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This simulation study evaluates the effectiveness of multiple imputation (MI) techniques for multilevel data. It compares the performance of traditional Multiple Imputation by Chained Equations (MICE) with tree-based methods such as Chained Random Forests with Predictive Mean Matching and Extreme Gradient Boosting. Adapted versions that include dummy variables for cluster membership are also included for the tree-based methods. Methods are evaluated for coefficient estimation bias, statistical power, and type I error rates on simulated hierarchical data with different cluster sizes (25 and 50) and levels of missingness (10\% and 50\%). Coefficients are estimated using random intercept and random slope models. The results show that while MICE is preferred for accurate rejection rates, Extreme Gradient Boosting is advantageous for reducing bias. Furthermore, the study finds that bias levels are similar across different cluster sizes, but rejection rates tend to be less favorable with fewer clusters (lower power, higher type I error). In addition, the inclusion of cluster dummies in tree-based methods improves estimation for Level 1 variables, but is less effective for Level 2 variables. When data become too complex and MICE is too slow, extreme gradient boosting is a good alternative for hierarchical data. Keywords: Multiple imputation; multi-level data; MICE; missRanger; mixgb

We present new convergence estimates of generalized empirical interpolation methods in terms of the entropy numbers of the parametrized function class. Our analysis is transparent and leads to sharper convergence rates than the classical analysis via the Kolmogorov n-width. In addition, we also derive novel entropy-based convergence estimates of the Chebyshev greedy algorithm for sparse n-term nonlinear approximation of a target function. This also improves classical convergence analysis when corresponding entropy numbers decay fast enough.

We propose a two-step Newton's method for refining an approximation of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.

We present a novel combination of dynamic embedded topic models and change-point detection to explore diachronic change of lexical semantic modality in classical and early Christian Latin. We demonstrate several methods for finding and characterizing patterns in the output, and relating them to traditional scholarship in Comparative Literature and Classics. This simple approach to unsupervised models of semantic change can be applied to any suitable corpus, and we conclude with future directions and refinements aiming to allow noisier, less-curated materials to meet that threshold.

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.

The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.

We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degenerate diffusion. The stabilisation is constructed so that the resulting method admits a \emph{numerical hypocoercivity} property, analogous to the corresponding property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting ``stronger-than-energy'' stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the ``time'' variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.

This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.

Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-|x|=b$. The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever $\rho(|A^{-1}|)<1/3$. We also present new results for the case where $A-I$ is a nonsingular $M$-matrix or an irreducible singular $M$-matrix. When $A-I$ is an irreducible singular $M$-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.