We present a specific-purpose globalized and preconditioned Newton-CG solver to minimize a metric-aware curved high-order mesh distortion. The solver is specially devised to optimize curved high-order meshes for high polynomial degrees with a target metric featuring non-uniform sizing, high stretching ratios, and curved alignment -- exactly the features that stiffen the optimization problem. To this end, we consider two ingredients: a specific-purpose globalization and a specific-purpose Jacobi-$\text{iLDL}^{\text{T}}(0)$ preconditioning with varying accuracy and curvature tolerances (dynamic forcing terms) for the CG method. These improvements are critical in stiff problems because, without them, the large number of non-linear and linear iterations makes curved optimization impractical. Finally, to analyze the performance of our method, the results compare the specific-purpose solver with standard optimization methods. For this, we measure the matrix-vector products indicating the solver computational cost and the line-search iterations indicating the total amount of objective function evaluations. When we combine the globalization and the linear solver ingredients, we conclude that the specific-purpose Newton-CG solver reduces the total number of matrix-vector products by one order of magnitude. Moreover, the number of non-linear and line-search iterations is mainly smaller but of similar magnitude.
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Electromagnetic (EM) body models designed to predict Radio-Frequency (RF) propagation are time-consuming methods which prevent their adoption in strict real-time computational imaging problems, such as human body localization and sensing. Physics-informed Generative Neural Network (GNN) models have been recently proposed to reproduce EM effects, namely to simulate or reconstruct missing data or samples by incorporating relevant EM principles and constraints. The paper discusses a Variational Auto-Encoder (VAE) model which is trained to reproduce the effects of human motions on the EM field and incorporate EM body diffraction principles. Proposed physics-informed generative neural network models are verified against both classical diffraction-based EM tools and full-wave EM body simulations.
This thesis is a corpus-based, quantitative, and typological analysis of the functions of Early Slavic participle constructions and their finite competitors ($jegda$-'when'-clauses). The first part leverages detailed linguistic annotation on Early Slavic corpora at the morphosyntactic, dependency, information-structural, and lexical levels to obtain indirect evidence for different potential functions of participle clauses and their main finite competitor and understand the roles of compositionality and default discourse reasoning as explanations for the distribution of participle constructions and $jegda$-clauses in the corpus. The second part uses massively parallel data to analyze typological variation in how languages express the semantic space of English $when$, whose scope encompasses that of Early Slavic participle constructions and $jegda$-clauses. Probabilistic semantic maps are generated and statistical methods (including Kriging, Gaussian Mixture Modelling, precision and recall analysis) are used to induce cross-linguistically salient dimensions from the parallel corpus and to study conceptual variation within the semantic space of the hypothetical concept WHEN.
Matching on a low dimensional vector of scalar covariates consists of constructing groups of individuals in which each individual in a group is within a pre-specified distance from an individual in another group. However, matching in high dimensional spaces is more challenging because the distance can be sensitive to implementation details, caliper width, and measurement error of observations. To partially address these problems, we propose to use extensive sensitivity analyses and identify the main sources of variation and bias. We illustrate these concepts by examining the racial disparity in all-cause mortality in the US using the National Health and Nutrition Examination Survey (NHANES 2003-2006). In particular, we match African Americans to Caucasian Americans on age, gender, BMI and objectively measured physical activity (PA). PA is measured every minute using accelerometers for up to seven days and then transformed into an empirical distribution of all of the minute-level observations. The Wasserstein metric is used as the measure of distance between these participant-specific distributions.
This letter proposes a few-shot physics-guided spatial temporal graph convolutional network (FPG-STGCN) to fast solve unit commitment (UC). Firstly, STGCN is tailored to parameterize UC. Then, few-shot physics-guided learning scheme is proposed. It exploits few typical UC solutions yielded via commercial optimizer to escape from local minimum, and leverages the augmented Lagrangian method for constraint satisfaction. To further enable both feasibility and continuous relaxation for integers in learning process, straight-through estimator for Tanh-Sign composition is proposed to fully differentiate the mixed integer solution space. Case study on the IEEE benchmark justifies that, our method bests mainstream learning ways on UC feasibility, and surpasses traditional solver on efficiency.
Solutions to the governing partial differential equations obtained from a discrete numerical scheme can have significant errors, especially near shocks when the discrete representation of the solution cannot fully capture the discontinuity in the solution. A recent approach to shock tracking [1, 2] has been to implicitly align the faces of mesh elements with the shock, yielding accurate solutions on coarse meshes. In engineering applications, the solution field is often used to evaluate a scalar functional of interest, such as lift or drag over an airfoil. While functionals are sensitive to errors in the flow solution, certain regions in the domain are more important for accurate evaluation of the functional than the rest. Using this fact, we formulate a goal-oriented implicit shock tracking approach that captures a segment of the shock that is important for evaluating the functional. Shock tracking is achieved using Lagrange-Newton-Krylov-Schur (LNKS) full space optimizer, with the objective of minimizing the adjoint-weighted residual error indicator. We also present a method to evaluate the sensitivity and the Hessian of the functional error. Using available block preconditioners for LNKS [3, 4] makes the full space approach scalable. The method is applied to test cases of two-dimensional advection and inviscid compressible flows to demonstrate functional-dependent shock tracking. Tracking the entire shock without using artificial dissipation results in the error converging at the orders of $\mathcal{O}(h^{p+1})$.
We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$\epsilon$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.
Single-particle entangled states (SPES) can offer a more secure way of encoding and processing quantum information than their multi-particle counterparts. The SPES generated via a 2D alternate quantum-walk setup from initially separable states can be either 3-way or 2-way entangled. This letter shows that the generated genuine three-way and nonlocal two-way SPES can be used as cryptographic keys to securely encode two distinct messages simultaneously. We detail the message encryption-decryption steps and show the resilience of the 3-way and 2-way SPES-based cryptographic protocols against eavesdropper attacks like intercept-and-resend and man-in-the-middle. We also detail how these protocols can be experimentally realized using single photons, with the three degrees of freedom being OAM, path, and polarization. These have unparalleled security for quantum communication tasks. The ability to simultaneously encode two distinct messages using the generated SPES showcases the versatility and efficiency of the proposed cryptographic protocol. This capability could significantly improve the throughput of quantum communication systems.
We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and Zivn\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We first show that we can reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then we show how to apply classic SDP-rounding tools in this case. We believe that the reduction to balanced hypergraphs is novel and could be of independent interest.
Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.
Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.
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