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Our goal is to highlight some of the deep links between numerical splitting methods and control theory. We consider evolution equations of the form $\dot{x} = f_0(x) + f_1(x)$, where $f_0$ encodes a non-reversible dynamic, so that one is interested in schemes only involving forward flows of $f_0$. In this context, a splitting method can be interpreted as a trajectory of the control-affine system $\dot{x}(t)=f_0(x(t))+u(t)f_1(x(t))$, associated with a control~$u$ which is a finite sum of Dirac masses. The general goal is then to find a control such that the flow of $f_0 + u(t) f_1$ is as close as possible to the flow of $f_0+f_1$. Using this interpretation and classical tools from control theory, we revisit well-known results concerning numerical splitting methods, and we prove a handful of new ones, with an emphasis on splittings with additional positivity conditions on the coefficients. First, we show that there exist numerical schemes of any arbitrary order involving only forward flows of $f_0$ if one allows complex coefficients for the flows of $f_1$. Equivalently, for complex-valued controls, we prove that the Lie algebra rank condition is equivalent to the small-time local controllability of a system. Second, for real-valued coefficients, we show that the well-known order restrictions are linked with so-called "bad" Lie brackets from control theory, which are known to yield obstructions to small-time local controllability. We use our recent basis of the free Lie algebra to precisely identify the conditions under which high-order methods exist.

We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.

While undulatory swimming of elongate limbless robots has been extensively studied in open hydrodynamic environments, less research has been focused on limbless locomotion in complex, cluttered aquatic environments. Motivated by the concept of mechanical intelligence, where controls for obstacle navigation can be offloaded to passive body mechanics in terrestrial limbless locomotion, we hypothesize that principles of mechanical intelligence can be extended to cluttered hydrodynamic regimes. To test this, we developed an untethered limbless robot capable of undulatory swimming on water surfaces, utilizing a bilateral cable-driven mechanism inspired by organismal muscle actuation morphology to achieve programmable anisotropic body compliance. We demonstrated through robophysical experiments that, similar to terrestrial locomotion, an appropriate level of body compliance can facilitate emergent swim through complex hydrodynamic environments under pure open-loop control. Moreover, we found that swimming performance depends on undulation frequency, with effective locomotion achieved only within a specific frequency range. This contrasts with highly damped terrestrial regimes, where inertial effects can often be neglected. Further, to enhance performance and address the challenges posed by nondeterministic obstacle distributions, we incorporated computational intelligence by developing a real-time body compliance tuning controller based on cable tension feedback. This controller improves the robot's robustness and overall speed in heterogeneous hydrodynamic environments.

An unexpected failure of a concrete gravity dam may cause unimaginable human suffering and massive economic losses. An earthquake is the main factor contributing to the concrete gravity dam's failure. In recent years, there has been a rise in efforts globally to make dams safe under dynamic loading. Numerical modeling of dams under earthquake loading yields substantial insights into dams' fracture and damage progression. In the present work, a particle-based computational framework is developed to investigate the failure of the Koyna dam, a concrete gravity dam in India exposed to dynamic loading. The dam-foundation system is considered here. The numerically obtained crack results in the concrete dam are compared with the available experimental results. The findings are consistent with one another.

We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

We show that differential privacy type guarantees can be obtained when using a Poisson synthesis mechanism to protect counts in contingency tables. Specifically, we show how to obtain $(\epsilon, \delta)$-probabilistic differential privacy guarantees via the Poisson distribution's cumulative distribution function. We demonstrate this empirically with the synthesis of an administrative-type confidential database.

To facilitate effective decision-making, gridded satellite precipitation products should include uncertainty estimates. Machine learning has been proposed for issuing such estimates. However, most existing algorithms for this purpose rely on quantile regression. Distributional regression offers distinct advantages over quantile regression, including the ability to model intermittency as well as a stronger ability to extrapolate beyond the training data, which is critical for predicting extreme precipitation. In this work, we introduce the concept of distributional regression for the engineering task of creating precipitation datasets through data merging. Building upon this concept, we propose new ensemble learning methods that can be valuable not only for spatial prediction but also for prediction problems in general. These methods exploit conditional zero-adjusted probability distributions estimated with generalized additive models for location, scale, and shape (GAMLSS), spline-based GAMLSS and distributional regression forests as well as their ensembles (stacking based on quantile regression, and equal-weight averaging). To identify the most effective methods for our specific problem, we compared them to benchmarks using a large, multi-source precipitation dataset. Stacking emerged as the most successful strategy. Three specific stacking methods achieved the best performance based on the quantile scoring rule, although the ranking of these methods varied across quantile levels. This suggests that a task-specific combination of multiple algorithms could yield significant benefits.

Gary Lorden provided several fundamental and novel insights into sequential hypothesis testing and changepoint detection. In this article, we provide an overview of Lorden's contributions in the context of existing results in those areas, and some extensions made possible by Lorden's work. We also mention some of Lorden's significant consulting work, including as an expert witness and for NASA, the entertainment industry, and Major League Baseball.

Weakly Supervised Semantic Segmentation (WSSS) employs weak supervision, such as image-level labels, to train the segmentation model. Despite the impressive achievement in recent WSSS methods, we identify that introducing weak labels with high mean Intersection of Union (mIoU) does not guarantee high segmentation performance. Existing studies have emphasized the importance of prioritizing precision and reducing noise to improve overall performance. In the same vein, we propose ORANDNet, an advanced ensemble approach tailored for WSSS. ORANDNet combines Class Activation Maps (CAMs) from two different classifiers to increase the precision of pseudo-masks (PMs). To further mitigate small noise in the PMs, we incorporate curriculum learning. This involves training the segmentation model initially with pairs of smaller-sized images and corresponding PMs, gradually transitioning to the original-sized pairs. By combining the original CAMs of ResNet-50 and ViT, we significantly improve the segmentation performance over the single-best model and the naive ensemble model, respectively. We further extend our ensemble method to CAMs from AMN (ResNet-like) and MCTformer (ViT-like) models, achieving performance benefits in advanced WSSS models. It highlights the potential of our ORANDNet as a final add-on module for WSSS models.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.