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We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022.

We extend the error bounds from [SIMAX, Vol. 43, Iss. 2, pp. 787-811 (2022)] for the Lanczos method for matrix function approximation to the block algorithm. Numerical experiments suggest that our bounds are fairly robust to changing block size and have the potential for use as a practical stopping criteria. Further experiments work towards a better understanding of how certain hyperparameters should be chosen in order to maximize the quality of the error bounds, even in the previously studied block-size one case.

Fractional Brownian trajectories (fBm) feature both randomness and strong scale-free correlations, challenging generative models to reproduce the intrinsic memory characterizing the underlying process. Here we test a diffusion probabilistic model on a specific dataset of corrupted images corresponding to incomplete Euclidean distance matrices of fBm at various memory exponents $H$. Our dataset implies uniqueness of the data imputation in the regime of low missing ratio, where the remaining partial graph is rigid, providing the ground truth for the inpainting. We find that the conditional diffusion generation stably reproduces the statistics of missing fBm-distributed distances for different values of $H$ exponent. Furthermore, while diffusion models have been recently shown to remember samples from the training database, we show that diffusion-based inpainting behaves qualitatively different from the database search with the increasing database size. Finally, we apply our fBm-trained diffusion model with $H=1/3$ for completion of chromosome distance matrices obtained in single-cell microscopy experiments, showing its superiority over the standard bioinformatics algorithms. Our source code is available on GitHub at //github.com/alobashev/diffusion_fbm.

We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.

This article analyses the simulation methodology for wall-modeled large-eddy simulations using solvers based on the spectral-element method (SEM). To that end, algebraic wall modeling is implemented in the popular SEM solver Nek5000. It is combined with explicit subgrid-scale (SGS) modeling, which is shown to perform better than the high-frequency filtering traditionally used with the SEM. In particular, the Vreman model exhibits a good balance in terms stabilizing the simulations, yet retaining good resolution of the turbulent scales. Some difficulties associated with SEM simulations on relatively coarse grids are also revealed: jumps in derivatives across element boundaries, lack of convergence for weakly formulated boundary conditions, and the necessity for the SGS model as a damper for high-frequency modes. In spite of these, state-of-the-art accuracy is achieved for turbulent channel flow and flat-plate turbulent boundary layer flow cases, proving the SEM to be a an excellent numerical framework for massively-parallel high-order WMLES.

We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions that have a unique zero. We can therefore show local quadratic convergence, and for certain extreme eigenvalues even global linear convergence of the method. Local definiteness is a weaker condition than right and left definiteness, which is often considered for multiparameter eigenvalue problems. These conditions are naturally satisfied for multiparameter Sturm-Liouville problems that arise when separation of variables can be applied to multidimensional boundary eigenvalue problems.

Domain decomposition (DD) methods are a natural way to take advantage of parallel computers when solving large scale linear systems. Their scalability depends on the design of the coarse space used in the two-level method. The analysis of adaptive coarse spaces we present here is quite general since it applies to symmetric and non symmetric problems, to symmetric preconditioners such the additive Schwarz method (ASM) and to the non-symmetric preconditioner restricted additive Schwarz (RAS), as well as to exact or inexact subdomain solves. The coarse space is built by solving generalized eigenvalues in the subdomains and applying a well-chosen operator to the selected eigenvectors.

Port-Hamiltonian systems provide an energy-based modeling paradigm for dynamical input-state-output systems. At their core, they fulfill an energy balance relating stored, dissipated and supplied energy. To accurately resolve this energy balance in time discretizations, we propose an adaptive grid refinement technique based on a posteriori error estimation. The evaluation of the error estimator includes the computation of adjoint sensitivities. To interpret this adjoint equation as a backwards-in-time equation, we show piecewise weak differentiability of the dual variable. Then, leveraging dissipativity of the port-Hamiltonian dynamics, we present a parallelizable approximation of the underlying adjoint system in the spirit of a block-Jacobi method to efficiently compute error indicators. We illustrate the performance of the proposed scheme by means of numerical experiments showing that it yields a smaller violation of the energy balance when compared to uniform refinements and traditional step-size controlled time stepping.

An interesting case of the well-known Dataset Shift Problem is the classification of Electroencephalogram (EEG) signals in the context of Brain-Computer Interface (BCI). The non-stationarity of EEG signals can lead to poor generalisation performance in BCI classification systems used in different sessions, also from the same subject. In this paper, we start from the hypothesis that the Dataset Shift problem can be alleviated by exploiting suitable eXplainable Artificial Intelligence (XAI) methods to locate and transform the relevant characteristics of the input for the goal of classification. In particular, we focus on an experimental analysis of explanations produced by several XAI methods on an ML system trained on a typical EEG dataset for emotion recognition. Results show that many relevant components found by XAI methods are shared across the sessions and can be used to build a system able to generalise better. However, relevant components of the input signal also appear to be highly dependent on the input itself.

Evaluations of model editing currently only use the `next few token' completions after a prompt. As a result, the impact of these methods on longer natural language generation is largely unknown. We introduce long-form evaluation of model editing (LEME) a novel evaluation protocol that measures the efficacy and impact of model editing in long-form generative settings. Our protocol consists of a machine-rated survey and a classifier which correlates well with human ratings. Importantly, we find that our protocol has very little relationship with previous short-form metrics (despite being designed to extend efficacy, generalization, locality, and portability into a long-form setting), indicating that our method introduces a novel set of dimensions for understanding model editing methods. Using this protocol, we benchmark a number of model editing techniques and present several findings including that, while some methods (ROME and MEMIT) perform well in making consistent edits within a limited scope, they suffer much more from factual drift than other methods. Finally, we present a qualitative analysis that illustrates common failure modes in long-form generative settings including internal consistency, lexical cohesion, and locality issues.