed inputs. We present a compression algorithm that identifies repetitive patterns at multiple length scales, incorporating a token-savings optimization criterion that ensures compression reduces costs by preventing dictionary overhead from exceeding savings. The algorithm achieves compression ratios up to 80$\%$ depending on dataset characteristics. To validate that LLM analytical accuracy is preserved under compression, we use decompression as a proxy task with unambiguous ground truth. Evaluation on the LogHub 2.0 benchmark using Claude 3.7 Sonnet demonstrates exact match rates exceeding 0.99 for template-based compression and average Levenshtein similarity scores above 0.91 for algorithmic compression, even at compression ratios of 60$\%$-80$\%$. Additionally, compression ratio explains less than 2$\%$ of variance in similarity metrics, indicating that decompression quality depends on dataset characteristics rather than compression intensity. This training-free approach works with API-based LLMs, directly addressing fundamental deployment constraints -- token limits and API costs -- and enabling cost-effective analysis of large-scale repetitive datasets, even as data patterns evolve over time.
Clobber is an alternate-turn two-player game introduced in 2001 by Albert, Grossman, Nowakowski and Wolfe. The board is a graph with each node colored black (x), white (o), or empty (-). Player Left has black stones, player Right has white stones. On a turn, a player takes one of their stones that is adjacent to an opponent stone and clobbers the opponent's stone (replaces it with theirs). Whoever cannot move loses. Linear clobber is clobber played on a path, for example, one row of a Go board. In 2004 Albert et al. conjectured that, for every even-length alternating-color linear clobber position except oxoxox, the first player has a winning strategy. We prove their conjecture.
This work presents a novel adaptive framework for simultaneously estimating spacecraft attitude and sensor misalignment. Uncorrected star tracker misalignment can introduce significant pointing errors that compromise mission objectives in GPS-denied environments. To address this challenge, the proposed architecture integrates a Bayesian Multiple-Model Adaptive Estimation (MMAE) framework operating over an N x N x N 3D hypothesis grid. Each hypothesis employs a 9-state Multiplicative Extended Kalman Filter (MEKF) to estimate attitude, angular velocity, and gyroscope bias using TRIAD-based vector measurements. A key contribution is the development of a robust grid refinement strategy that uses hypothesis diversity and weighted-mean grid centering to prevent the premature convergence commonly encountered in classical, dominant model-based refinement triggers. Extensive Monte Carlo simulations demonstrate that the proposed method reduces the final misalignment RMSE relative to classical approaches, achieving arcsecond-level accuracy. The resulting framework offers a computationally tractable and statistically robust solution for in-flight calibration, enhancing the navigational autonomy of resource-constrained spacecraft.
Existence of the fundamental solution of the logarithmic Laplacian (in dimensions $d \geq 3$) was established by Huyuan Chen and Laurent Véron (2024). In this note, we present an alternative approach, based on a modification on the classical division problem. This is inspired by the theory of fundamental solutions by Malgrange and Ehrenpreis. Moreover, we give a variant of the Liouville theorem for the logarithmic Laplacian and give some further clarification regarding a conjecture posed by Chen and Véron regarding the behavior of solutions in dimensions 1 and 2.
In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.
We introduce a method of obtaining a higher order extension problem, á la Caffarelli-Silvestre, utilizing ideas from renormalization. Moreover, we give an alternative perspective of the recently developed extension problem for the logarithmic laplacian developed by Chen, Hauer and Weth (2023) [arXiv:2312.15689].
Moist thermodynamics is a fundamental driver of atmospheric dynamics across all scales, making accurate modeling of these processes essential for reliable weather forecasts and climate change projections. However, atmospheric models often make a variety of inconsistent approximations in representing moist thermodynamics. These inconsistencies can introduce spurious sources and sinks of energy, potentially compromising the integrity of the models. Here, we present a thermodynamically consistent and structure preserving formulation of the moist compressible Euler equations. When discretised with a summation by parts method, our spatial discretisation conserves: mass, water, entropy, and energy. These properties are achieved by discretising a skew symmetric form of the moist compressible Euler equations, using entropy as a prognostic variable, and the summation-by-parts property of discrete derivative operators. Additionally, we derive a discontinuous Galerkin spectral element method with energy and tracer variance stable numerical fluxes, and experimentally verify our theoretical results through numerical simulations.
Implicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners this typically involves the factorisation of the (approximate) Jacobian resulting from linearization of the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux form representation of the potential temperature on the Lorenz grid using mixed finite elements. This formulation allows for spatial discretisations that conserve both energy and potential temperature variance. By introducing the dry thermodynamic entropy as an auxiliary variable for the solution of the algebraic system, the resulting preconditioner is shown to have a similar block structure to an existing preconditioner for the material form transport of potential temperature on the Charney-Phillips grid. This new formulation is also shown to be more efficient and stable than both the material form transport of potential temperature on the Charney-Phillips grid, and a previous Helmholtz preconditioner for the flux form transport of density weighted potential temperature on the Lorenz grid for a 1D thermal bubble configuration. The new preconditioner is further verified against standard two dimensional test cases in a vertical slice geometry.
In this paper, we describe the wide-field spectroscopic survey telescope (WST) project. WST is a 12-metre wide-field spectroscopic survey telescope with simultaneous operation of a large field-of-view (3 sq. degree), high-multiplex (20,000) multi-object spectrograph (MOS), with both a low and high-resolution modes, and a giant 3x3 arcmin2 integral field spectrograph (IFS). In scientific capability, these specifications place WST far ahead of existing and planned facilities. In only 5 years of operation, the MOS would target 250 million galaxies and 25 million stars at low spectral resolution, plus 2 million stars at high resolution. Without need for pre-imaged targets, the IFS would deliver 4 billion spectra offering many serendipitous discoveries. Given the current investment in deep imaging surveys and noting the diagnostic power of spectroscopy, WST will fill a crucial gap in astronomical capability and work in synergy with future ground and space-based facilities. We show how it can address outstanding scientific questions in the areas of cosmology; galaxy assembly, evolution, and enrichment, including our own Milky Way; the origin of stars and planets; and time domain and multi-messenger astrophysics. WST's uniquely rich dataset may yield unforeseen discoveries in many of these areas. The telescope and instruments are designed as an integrated system and will mostly use existing technology, with the aim to minimise the carbon footprint and environmental impact. We will propose WST as the next European Southern Observatory (ESO) project after completion of the 39-metre ELT.
WST - Widefield Spectroscopic Telescope: We summarise the design challenges of instrumentation for a proposed 12m class Telescope that aims to provide a large (>2.5 square degree) field of view and enable simultaneous Multi-object (> 20,000 objects) and Integral Field spectroscopy (inner 3x3 arcminutes field of view), initially at visible wavelengths. For the MOS mode, instrumentation includes the fiber positioning units, fiber runs and the high (R~40,000) and low (R~3,000 - 4,000) resolution spectrographs. For the MUSE like Integral Field Spectrograph, this includes the relay from the Telescope Focal Plane, the multi-stage splitting and slicing and almost 150 identical spectrographs. We highlight the challenge of mass production at a credible cost and the issues of maintenance and sustainable operation.
The Mid-Infrared Instrument (MIRI) aboard the James Webb Space Telescope (JWST) provides the observatory with a huge advance in mid-infrared imaging and spectroscopy covering the wavelength range of 5 to 28 microns. This paper describes the performance and characteristics of the MIRI imager as understood during observatory commissioning activities, and through its first year of science operations. We discuss the measurements and results of the imager's point spread function, flux calibration, background, distortion and flat fields as well as results pertaining to best observing practices for MIRI imaging, and discuss known imaging artefacts that may be seen during or after data processing. Overall, we show that the MIRI imager has met or exceeded all its pre-flight requirements, and we expect it to make a significant contribution to mid-infrared science for the astronomy community for years to come.
Methods for upwinding the potential vorticity in a compatible finite element discretisation of the rotating shallow water equations are studied. These include the well-known anticipated potential vorticity method (APVM), streamwise upwind Petrov-Galerkin (SUPG) method, and a recent approach where the trial functions are evaluated downstream within the reference element. In all cases the upwinding scheme conserves both potential vorticity and energy, since the antisymmetric structure of the equations is preserved. The APVM leads to a symmetric definite correction to the potential enstrophy that is dissipative and inconsistent, resulting in a turbulent state where the potential enstrophy is more strongly damped than for the other schemes. While the SUPG scheme is widely known to be consistent, since it modifies the test functions only, the downwinded trial function formulation results in the advection of downwind corrections. Results of the SUPG and downwinded trial function schemes are very similar in terms of both potential enstrophy conservation and turbulent spectra. The main difference between these schemes is in the energy conservation and residual errors. If just two nonlinear iterations are applied then the energy conservation errors are improved for the downwinded trial function formulation, reflecting a smaller residual error than for the SUPG scheme. We also present formulations by which potential enstrophy is exactly integrated at each time level. Results using these formulations are observed to be stable in the absence of dissipation, despite the uncontrolled aliasing of grid scale turbulence. Using such a formulation and the APVM with a coefficient $\mathcal{O}(100)$ times smaller that its regular value leads to turbulent spectra that are greatly improved at the grid scale over the SUPG and downwinded trial function formulations with unstable potential enstrophy errors.
We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.
We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.
Atmospheric systems incorporating thermal dynamics must be stable with respect to both energy and entropy. While energy conservation can be enforced via the preservation of the skew-symmetric structure of the Hamiltonian form of the equations of motion, entropy conservation is typically derived as an additional invariant of the Hamiltonian system, and satisfied via the exact preservation of the chain rule. This is particularly challenging since the function spaces used to represent the thermodynamic variables in compatible finite element discretisations are typically discontinuous at element boundaries. In the present work we negate this problem by constructing our equations of motion via weighted averages of skew-symmetric formulations using both flux form and material form advection of thermodynamic variables, which allow for the necessary cancellations required to conserve entropy without the chain rule. We show that such formulations allow for stable simulations of both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.
Non-hydrostatic atmospheric models often use semi-implicit temporal discretisations in order to negate the time step limitation of explicitly resolving the fast acoustic and gravity waves. Solving the resulting system to machine precision using Newton's method is considered prohibitively expensive, and so the non-linear solver is typically truncated to a fixed number of iterations, using an approximate Jacobian matrix that is reassembled only once per time step. The present article studies the impact of using various third-order, four stage Rosenbrock-Wanner schemes, where integration weights are chosen to meet specific stability and order conditions, in comparison to a Crank-Nicolson time discretisation, as is done in the UK Met Office's LFRic model. Rosenbrock-Wanner schemes present a promising alternative on account of their ability to preserve their temporal order with only an approximate Jacobian, and may be constructed to be stiffly-stable, so as to ensure the decay of fast unresolved modes. These schemes are compared for the 2D rotating shallow water equations and the 3D compressible Euler equations at both planetary and non-hydrostatic scales and are shown to exhibit improved results in terms of their energetic profiles and stability. Results in terms of computational performance are mixed, with the Crank-Nicolson method allowing for longer time steps and faster time to solution for the baroclinic instability test case at planetary scales, and the Rosenbrock-Wanner methods allowing for longer time steps and faster time to solution for a rising bubble test case at non-hydrostatic scales.