This work addresses the interpolation of probability measures within a spatial statistics framework. We develop a Kriging approach in the Wasserstein space, leveraging the quantile function representation of the one-dimensional Wasserstein distance. To mitigate the inaccuracies in semivariogram estimation that arise from sparse datasets, we combine this formulation with cross-validation techniques. In particular, we introduce a variant of the virtual cross-validation formulas tailored to quantile functions. The effectiveness of the proposed method is demonstrated on a controlled toy problem as well as on a real-world application from nuclear safety.

This paper studies the d-dimensional extension of a fictitious domain penalization technique that we previously proposed for Neumann or Robin boundary conditions. We apply Droniou's approach for non-coercive linear elliptic problems to obtain the existence and uniqueness of the solution of the penalized problem, and we derive a boundary layer approach to establish the convergence of the penalization method. The developed boundary layer approach is adapted from the one used for Dirichlet boundary conditions, but in contrast to the latter where coercivity enables a straightforward estimate of the remainders, we reduce the convergence of the penalization method to the existence of suitable supersolutions of a dual problem. These supersolutions are then constructed as approximate solutions of the dual problem using an additional formal boundary layer approach. The proposed approach results in an advection-dominated problem, requiring the use of appropriate numerical methods suitable for singular perturbation problems. Numerical experiments, using upwind finite differences, validate both the convergence rate and the boundary layer thickness, illuminating the theoretical results.

This paper deals with the approximation of discrete probability measure-valued data by a new subdivision scheme. Its construction relies on a coupling between linear subdivision and optimal transport. A mathematical analysis is performed to study its convergence. Two test cases are finally described to emphasize its capability: the first one is related to point cloud interpolation while the second one is a first attempt in the framework of image approximation.

We define a metric in the space of positive finite positive measures that extends the 2-Wasserstein metric, i.e. its restriction to the set of probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic formulation and extend the gradient flow machinery of the Wasserstein space. In addition, we relate the barycenter in this space to the barycenter in the Wasserstein space of the normalized measures.

This work presents an algorithm to sample from the Wasserstein barycenter of absolutely continuous measures. Our method is based on the gradient flow of the multimarginal formulation of the Wasserstein barycenter, with an additive penalization to account for the marginal constraints. We prove that the minimum of this penalized multimarginal formulation is achieved for a coupling that is close to the Wasserstein barycenter. The performances of the algorithm are showcased in several settings.

This paper is devoted to the construction of a new fast-to-evaluate model for the prediction of 2D crack paths in concrete-like microstructures. The model generates piecewise linear cracks paths with segmentation points selected using a Markov chain model. The Markov chain kernel involves local indicators of mechanical interest and its parameters are learnt from numerical full-field 2D simulations of cracking using a cohesive-volumetric finite element solver called XPER. This model does not include any mechanical elements. It is the database, derived from the XPER crack, that contains the mechanical information and optimizes the probabilistic model. The resulting model exhibits a drastic improvement of CPU time in comparison to simulations from XPER.

Converging approximation of a regular curve by polygonal lines in the uniform norm does not imply the convergence of the discrete differentials to their smooth counterpart. In this paper, we provide a constructive approach that, given a converging polygon sequence and an approximation of its distance to the objective curve, provides another sequence of polygons for which convergence of discrete differentials occurs as well. This approach is based on the notion of local scale of a polygon and uses multi-resolution decomposition as well as a non linear smoothing process. We provide the proof of the convergence and some numerical evidence of it, with application to the evaluation of solid friction in a pipe.

Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

We make a comparison of the finite difference and multiresolution method for solving the elliptic equations on irregular domains. The Dirichlet boundary condition is treated by the ghost fluid method (GFM) for the finite difference method and the Lagrange multiplier for the multiresolution method. Numerical results illustrate the improved convergence rate of errors and their gradients with the multiresolution method, up to eighth-order. Moreover, the wavelet-based multiresolution has an advantage of preconditioning of the linear system arising from the discretization of elliptic equation over the finite difference method. (C) 2021 Elsevier Inc. All rights reserved.

Highly anisotropic elliptic problems occur in many physical models that need to be solved numerically. A direction of dominant diffusion is thus introduced (called here parallel direction) along which the diffusion coefficient is several orders larger of magnitude than in the perpendicular one. In this case, finite-difference methods based on misaligned stencils are generally not designed to provide an optimal discretization, and may lead the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. This paper proposes an original scheme using non-aligned Cartesian grids and interpolations aligned along a parallel diffusion direction. Here, this direction is assumed to be supported by a divergence-free vector field which never vanishes and it is supposed to be stationary in time. Based on the Support Operator Method (SOM), the self-adjointness property of the parallel diffusion operator is maintained on the discrete level. Compared with existing methods, the present formulation further guarantees the conservativity of the fluxes in both parallel and perpendicular directions. In addition, when the flow intercepts a boundary in the parallel direction, an accurate discretization of the boundary condition is presented that avoids the uncertainties of extrapolated far ghost points classically used and ensures a better accuracy of the solution. Numerical tests based on manufactured solutions show the method is able to provide accurate and stable numerical approximations in both periodic and bounded domains with a drastically reduced number of degrees of freedom with respect to non-aligned approaches.

La construction d'un materiau numerique repose sur des descripteurs morphologiques qui caracterisent un materiau reel a l'aide d'informations statistiques et geometriques. La microstructure numerique generee est qualifie statistiquement semblable par rapport a la realite. Or deux microstructures statistiquement semblables peuvent presenter des comportements de fissuration differents. Ces differences peuvent avoir une influence sur la permeabilite des enceintes. Pour contourner cette limitation, nous proposons dans ce travail un nouvelle approche qui combine descripteurs morphologiques et modele de prediction de fissure pour evaluer la similarite entre 2 microstructures.

Subdivision schemes have been extensively developed since the eighties with very powerful applications for surface generation. To be implemented for compression, subdivision schemes have to be coupled with decimation operators sharing some consistency relation and with detail operators. The flexibility of subdivision schemes (they can be non-stationary, position or zone dependent, non-linear,…) makes that the construction of consistent decimation operators is a difficult task. In this paper, following the first results introduced in Kui et al. (On the coupling of decimation operator with subdivision schemes for multi-scale analysis. In: Lecture notes in computer science, vol. 10521. Springer, Berlin, pp. 162–185, 2016), we present the construction of multiresolution analyses connected to general subdivision schemes with detailed application to a non-interpolatory linear scheme called shifted Lagrange (Dyn et al., A C2 four-point subdivision scheme with fourth order accuracy and its extensions. In: Mathematical methods for curves and surfaces: Tromsø 2004. Citeseer, 2005) and its non-linear version called shifted PPH (Amat et al., Math. Comput. 80:959–959, 2011).

Drilling monitoring aims at anticipating and detecting any drill string failures during well construction. A key element for the monitoring activity is the estimation of friction along the wellbore trajectory. Friction models require the evaluation of the actual wellbore trajectory. This evaluation is performed applying any of various reconstruction methods available in the industry to discrete deviation measurements. Although all these methods lead to nearly identical bit location, friction estimations are highly dependent on reconstruction methods due to huge dierences in the trajectory derivatives. To control this instability, a new reliable estimation of wellbore friction using a nonlinear trajectory smoothing process is introduced. This process uses a multi-scale approach and a specic nonlinear smoothing through subdivision schemes and their related decimation schemes. Two smoothing processes are compared: one using an interpolatory subdivision operator, and the other, a non-interpolatory subdivision operator. Validation has been performed on a synthetic plane noisy trajectory. The non-interpolatory process provides trajectory derivatives estimate much closer to those of the initial trajectory. Both processes have been applied to a real three-dimensional wellbore trajectory, improving signicantly the friction estimates.

This paper is devoted to the construction and analysis of a new family of three-points nonlinear cell-average subdivision (multiresolution) schemes. They are based on a centered piecewise nonlinear reconstruction adapted to discontinuities. Some theoretical properties of these schemes (convergence, order of approximation, preservation of the monotonicity in the data, stability or absence of the Gibbs phenomenon) are analyzed. Finally, various numerical examples are presented.

Gaussian process (GP) models have become popular for approximating and exploring nonlinear systems using scarce input/output samples and prior hypotheses done through mean and covariance functions. While it is common to make stationarity assumptions and use variance-based criteria for exploration, in realistic cases it is not rare that systems under study exhibit a heterogeneous behavior depending on regions of the parameter space. We consider a class of problems where high variations occur along unknown noncanonical directions and we tackle the problem of accommodating nonstationarity from two angles. First we define a novel class of covariances (WaMI-GP) that simultaneously generalizes kernels of multiple index and of tensorized warped GPs, and second, we introduce derivative-based sampling criteria dedicated to the exploration of high-variation regions. The novel GP class is investigated through both mathematical analysis and numerical experiments, and it is shown that it allows encoding much expressiveness while keeping the number of parameters to be inferred moderate. Criteria and models are compared on a mechanical test case from safety studies conducted by IRSN. On this application some of the proposed criteria outperform usual variance-based criteria in the case of a stationary GP model; however, variance-based criteria with WaMI-GP perform even better. Our method is also compared with the treed Gaussian processes (TGP) on this application and on a NASA test case. In the IRSN application, WaMI-GP dominates TGP in static and sequential settings. In the NASA application, while TGP clearly dominates in the static case, for small designs it is outperformed by WaMI-GP in the sequential setup.

This paper is devoted to the construction of multiresolution frameworks related to general but linear subdivision schemes applied on sequences of finite length. Thanks to the flexible properties of subdivision schemes, a subdivision-based multiresolution is a promising powerful tool for compression, control and analysis of data. However, its construction is not straightforward either for non-interpolatory subdivisions, or for sequences of finite length. In this paper, given a subdivision for finite length sequences, we provide a first approach to construct compatible multiresolutions thanks to an extension of some classical wavelet results. As it will become clear that this type of approach can become computationally costly in practice, a new edge-adapted method that combines local consistent decimation is then developed to circumvent this limitation. An illustration in the case of the 4-point shifted Lagrange subdivision scheme, for which there is no available multiresolution up to now, is then provided. Finally, some properties of the new multiresolutions are analyzed and their performances are evaluated in the framework of image approximation.

Subdivision schemes [5, 11] are powerful tools for the fast generation of refined sequences ultimately representing curves or surfaces. Coupled with decimation operators, they generate multi-scale transforms largely used in signal/image processing [1, 3] that generalize the multi-resolution analysis/wavelet framework [8]. The flexibility of subdivision schemes (a subdivision scheme can be non-stationary, non-homogeneous, position-dependent, interpolating, approximating, non-linear...) (e.g. [3]) is balanced, as a counterpart, by the fact that the construction of suitable consistent decimation operators is not direct and easy. In this paper, we first propose a generic approach for the construction of decimation operators consistent with a given linear subdivision. A study of the so-called prediction error within the multi-scale framework is then performed and a condition on the subdivision mask to ensure a fast decay of this error is established. Finally, the cases of homogeneous Lagrange interpolatory subdivision, spline subdivision, subdivision related to Daubechies scaling functions (and wavelets) and some recently developed non stationary non interpolating schemes are revisited.

We tackle the problem of reconstructing functions possessing highly heterogeneous behaviour across the input space from scattered evaluations. Our main approach combines non-stationary Gaussian process (GP) modelling with wavelet local analysis. A warped GP model is assumed, and a novel stationarization algorithm is proposed that relies on successive inverse warpings based on local scale estimation. The approach is applied to two mechanical case studies highlighting promising prediction performance compared to state-of-the-art methods.