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Multiple imputation (MI) models can be improved by including auxiliary covariates (AC), but their performance in high-dimensional data is not well understood. We aimed to develop and compare high-dimensional MI (HDMI) approaches using structured and natural language processing (NLP)-derived AC in studies with partially observed confounders. We conducted a plasmode simulation study using data from opioid vs. non-steroidal anti-inflammatory drug (NSAID) initiators (X) with observed serum creatinine labs (Z2) and time-to-acute kidney injury as outcome. We simulated 100 cohorts with a null treatment effect, including X, Z2, atrial fibrillation (U), and 13 other investigator-derived confounders (Z1) in the outcome generation. We then imposed missingness (MZ2) on 50% of Z2 measurements as a function of Z2 and U and created different HDMI candidate AC using structured and NLP-derived features. We mimicked scenarios where U was unobserved by omitting it from all AC candidate sets. Using LASSO, we data-adaptively selected HDMI covariates associated with Z2 and MZ2 for MI, and with U to include in propensity score models. The treatment effect was estimated following propensity score matching in MI datasets and we benchmarked HDMI approaches against a baseline imputation and complete case analysis with Z1 only. HDMI using claims data showed the lowest bias (0.072). Combining claims and sentence embeddings led to an improvement in the efficiency displaying the lowest root-mean-squared-error (0.173) and coverage (94%). NLP-derived AC alone did not perform better than baseline MI. HDMI approaches may decrease bias in studies with partially observed confounders where missingness depends on unobserved factors.

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.

We construct a simple and robust finite volume discretization for linearized mechanics, Stokes and poromechanics, based only on co-located, cell-centered variables. The discretization has a minimal stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We fully justify the method theoretically in terms of stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments support the theoretical results, and shed light on grid families not explicitly treated by the theoretical results.

This paper studies linear reconstruction of partially observed functional data which are recorded on a discrete grid. We propose a novel estimation approach based on approximate factor models with increasing rank taking into account potential covariate information. Whereas alternative reconstruction procedures commonly involve some preliminary smoothing, our method separates the signal from noise and reconstructs missing fragments at once. We establish uniform convergence rates of our estimator and introduce a new method for constructing simultaneous prediction bands for the missing trajectories. A simulation study examines the performance of the proposed methods in finite samples. Finally, a real data application of temperature curves demonstrates that our theory provides a simple and effective method to recover missing fragments.

In the present paper, we introduce new tensor krylov subspace methods for solving large Sylvester tensor equations. The proposed method uses the well-known T-product for tensors and tensor subspaces. We introduce some new tensor products and the related algebraic properties. These new products will enable us to develop third-order the tensor FOM (tFOM), GMRES (tGMRES), tubal Block Arnoldi and the tensor tubal Block Arnoldi method to solve large Sylvester tensor equation. We give some properties related to these method and present some numerical experiments.

In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determinantal point process, we build an unbiased Monte Carlo estimator of the integral of any Lipschitz function, and show that the estimator satisfies a central limit theorem, with a faster rate than under independent sampling. In particular, seeing a complex manifold of dimension $d$ as a real manifold of dimension $d_{\mathbb{R}}=2d$, the mean squared error for $N$ quadrature nodes decays as $N^{-1-2/d_{\mathbb{R}}}$; this is faster than previous DPP-based quadratures and reaches the optimal worst-case rate investigated by [Bakhvalov 1965] in Euclidean spaces. The determinantal point process we use is characterized by its kernel, which is the Bergman kernel of a holomorphic Hermitian line bundle, and we strongly build upon the work of Berman that led to the central limit theorem in [Berman, 2018].We provide numerical illustrations for the Riemann sphere.

In this paper, a force-based beam finite element model based on a modified higher-order shear deformation theory is proposed for the accurate analysis of functionally graded beams. In the modified higher-order shear deformation theory, the distribution of transverse shear stress across the beam's thickness is obtained from the differential equilibrium equation, and a modified shear stiffness is derived to take the effect of transverse shear stress distribution into consideration. In the proposed beam element model, unlike traditional beam finite elements that regard generalized displacements as unknown fields, the internal forces are considered as the unknown fields, and they are predefined by using the closed-form solutions of the differential equilibrium equations of higher-order shear beam. Then, the generalized displacements are expressed by the internal forces with the introduction of geometric relations and constitutive equations, and the equation system of the beam element is constructed based on the equilibrium conditions at the boundaries and the compatibility condition within the element. Numerical examples underscore the accuracy and efficacy of the proposed higher-order beam element model in the static analysis of functionally graded sandwich beams, particularly in terms of true transverse shear stress distribution.

This paper presents a regularized recursive identification algorithm with simultaneous on-line estimation of both the model parameters and the algorithms hyperparameters. A new kernel is proposed to facilitate the algorithm development. The performance of this novel scheme is compared with that of the recursive least squares algorithm in simulation.

This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.

In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal $L^2$ error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.