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A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.

In the context of cubic splines, the authors have contributed to a recent paper dealing with the computation of nonlinear derivatives at the interior nodes so that monotonicity is enforced while keeping the order of approximation of the spline as high as possible. During the review process of that paper, one of the reviewers raised the question of whether a cubic spline interpolating monotone data could be forced to preserve monotonicity by imposing suitable values of the first derivative at the endpoints. Albeit a negative answer appears to be intuitive, we have found no results regarding this fact. In this short work we prove that the answer to that question is actually negative.

Most of the existing Mendelian randomization (MR) methods are limited by the assumption of linear causality between exposure and outcome, and the development of new non-linear MR methods is highly desirable. We introduce two-stage prediction estimation and control function estimation from econometrics to MR and extend them to non-linear causality. We give conditions for parameter identification and theoretically prove the consistency and asymptotic normality of the estimates. We compare the two methods theoretically under both linear and non-linear causality. We also extend the control function estimation to a more flexible semi-parametric framework without detailed parametric specifications of causality. Extensive simulations numerically corroborate our theoretical results. Application to UK Biobank data reveals non-linear causal relationships between sleep duration and systolic/diastolic blood pressure.

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, the spatial domain of its input function needs to be identical to its output, which limits its applicability. For instance, the widely used Fourier neural operator (FNO) fails to approximate the operator that maps the boundary condition to the PDE solution. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas NO fail. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.

Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite-dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We thus propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particular, our distance 1) metrizes weak convergence; 2) can be estimated numerically through samples with low complexity; 3) can be bounded analytically from above and below. The main ingredient are integral probability metrics, which lead to the name hierarchical IPM.

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some $4$-dimensional manifolds without and with boundary.

The human cerebral cortex has many bumps and grooves called gyri and sulci. Even though there is a high inter-individual consistency for the main cortical folds, this is not the case when we examine the exact shapes and details of the folding patterns. Because of this complexity, characterizing the cortical folding variability and relating them to subjects' behavioral characteristics or pathologies is still an open scientific problem. Classical approaches include labeling a few specific patterns, either manually or semi-automatically, based on geometric distances, but the recent availability of MRI image datasets of tens of thousands of subjects makes modern deep-learning techniques particularly attractive. Here, we build a self-supervised deep-learning model to detect folding patterns in the cingulate region. We train a contrastive self-supervised model (SimCLR) on both Human Connectome Project (1101 subjects) and UKBioBank (21070 subjects) datasets with topological-based augmentations on the cortical skeletons, which are topological objects that capture the shape of the folds. We explore several backbone architectures (convolutional network, DenseNet, and PointNet) for the SimCLR. For evaluation and testing, we perform a linear classification task on a database manually labeled for the presence of the "double-parallel" folding pattern in the cingulate region, which is related to schizophrenia characteristics. The best model, giving a test AUC of 0.76, is a convolutional network with 6 layers, a 10-dimensional latent space, a linear projection head, and using the branch-clipping augmentation. This is the first time that a self-supervised deep learning model has been applied to cortical skeletons on such a large dataset and quantitatively evaluated. We can now envisage the next step: applying it to other brain regions to detect other biomarkers.

Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous improvement over existing software for certified path tracking.

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the sufficiently smooth solution, we first clarify that the standard graded meshes are worse than uniform meshes and may even lead to divergence; instead, an optimal convergence rate arises in so-called anomalous graded meshes. Furthermore, under low regularity solutions, it may suffer from a severe order reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174). In this case, conversely, a sharp error estimates appears in standard graded meshes, but offering far less than first-order accuracy. For the time-dependent case, however, second-order convergence can be achieved on graded meshes. The related analysis are easily extended for certain multidimensional problems. Numerical results are provided that confirm the sharpness of the error estimates.