1,721,109 research outputs found
On the convergence of the rescaled localized radial basis function method
Full text linkThe rescaled localized RBF method was introduced in Deparis, Forti, and Quarteroni (2014) for scattered data interpolation. It is a rational approximation method based on interpolation with compactly supported radial basis functions. It requires the solution of two linear systems with the same sparse matrix, which has a small condition number, due to the scaling of the basis function. Hence, it can be computed using an unpreconditioned conjugate gradient method in linear time. Numerical evidence provided in Deparis, Forti, and Quarteroni (2014) shows that the method produces good approximations for many examples but no theoretical results were provided. In this paper, we discuss the convergence of the rescaled localized RBF method in the case of quasi-uniform data and stationary scaling. As the method is not only interpolatory but also reproduces constants exactly, linear convergence is expected. We can show this linear convergence up to a certain conjecture
Fast computation of orthonormal basis for RBF spaces through Krylov space methods
No full textIn recent years, in the setting of radial basis function, the study of approximation algorithms has particularly focused on the construction of (stable) bases for the associated Hilbert spaces. One of the ways of describing such spaces and their properties is the study of a particular integral operator and its spectrum. We proposed in a recent work the so-called WSVD basis, which is strictly connected to the eigen-decomposition of this operator and allows to overcome some problems related to the stability of the computation of the approximant for a wide class of radial kernels. Although effective, this basis is computationally expensive to compute. In this paper we discuss a method to improve and compute in a fast way the basis using methods related to Krylov subspaces. After reviewing the connections between the two bases, we concentrate on the properties of the new one, describing its behavior by numerical tests
A new stable basis for radial basis function interpolation
No full textIt is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space NΦ(Ω) associated to a kernel Φ on a domain Ω© 2013 Elsevier B.V. All rights reserved
Subendotyping of Dermatophagoides pteronyssinus–Induced Rhinitis and Its Impact on Respiratory Comorbidities
No full textBackground: The impact of delayed hypersensitivity to Dermatophagoides pteronyssinus (DP) on comorbidities of allergic rhinitis (AR) is unknown. Objective: The primary end point was to test the hypothesis that DP-induced AR could be divided into 2 subendotypes on the basis of presence or absence of a delayed-type mite sensitization detected by the positive result of atopy patch test for DP (DP-APT). The second end point was to evaluate differences in the long-term risk of respiratory comorbidities and nasal airway response to mite exposure. Methods: In a prospective observational study, we included 472 patients with DP-induced AR. A total of 343 patients had positive results of skin prick test/serum specific IgE and DP-APT and were assigned to a subendotype with both IgE- and T-cell–mediated mite sensitization (BMSS). The remaining 129 patients without delayed-type mite sensitization were included in the subendotype with only IgE-mediated mite sensitization. Nasal allergen provocation test with active anterior rhinomanometry, paranasal sinuses computed tomography scan, nasal endoscopy, and spirometry were performed. Results: At baseline, BMSS showed a larger increase in nasal airway resistance, total nasal score, and visual analogue scale score to mite exposure. During a 15-year follow-up, 56 patients developed chronic rhinosinusitis with nasal polyps, with higher incidence in BMSS than in the subendotype with only IgE-mediated mite sensitization (50 patients, 14.6% vs 6 patients, 12.4%; P < .001). BMSS also showed a higher incidence of conjunctivitis (25.7% vs 12.4%; P < .01). The rate of adult-onset asthma did not differ between groups, but patients with BMSS showed a more frequent link to chronic rhinosinusitis with nasal polyps (6 of 29 patients, 20.7% vs 0 of 10 patients, 0%). DP-APT independently predicted chronic rhinosinusitis with nasal polyps and conjunctivitis. Conclusions: Two subendotypes with significantly different clinical outcome can be identified among patients with DP-induced AR according to the presence of delayed-type mite sensitization detected by positive DP-APT result
Risk of Chronic Rhinosinusitis with Nasal Polyps in Endotypes of Dermatophagoides pteronyssinus-Induced Rhinitis
No full textBackground: Observation of the natural history of two emerging endotypes of allergic rhinitis, local-sensitization rhinitis (LAR) and dual-allergic rhinitis (DAR), compared with systemic-sensitization rhinitis (AR), could improve knowledge of the role of allergy in chronic rhinosinusitis with nasal polyps (CRSwNP). Objective: To test the hypothesis that endotypes of Dermatophagoides pteronyssinus (DP)-induced rhinitis were risk factors for CRSwNP and adult-onset asthma and to investigate whether delayed hypersensitivity to DP, assessed by atopy patch test, could be a contributing factor. Methods: We conducted a prospective observational study over 15 years on a cohort of 999 patients: 468 with AR, 333 with LAR, and 198 with DAR. The latter endotype was characterized by the coexistence of seasonal disease caused by systemic sensitization to pollen in patients with DP-induced LAR. The study design included a physical visit; ear, nose, and throat examination with anterior rhinoscopy; skin prick test; serum-specific IgE; DP-atopy patch test; nasal allergen provocation test with DP; paranasal sinuses computed tomography scan; nasal endoscopy; and spirometry. Results: During 15 years of follow-up, 194 patients developed CRSwNP with a higher rate of LAR (28.2%) and DAR (22.2%) than AR (12%). For LAR and DAR, 7.5% and 10.6% of patients developed adult-onset asthma temporally linked to CRSwNP in 68% and 71.4% of cases, respectively. A total of 858 patients with rhinitis had delayed hypersensitivity to DP. Moreover, DP-ATP was an independent predictive factor for CRSwNP and had elevated positive and negative predictive values for localized allergic disease of the nasal mucosa. Conclusions: Endotypes of DP-induced allergic rhinitis represent risk factors for CRSwNP. Patients with local-sensitization rhinitis and DAR are more at risk than those with AR. In these emerging endotypes, progression toward CRSwNP is often associated with the development of adult-onset asthma. Chronic rhinosinusitis with nasal polyps shows several possible indicators for type 2 endotype. Delayed hypersensitivity to DP is an independent predictive factor for CRSwNP
A mixed interpolation-regression approximation operator on the triangle
Full text linkIn several applications, ranging from computational geometry and finite element analysis to computer graphics, there is a need to approximate functions defined on triangular domains rather than rectangular ones. For this purpose, frequently used interpolation methods include barycentric interpolation, piecewise linear interpolation, and polynomial interpolation. However, the use of polynomial interpolation methods may suffer from the Runge phenomenon, affecting the accuracy of the approximation in the presence of equidistributed data. In these situations, the constrained mock-Chebyshev least squares approximation on rectangular domains was shown to be a successful approximation tool. In this paper, we extend it to triangular domains, by using both Waldron and discrete Leja points. This paper is dedicated to Len Bos on the occasion of his retirement. Len, for us, is a master of mathematics and also a big friend. He introduced us to the fascinating world of "finding good interpolation no..
Fast and stable rational RBF-based partition of unity interpolation
Full text linkWe perform a local computation via the Partition of Unity (PU) method of rational Radial Basis Function (RBF) interpolants. We investigate the well-posedness of the problem and we provide error bounds. The resulting scheme, efficiently implemented by means of the Deflation Accelerated Conjugate Gradient (DACG), enables us to deal with huge data sets and, thanks to the use of Variably Scaled Kernels (VSKs), it turns out to be stable. For functions with steep gradients or discontinuities, which are truly common in applications, the results show that the new proposed method outperforms the classical and rescaled PU schemes
Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization
Full text linkThe aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains
A rescaled method for RBF approximation
No full textIn the recent paper [1], a new method to compute stable kernel-based interpolants has been presented. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allows us to consider its error and stability properties
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