Log-concavity of B-splines Floater, Michael S.
Journal of approximation theory,
June 2024, 2024-06-00, Letnik:
300
Journal Article
Recenzirano
Odprti dostop
Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave ...property using only recursion formulas for B-splines and their first and second derivatives.
We present an adaptive multi-patch isogeometric analysis on the basis of truncated hierarchical B-splines (THB-splines) for solving two-dimensional complex isotropic/orthotropic elasticity. The ...THB-splines have local refinement property and their basis functions exhibit linear independence, so these features are highly applicable for adaptive isogeometric analysis. Guided by a posterior error estimator based on stress recovery, the adaptive algorithm is utilized in isogeometric analysis. In order to further extend the proposed method to solve complex geometry problems, the multi-patch technique is adopted to achieve exact modeling with Nitsche’s method as a multi-patch coupling approach. An isotropic numerical example with exact analytical solutions and three orthotropic numerical examples are presented to verify the effectiveness and accuracy of the developed method. Numerical solutions show that the developed adaptive isogeometric analysis method has high computational efficiency.
•An adaptive multi-patch isogeometric analysis method on the basis of THB-splines is presented.•Multi-patch technique is used to accurately describe complex geometric shapes.•The Nitsche’s method is employed as a multi-patch coupling approach.•An adaptive local refinement method based on posterior error estimation is proposed.•The proposed method is suitable for solving complex isotropic/orthotropic elasticity.
In this paper, we apply statistical methods for functional data to explore the heterogeneity in the registered number of deaths of COVID‐19, over time. The cumulative daily number of deaths in ...regions across Brazil is treated as continuous curves (functional data). The first stage of the analysis applies clustering methods for functional data to identify and describe potential heterogeneity in the curves and their functional derivatives. The estimated clusters are labeled with different “levels of alert” to identify cities in a possible critical situation. In the second stage of the analysis, we apply a functional quantile regression model for the death curves to explore the associations with functional rates of vaccination and stringency and also with several scalar geographical, socioeconomic and demographic covariates. The proposed model gave a better curve fit at different levels of the cumulative number of deaths when compared to a functional regression model based on ordinary least squares. Our results add to the understanding of the development of COVID‐19 death counts.
In this article we analyze the error produced by the removal of an arbitrary knot from a spline function; we consider the L2-, the H1- and the L∞-errors. When a knot has multiplicity greater than ...one, this implies a reduction of its multiplicity by one unit. In particular, we deduce a very simple formula to compute the error in terms of some neighboring knots and a few coefficients of the considered spline. Furthermore, we show precisely how this error is related to the jump of a derivative of the spline at the knot. We then use the developed theory to propose efficient and very low-cost local error indicators and adaptive coarsening algorithms. Finally, we present some numerical experiments to illustrate their performance and show some applications.
•We make an in deep quantitative analysis for the error produced by the removal of an arbitrary knot from a spline function.•We deduce a simple formula to compute such error in terms of neighboring knots and few coefficients of the considered spline.•We establish the precise relationship between the jump of a suitable derivative and the error mentioned in the previous item.•We develop efficient and very low-cost algorithms for automatic adaptive coarsening tailored to the L2, L∞, or H1-error.
We consider the stiffness matrices arising from the Galerkin B-spline isogeometric analysis discretization of classical elliptic problems. By exploiting their specific spectral properties, compactly ...described by a symbol, we design an efficient multigrid method for the fast solution of the related linear systems. The convergence rate of general-purpose multigrid methods, based on classical stationary smoothers, is optimal (i.e., bounded independently of the matrix size), but it also worsens exponentially with respect to the spline degree. The symbol allows us to give a detailed theoretical explanation of this exponential worsening in the case of the two-grid scheme. In addition, thanks to a specific factorization of the symbol, we are able to design an ad hoc multigrid method with an effective preconditioned CG or GMRES smoother at the finest level, in the spirit of the multi-iterative idea. The convergence rate of this multi-iterative multigrid method is not only optimal but also robust (i.e., bounded substantially independently of the spline degree). This can again be explained by the symbol, in combination with the theory of generalized locally Toeplitz sequences. A selected set of numerical experiments confirms our symbol-based analysis, as well as the effectiveness of the proposed multi-iterative multigrid method, also for larger spline degree.
Purpose
Discretizing tomographic forward and backward operations is a crucial step in the design of model‐based reconstruction algorithms. Standard projectors rely on linear interpolation, whose ...adjoint introduces discretization errors during backprojection. More advanced techniques are obtained through geometric footprint models that may present a high computational cost and an inner logic that is not suitable for implementation on massively parallel computing architectures. In this work, we take a fresh look at the discretization of resampling transforms and focus on the issue of magnification‐induced local sampling variations by introducing a new magnification‐driven interpolation approach for tomography.
Methods
Starting from the existing literature on spline interpolation for magnification purposes, we provide a mathematical formulation for discretizing a one‐dimensional homography. We then extend our approach to two‐dimensional representations in order to account for the geometry of cone‐beam computed tomography with a flat panel detector. Our new method relies on the decomposition of signals onto a space generated by nonuniform B‐splines so as to capture the spatially varying magnification that locally affects sampling. We propose various degrees of approximations for a rapid implementation of the proposed approach. Our framework allows us to define a novel family of projector/backprojector pairs parameterized by the order of the employed B‐splines. The state‐of‐the‐art distance‐driven interpolation appears to fit into this family thus providing new insight and computational layout for this scheme. The question of data resampling at the detector level is handled and integrated with reconstruction in a single framework.
Results
Results on both synthetic data and real data using a quality assurance phantom, were performed to validate our approach. We show experimentally that our approximate implementations are associated with reduced complexity while achieving a near‐optimal performance. In contrast with linear interpolation, B‐splines guarantee full usage of all data samples, and thus the X‐ray dose, leading to more uniform noise properties. In addition, higher‐order B‐splines allow analytical and iterative reconstruction to reach higher resolution. These benefits appear more significant when downsampling frames acquired by X‐ray flat‐panel detectors with small pixels.
Conclusions
Magnification‐driven B‐spline interpolation is shown to provide high‐accuracy projection operators with good‐quality adjoints for iterative reconstruction. It equally applies to backprojection for analytical reconstruction and detector data downsampling.
In this paper, we consider a time‐dependent singularly perturbed differential‐difference equation with small shifts arising in the field of neuroscience. The terms containing the delay and advance ...parameters are approximated by using the Taylor's series expansion. The continuous problem is semi‐discretized using the Crank–Nicolson finite difference method in the time direction on uniform mesh and quadratic B‐spline collocation method in the space direction on exponentially graded mesh. The method is shown to be second‐order uniformly convergent in space and time direction. Theoretical estimates are carried out which support the obtained numerical experiments.
Summary
Numerical simulation of stretch‐induced wrinkling in thin elastic sheets is a challenging problem due to a vanishing bending stiffness and the coexistence of superabundant equilibrium ...solutions. In this work, we present a computational framework to capture the morphological evolution of stretch‐induced wrinkles. The application of modified Föppl‐von Kármán plate model results in a fourth‐order partial differential equation. The convergence of finite‐element solutions necessitates C1$$ {C}^1 $$‐continuous approximations. Herein, Powell‐Sabin B‐splines, which are based on triangles, are utilized for both the approximation of the field variables and the description of the geometry. To trace the wrinkling behavior in thin sheets, a path‐following technique using asymptotic numerical method is considered. The advantage of this method is an adaptive step length, which works incredibly well near the bifurcation points and allows for the computation of the post‐bifurcation diagrams with a quite small perturbation. The versatility and accuracy of the developed computational approach are assessed in three case studies, featuring wrinkling in highly stretched rectangular and annular thin sheets.
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