Automatically Bounding the Taylor Remainder Series: Tighter Bounds and New Applications

We present a new algorithm for automatically bounding the Taylor remainder series. In the special case of a scalar function $f: \mathbb{R} \to \mathbb{R}$, our algorithm takes as input a reference point $x_0$, trust region $[a, b]$, and integer $k \ge 1$, and returns an interval $I$ such that $f(x) - \sum_{i=0}^{k-1} \frac {1} {i!} f^{(i)}(x_0) (x - x_0)^i \in I (x - x_0)^k$ for all $x \in [a, b]$. As in automatic differentiation, the function $f$ is provided to the algorithm in symbolic form, and must be composed of known atomic functions. At a high level, our algorithm has two steps. First, for a variety of commonly-used elementary functions (e.g., $\exp$, $\log$), we use recently-developed theory to derive sharp polynomial upper and lower bounds on the Taylor remainder series. We then recursively combine the bounds for the elementary functions using an interval arithmetic variant of Taylor-mode automatic differentiation. Our algorithm can make efficient use of machine learning hardware accelerators, and we provide an open source implementation in JAX. We then turn our attention to applications. Most notably, in a companion paper we use our new machinery to create the first universal majorization-minimization optimization algorithms: algorithms that iteratively minimize an arbitrary loss using a majorizer that is derived automatically, rather than by hand. We also show that our automatically-derived bounds can be used for verified global optimization and numerical integration, and to prove sharper versions of Jensen's inequality.Comment: Previous version has been split into 3 articles: arXiv:2308.00679, arXiv:2308.00190, and this articl

Numerical comparison of pipe-column-separation models

Results comparing six column-separation numerical models for simulating localized vapor cavities and distributed vaporous cavitation in pipelines are presented. The discrete vapor-cavity model (DVCM) is shown to be quite sensitive to selected input parameters. For short pipeline systems, the maximum pressure rise following column separation can vary markedly for small changes in wave speed, friction factor, diameter, initial velocity, length of pipe, or pipe slope. Of the six numerical models, three perform consistently over a broad number of reaches. One of them, the discrete gas-cavity model, is recommended for general use as it is least sensitive to input parameters or to the selected discretization of the pipeline. Three models provide inconsistent estimates of the maximum pressure rise as the number of reaches is increased; however, these models do give consistent results provided the ratio of maximum cavity size to reach volume is kept below 10%.Angus R. Simpson and Anton Bergan

Wakefield-Induced Ionization injection in beam-driven plasma accelerators

We present a detailed analysis of the features and capabilities of Wakefield-Induced Ionization (WII) injection in the blowout regime of beam driven plasma accelerators. This mechanism exploits the electric wakefields to ionize electrons from a dopant gas and trap them in a well-defined region of the accelerating and focusing wake phase, leading to the formation of high-quality witness-bunches [Martinez de la Ossa et al., Phys. Rev. Lett. 111, 245003 (2013)]. The electron-beam drivers must feature high-peak currents ($I_b^0\gtrsim 8.5~\mathrm{kA}$) and a duration comparable to the plasma wavelength to excite plasma waves in the blowout regime and enable WII injection. In this regime, the disparity of the magnitude of the electric field in the driver region and the electric field in the rear of the ion cavity allows for the selective ionization and subsequent trapping from a narrow phase interval. The witness bunches generated in this manner feature a short duration and small values of the normalized transverse emittance ($k_p\sigma_z \sim k_p\epsilon_n \sim 0.1$). In addition, we show that the amount of injected charge can be adjusted by tuning the concentration of the dopant gas species, which allows for controlled beam loading and leads to a reduction of the total energy spread of the witness beams. Electron bunches, produced in this way, fulfil the requirements to drive blowout regime plasma wakes at a higher density and to trigger WII injection in a second stage. This suggests a promising new concept of self-similar staging of WII injection in steps with increasing plasma density, giving rise to the potential of producing electron beams with unprecedented energy and brilliance from plasma-wakefield accelerators

Discovering Valuable Items from Massive Data

Suppose there is a large collection of items, each with an associated cost and an inherent utility that is revealed only once we commit to selecting it. Given a budget on the cumulative cost of the selected items, how can we pick a subset of maximal value? This task generalizes several important problems such as multi-arm bandits, active search and the knapsack problem. We present an algorithm, GP-Select, which utilizes prior knowledge about similarity be- tween items, expressed as a kernel function. GP-Select uses Gaussian process prediction to balance exploration (estimating the unknown value of items) and exploitation (selecting items of high value). We extend GP-Select to be able to discover sets that simultaneously have high utility and are diverse. Our preference for diversity can be specified as an arbitrary monotone submodular function that quantifies the diminishing returns obtained when selecting similar items. Furthermore, we exploit the structure of the model updates to achieve an order of magnitude (up to 40X) speedup in our experiments without resorting to approximations. We provide strong guarantees on the performance of GP-Select and apply it to three real-world case studies of industrial relevance: (1) Refreshing a repository of prices in a Global Distribution System for the travel industry, (2) Identifying diverse, binding-affine peptides in a vaccine de- sign task and (3) Maximizing clicks in a web-scale recommender system by recommending items to users

Pipeline column separation flow regimes

A generalized set of pipeline column separation equations is presented describing all conventional types of low-pressure regions. These include water hammer zones, distributed vaporous cavitation, vapor cavities, and shocks (that eliminate distributed vaporous cavitation zones). Numerical methods for solving these equations are then considered, leading to a review of three numerical models of column separation. These include the discrete vapor cavity model, the discrete gas cavity model, and the generalized interface vaporous cavitation model. The generalized interface vaporous cavitation model enables direct tracking of actual column separation phenomena (e.g., discrete cavities, vaporous cavitation zones), and consequently, better insight into the transient event. Numerical results from the three column separation models are compared with results of measurements for a number of flow regimes initiated by a rapid closure of a downstream valve in a sloping pipeline laboratory apparatus. Finally, conclusions are drawn about the accuracy of the modeling approaches. A new classification of column separation (active or passive) is proposed based on whether the maximum pressure in a pipeline following column separation results in a short-duration pressure pulse that exceeds the magnitude of the Joukowsky pressure rise for rapid valve closure.Anton Bergant and Angus R. Simpso

State Legislative Response to the Housing Crisis

Great public attention has recently been focused on the crisis in housing facing all major urban areas in this country. This article has been prepared to bring close attention to one segment of the hoped for solution-legislative action needed on the state level

Measuring fast electron spectra and laser absorption in relativistic laser-solid interactions using differential bremsstrahlung photon detectors

A photon detector suitable for the measurement of bremsstrahlung spectra generated in relativistically-intense laser-solid interactions is described. The Monte Carlo techniques used to back-out the fast electron spectrum and laser energy absorbed into fast electrons are detailed. A relativistically-intense laser-solid experiment using frequency doubled laser light is used to demonstrate the effective operation of the detector. The experimental data was interpreted using the 3-spatial-dimension Monte Carlo code MCNPX (Pelowitz 2008), and the fast electron temperature found to be 125 keV

CD25 expression distinguishes functionally distinct alloreactive CD4+ CD134+ (OX40+) T-cell subsets in acute graft-versus-host disease

AbstractCD134 (OX40) is expressed on activated CD4+ donor T cells in allogeneic stem cell transplant recipients with acute graft-versus-host disease. The data presented here reveal that differential expression of CD25 by CD4+ CD134+ T cells allows separation of these activated cells into 2 phenotypically and functionally distinct alloreactive T-cell subsets. These subsets exhibit distinct tissue associations, with CD4+ CD134+ CD25− T cells preferentially found in lymphoid tissues and CD4+ CD134+ CD25+ T cells located in lymphoid tissues and inflamed extralymphoid tissues. The CD25− T-cell subset exhibited potent proliferative responses to both concanavalin A and allogeneic host leukocytes. By contrast, the CD25+ T-cell subset proliferated minimally in response to either treatment and inhibited alloantigen-induced proliferation of the CD25− subset. Proliferative unresponsiveness associated with the CD25+ T-cell subset did not extend to cytokine secretion. When stimulated with alloantigen, both CD4+ CD134+ T-cell subsets responded by secreting interferon-γ and interleukin (IL)-10, and neither T-cell subset produced detectable levels of IL-2 or IL-4. Three-day treatment of the CD25+ T-cell subset with IL-2 restored the proliferative responsiveness of these cells to host alloantigens, suggesting that the proliferative unresponsiveness associated with this T-cell subset reflected a requirement for IL-2. The preferential tissue associations and distinct functional properties associated with these separable alloreactive CD4+ CD134+ T-cell subsets suggest that they participate differentially in clinical graft-versus-host disease