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We introduce a new technique called Drapes to enhance the sensitivity in searches for new physics at the LHC. By training diffusion models on side-band data, we show how background templates for the signal region can be generated either directly from noise, or by partially applying the diffusion process to existing data. In the partial diffusion case, data can be drawn from side-band regions, with the inverse diffusion performed for new target conditional values, or from the signal region, preserving the distribution over the conditional property that defines the signal region. We apply this technique to the hunt for resonances using the LHCO di-jet dataset, and achieve state-of-the-art performance for background template generation using high level input features. We also show how Drapes can be applied to low level inputs with jet constituents, reducing the model dependence on the choice of input observables. Using jet constituents we can further improve sensitivity to the signal process, but observe a loss in performance where the signal significance before applying any selection is below 4$\sigma$.

This thesis explores the application of Plane Wave Discontinuous Galerkin (PWDG) methods for the numerical simulation of electromagnetic scattering by periodic structures. Periodic structures play a pivotal role in various engineering and scientific applications, including antenna design, metamaterial characterization, and photonic crystal analysis. Understanding and accurately predicting the scattering behavior of electromagnetic waves from such structures is crucial in optimizing their performance and advancing technological advancements. The thesis commences with an overview of the theoretical foundations of electromagnetic scattering by periodic structures. This theoretical dissertation serves as the basis for formulating the PWDG method within the context of wave equation. The DtN operator is presented and it is used to derive a suitable boundary condition. The numerical implementation of PWDG methods is discussed in detail, emphasizing key aspects such as basis function selection and boundary conditions. The algorithm's efficiency is assessed through numerical experiments. We then present the DtN-PWDG method, which is discussed in detail and is used to derive numerical solutions of the scattering problem. A comparison with the finite element method (FEM) is presented. In conclusion, this thesis demonstrates that PWDG methods are a powerful tool for simulating electromagnetic scattering by periodic structures.

Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.

The Neumann--Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann--Neumann methods that have better convergence properties and require less computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov--Poincar\'e operators.

This paper presents a learnable solver tailored to solve discretized linear partial differential equations (PDEs). This solver requires only problem-specific training data, without using specialized expertise. Its development is anchored by three core principles: (1) a multilevel hierarchy to promote rapid convergence, (2) adherence to linearity concerning the right-hand side of equations, and (3) weights sharing across different levels to facilitate adaptability to various problem sizes. Built on these foundational principles, we introduce a network adept at solving PDEs discretized on structured grids, even when faced with heterogeneous coefficients. The cornerstone of our proposed solver is the convolutional neural network (CNN), chosen for its capacity to learn from structured data and its similar computation pattern as multigrid components. To evaluate its effectiveness, the solver was trained to solve convection-diffusion equations featuring heterogeneous diffusion coefficients. The solver exhibited swift convergence to high accuracy over a range of grid sizes, extending from $31 \times 31$ to $4095 \times 4095$. Remarkably, our method outperformed the classical Geometric Multigrid (GMG) solver, demonstrating a speedup of approximately 3 to 8 times. Furthermore, we explored the solver's generalizability to untrained coefficient distributions. The findings showed consistent reliability across various other coefficient distributions, revealing that when trained on a mixed coefficient distribution, the solver is nearly as effective in generalizing to all types of coefficient distributions.

We consider logistic regression including two sets of discrete or categorical covariates that are missing at random (MAR) separately or simultaneously. We examine the asymptotic properties of two multiple imputation (MI) estimators, given in the study of Lee at al. (2023), for the parameters of the logistic regression model with both sets of discrete or categorical covariates that are MAR separately or simultaneously. The proposed estimated asymptotic variances of the two MI estimators address a limitation observed with Rubin's type estimated variances, which lead to underestimate the variances of the two MI estimators (Rubin, 1987). Simulation results demonstrate that our two proposed MI methods outperform the complete-case, semiparametric inverse probability weighting, random forest MI using chained equations, and stochastic approximation of expectation-maximization methods. To illustrate the methodology's practical application, we provide a real data example from a survey conducted in the Feng Chia night market in Taichung City, Taiwan.

In this paper, a new two-relaxation-time regularized (TRT-R) lattice Boltzmann (LB) model for convection-diffusion equation (CDE) with variable coefficients is proposed. Within this framework, we first derive a TRT-R collision operator by constructing a new regularized procedure through the high-order Hermite expansion of non-equilibrium. Then a first-order discrete-velocity form of discrete source term is introduced to improve the accuracy of the source term. Finally and most importantly, a new first-order space-derivative auxiliary term is proposed to recover the correct CDE with variable coefficients. To evaluate this model, we simulate a classic benchmark problem of the rotating Gaussian pulse. The results show that our model has better accuracy, stability and convergence than other popular LB models, especially in the case of a large time step.

The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.

Due to their cost, experiments for inertial confinement fusion (ICF) heavily rely on numerical simulations to guide design. As simulation technology progresses, so too can the fidelity of models used to plan for new experiments. However, these high-fidelity models are by themselves insufficient for optimal experimental design, because their computational cost remains too high to efficiently and effectively explore the numerous parameters required to describe a typical experiment. Traditionally, ICF design has relied on low-fidelity modeling to initially identify potentially interesting design regions, which are then subsequently explored via selected high-fidelity modeling. In this paper, we demonstrate that this two-step approach can be insufficient: even for simple design problems, a two-step optimization strategy can lead high-fidelity searching towards incorrect regions and consequently waste computational resources on parameter regimes far away from the true optimal solution. We reveal that a primary cause of this behavior in ICF design problems is the presence of low-fidelity optima in distinct regions of the parameter space from high-fidelity optima. To address this issue, we propose an iterative multifidelity Bayesian optimization method based on Gaussian Process Regression that leverages both low- and high-fidelity modelings. We demonstrate, using both two- and eight-dimensional ICF test problems, that our algorithm can effectively utilize low-fidelity modeling for exploration, while automatically refining promising designs with high-fidelity models. This approach proves to be more efficient than relying solely on high-fidelity modeling for optimization.

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.