Near-best bivariate spline quasi-interpolants on a four-directional mesh of the plane

Spline quasi-interpolants (QIs) are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete and integral quasi-interpolants which are based on $\Omega$-~splines, i.e. B-splines with regular lozenge supports on the uniform four directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinity norms which depend on a finite number of free parameters. We show that this problem has always a solution, which is not unique in general. Concrete examples of these types of quasi-interpolants are given in the last section

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Interpolation Based Parametric Model Order Reduction

In this thesis, we consider model order reduction of parameter-dependent large-scale dynamical systems. The objective is to develop a methodology to reduce the order of the model and simultaneously preserve the dependence of the model on parameters. We use the balanced truncation method together with spline interpolation to solve the problem. The core of this method is to interpolate the reduced transfer function, based on the pre-computed transfer function at a sample in the parameter domain. Linear splines and cubic splines are employed here. The use of the latter, as expected, reduces the error of the method. The combination is proven to inherit the advantages of balanced truncation such as stability preservation and, based on a novel bound for the infinity norm of the matrix inverse, the derivation of error bounds. Model order reduction can be formulated in the projection framework. In the case of a parameter-dependent system, the projection subspace also depends on parameters. One cannot compute this parameter-dependent projection subspace, but has to approximate it by interpolation based on a set of pre-computed subspaces. It turns out that this is the problem of interpolation on Grassmann manifolds. The interpolation process is actually performed on tangent spaces to the underlying manifold. To do that, one has to invoke the exponential and logarithmic mappings which involve some singular value decompositions. The whole procedure is then divided into the offline and online stage. The computation time in the online stage is a crucial point. By investigating the formulation of exponential and logarithmic mappings and analyzing the structure of sums of singular value decompositions, we succeed to reduce the computational complexity of the online stage and therefore enable the use of this algorithm in real time

Model estimation, identification and inference for next-generation functional data and spatial data

This dissertation is composed of three research projects focused on model estimation, identification, and inference for next-generation functional data and spatial data. The first project deals with data that are collected on a count or binary response with spatial covariate information. In this project, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies and the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. In the second project, motivated by recent work of analyzing data in the biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose to use flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images and other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-$n$ consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals (PCIs) and data-driven simultaneous confidence corridors (SCCs) for the coefficient functions are constructed. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to the spatially normalized Positron Emission Tomography (PET) data of Alzheimer\u27s Disease Neuroimaging Initiative (ADNI). In the third project, we propose a heterogeneous functional linear model to simultaneously estimate multiple coefficient functions and identify groups, such that coefficient functions are identical within groups and distinct across groups. By borrowing information from relevant subgroups, our method enhances estimation efficiency while preserving heterogeneity. We use an adaptive fused lasso penalty to shrink subgroup coefficients to shared common values within each group. We also establish the theoretical properties of our adaptive fused lasso estimators. To enhance the computation efficiency and incorporate neighborhood information, we propose to use a graph-constrained adaptive lasso. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to a dataset of hybrid maize grain yields from the Genomes to Fields consortium

Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio

Blending techniques in Curve and Surface constructions

Source at https://www.geofo.no/geofoN.html. <p

Trivariate C1-Splines auf gleichmäßigen Partitionen

In der vorliegenden Dissertation werden Splines auf gleichmäßigen Partitionen untersucht. Ziel der Arbeit ist die Analyse von multivariaten Splineräumen und die Entwicklung von neuen Methoden zur Lösung von Interpolations- und Approximationsproblemen mit trivariaten C1-Splines. Die entwickelten Methoden werden in Hinblick auf Lokalität, Stabilität und Approximationsordnung untersucht und die Ergebnisse dem Stand der Technik gegenübergestellt. Erstmalig kann dabei eine Quasi-Interpolationsmethode für trivariate C1-Splines vom totalen Grad zwei entwickelt werden und zur interaktiven Volumenvisualisierung mit Raycasting Techniken effizient eingesetzt werden

Preconditioning for linear systems arising from discretization of the Navier-Stokes equations using isogeometric analysis

Tato práce se zabývá iteračním řešením sedlobodových soustav lineárních algebraických rovnic získaných diskretizací Navierových--Stokesových rovnic pro nestlačitelné proudění pomocí isogemetrické analýzy (IgA). Konkrétně se zaměřuje na předpodmiňovače pro krylovovské metody. Jedním z cílů práce je prozkoumat efektivitu moderních blokových předpodmiňovačů pro různé isogeometrické diskretizace, tj. pro B-spline bázové funkce různého stupně a spojitosti, a poskytnout přehled o jejich chování v závislosti na různých parametrech úlohy. Hlavním cílem je na základě této studie navrhnout vhodný přístup k řešení těchto soustav s případnými úpravami, které by zlepšily vlastnosti dané metody pro soustavy získané isogeometrickou analýzou. Práce má dvě části. V první části jsou představeny úlohy pro nestlačitelné vazké proudění a metoda diskretizace pomocí isogeometrické analýzy. Dále uvádíme podrobný přehled metod řešení sedlobodových soustav lineárních rovnic, ve kterém se zaměřujeme především na blokové předpodmiňovače. Druhá část je věnována numerickým experimentům. Provádíme srovnání vybraných předpodmiňovačů pro několik stacionárních a nestacionarních úloh ve dvou a třech dimenzích. Zvláštní pozornost je věnována aproximaci matice hmotnosti, jejíž volba se ukazuje být v kontextu IgA důležitá, a okrajovým podmínkám pro PCD předpodmiňovač. Navrhujeme vhodnou kombinaci varianty PCD, okrajových podmínek a jejich škálování, abychom získali efektivní předpodmiňovač, který je robustní vzhledem k stupni a spojitosti diskretizace. V mnoha případech se tato volba ukazuje jako nejefektivnější z uvažovaných metod.ObhájenoThis doctoral thesis deals with iterative solution of the saddle-point linear systems obtained from discretization of the incompressible Navier--Stokes equations using the isogeometric analysis (IgA) approach. Specifically, it is focused on preconditioners for Krylov subspace methods. One of the goals of the thesis is to investigate the performance of the state-of-the-art block preconditioners for various IgA discretizations, i.e., for B-spline discretization bases of varying polynomial degree and interelement continuity, and provide an overview of their behavior depending on different problem parameters. The main goal is, based on the this study, to propose suitable solution approach to the considered linear systems with possible modifications that would improve the performance for IgA discretizations in particular. The thesis is basically divided into two parts. In the first part we introduce the mathematical model of incompressible viscous flow and the isogeometric analysis discretization method. Then we provide a detailed overview of the solution techniques for saddle-point linear systems, especially aimed at the family of block preconditioners. The second part is devoted to numerical experiments. We present a comparison of the selected preconditioners for several steady-state and time-dependent test problems in two and three dimensions. A particular attention is devoted to mass matrix approximation within the preconditioners, which appears to be important in the context of IgA, and to the boundary conditions for the pressure convection--diffusion (PCD) preconditioner. A suitable combination of PCD variant, boundary conditions and their appropriate scaling is proposed, leading to an effective preconditioner which is robust with respect to the discretization degree and continuity. In many cases, this choice of preconditioner proves to be the most efficient among all considered methods

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions