13 research outputs found
Eliminating the Gibbs phenomenon: the non-linear Petrov-Galerkin method for the convection-diffusion-reaction equation
In this thesis we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary and interior layers typically present in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The aim of this thesis is to develop a numerical method that eliminates Gibbs phenomena in the numerical approximation.
We consider a weak formulation of the partial differential equation of interest in L^q-type Sobolev spaces, with 1<q<â. We then apply a non-standard, non-linear Petrov-Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. By replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces, paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties.
For the convection-diffusion-reaction equation, we obtain a generalization of a similar approach from the L^2-setting to the L^q-setting and discuss the choices we have made regarding the continuous and discrete test spaces and the corresponding norms. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.
We furthermore demonstrate that the approximations obtained with our scheme qualitatively behave like the L^q-best approximation of the analytical solution in the same finite element space. We use this observation to study more closely in which cases the oscillations in the numerical approximation vanish as q tends to 1. To this end, we investigate Gibbs phenomena in the context of the L^q-best approximation of discontinuities in finite element spaces with 1â€qâ. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as q tends to 1. We then use these results to design the underlying meshes of the finite element spaces employed for our numerical scheme for the convection-diffusion-reaction equation such that Gibbs phenomena in the numerical approximation are eliminated.
While it is classical in the context of finite element methods to consider the solution of the convection diffusion-reaction equation in the Hilbert space H_0^1(Ω)$, the Banach Sobolev space W^{1,q}_0(Ω), 1<q<â, has received very little attention in this context. However, it is more general allowing for less regular solutions and, moreover, it allows us to consider the non-linear Petrov-Galerkin method that forms the centre of this research. In this thesis, we therefore also present a well-posedness theory for the convection-diffusion-reaction equation in the W^{1,q}_0(Ω)-W_0^{1,q'}(Ω) functional setting, 1/q+1/q'=1. The theory is based on directly establishing the inf-sup conditions which are essential to the analysis of the non-linear Petrov-Galerkin method. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplace operator
Gibbs Phenomena for -Best Approximation in Finite Element Spaces -- Some Examples
Recent developments in the context of minimum residual finite element methods
are paving the way for designing finite element methods in non-standard
function spaces. This, in particular, permits the selection of a solution space
in which the best approximation of the solution has desirable properties. One
of the biggest challenges in designing finite element methods are non-physical
oscillations near thin layers and jump discontinuities. In this article we
investigate Gibbs phenomena in the context of -best approximation of
discontinuities in finite element spaces with . Using carefully
selected examples, we show that on certain meshes the Gibbs phenomenon can be
eliminated in the limit as tends to . The aim here is to show the
potential of as a solution space in connection with suitably designed
meshes
Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation
In this article we consider the numerical approximation of the
convection-diffusion-reaction equation. One of the main challenges of designing
a numerical method for this problem is that boundary layers occurring in the
convection-dominated case can lead to non-physical oscillations in the
numerical approximation, often referred to as Gibbs phenomena. The idea of this
article is to consider the approximation problem as a residual minimization in
dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a
non-standard, non-linear PetrovGalerkin discretization, that is applicable to
reflexive Banach spaces such that the space itself and its dual are strictly
convex. Similar to discontinuous Petrov-Galerkin methods, this method is based
on minimizing the residual in a dual norm. Replacing the intractable dual norm
by a suitable discrete dual norm gives rise to a non-linear inexact mixed
method. This generalizes the Petrov-Galerkin framework developed in the context
of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the
convection-diffusion-reaction equation, this yields a generalization of a
similar approach from the L2-setting to the Lq-setting. A key advantage of
considering a more general Banach space setting is that, in certain cases, the
oscillations in the numerical approximation vanish as q tends to 1, as we will
demonstrate using a few simple numerical examples
Gibbs phenomena for Lq-best approximation in finite element spaces
Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as L q-type Sobolev spaces. For q â 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for L q-best approximations of discontinuities in finite element spaces with 1 †q < â. We prove sufficient conditions on meshes in one and two dimensions such that over-and undershoots vanish in the limit q â 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon
Eliminating Gibbs phenomena: A non-linear PetrovâGalerkin method for the convectionâdiffusionâreaction equation
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a non-standard, non-linear PetrovGalerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples
The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkinâs Method
While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H10(Ω), the Banach Sobolev space W1,q0(Ω), 1 less than â , is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the W1,q0(Ω)-W1,qâČ0(Ω) functional setting, 1q+1qâČ=1. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkinâs method in this setting, for a diffusion-dominated case and under the assumption of W1,qâČ-stability of the H10-projector
Eliminating the Gibbs phenomenon: the non-linear Petrov-Galerkin method for the convection-diffusion-reaction equation
In this thesis we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary and interior layers typically present in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The aim of this thesis is to develop a numerical method that eliminates Gibbs phenomena in the numerical approximation.
We consider a weak formulation of the partial differential equation of interest in L^q-type Sobolev spaces, with 1<q<â. We then apply a non-standard, non-linear Petrov-Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. By replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces, paving the way for designing finite element methods in non-standard function spaces. This, in particular, permits the selection of a solution space in which the best approximation of the solution has desirable properties.
For the convection-diffusion-reaction equation, we obtain a generalization of a similar approach from the L^2-setting to the L^q-setting and discuss the choices we have made regarding the continuous and discrete test spaces and the corresponding norms. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.
We furthermore demonstrate that the approximations obtained with our scheme qualitatively behave like the L^q-best approximation of the analytical solution in the same finite element space. We use this observation to study more closely in which cases the oscillations in the numerical approximation vanish as q tends to 1. To this end, we investigate Gibbs phenomena in the context of the L^q-best approximation of discontinuities in finite element spaces with 1â€qâ. Using carefully selected examples, we show that on certain meshes the Gibbs phenomenon can be eliminated in the limit as q tends to 1. We then use these results to design the underlying meshes of the finite element spaces employed for our numerical scheme for the convection-diffusion-reaction equation such that Gibbs phenomena in the numerical approximation are eliminated.
While it is classical in the context of finite element methods to consider the solution of the convection diffusion-reaction equation in the Hilbert space H_0^1(Ω)$, the Banach Sobolev space W^{1,q}_0(Ω), 1<q<â, has received very little attention in this context. However, it is more general allowing for less regular solutions and, moreover, it allows us to consider the non-linear Petrov-Galerkin method that forms the centre of this research. In this thesis, we therefore also present a well-posedness theory for the convection-diffusion-reaction equation in the W^{1,q}_0(Ω)-W_0^{1,q'}(Ω) functional setting, 1/q+1/q'=1. The theory is based on directly establishing the inf-sup conditions which are essential to the analysis of the non-linear Petrov-Galerkin method. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplace operator
Einfluss verschiedener SpĂŒllösungen auf den Haftverbund des Sealers Apexit Plus zum Wurzelkanaldentin
Der Misserfolg einer Wurzelkanalbehandlung hat viele Ursachen, die nicht immer eindeutig
sind. Dem Behandler stehen zahlreiche Herangehensweisen und Medikamente
zur VerfĂŒgung, um eine endodontische Behandlung durchzufĂŒhren.
Sie umfasst das mechanische Aufbereiten des Kanals, das SpĂŒlen mit geeigneter FlĂŒssigkeit
sowie einen bakterien- und flĂŒssigkeitsdichten Verschluss. Zum einen sollte
dieses Material eine Haftkraft zum umgebenden Wurzeldentin ausbilden, zum anderen
sollte zum gewÀhlten Kernmaterial ebenfalls eine AdhÀsion entstehen [101].
In dieser In-vitro-Studie sollte der Einfluss verschiedener SpĂŒllösungen auf den Haftverbund
zwischen Wurzelkanaldentin und dem calciumsalicylatbasierten Sealers
Apexit Plus untersucht werden.
In dieser Studie wurden 60 extrahierte, kariesfreie und unbehandelte Zahnwurzeln auf
fĂŒnf Gruppen randomisiert verteilt (n=12). Sie wurden dekapitiert und mit dem maschinellen
Feilensystem BioRaCe bei einer ArbeitslÀnge von 8 mm bis .02#60 aufbereitet.
Die zuvor gebildeten Gruppen wurden einer SpĂŒllösung zugeteilt: A =
Chlorhexidindiglukonat (CHX) 2%, B = Ethylendiamintetraacetat (EDTA) 16%, C =
Natriumhypochlorit (NaOCl) 3%, D = ZitronensÀure 40%, E = Aqua dest.
Nachdem die Proben mit ihrer zugeordneten FlĂŒssigkeit gespĂŒlt wurden, erfolgte die
sorgfĂ€ltige Trocknung mit Papierspitzen und anschlieĂendem Einbringen des vorbereiteten
Stahlspreaders mit dem zu untersuchenden Sealer.
Die Wurzeln lagerten 14 Tage unter feuchten Bedingungen. AnschlieĂend fanden die
Pullout-Tests in einer UniversalprĂŒfmaschine statt. Es wurde die maximale Kraft bis
zum adhĂ€siven Versagen bei einer PrĂŒfgeschwindigkeit von 2 mm/min ermittelt. ZusĂ€tzlich
erfolgte eine Auswertung der Frakturmodi an den experimentellen Spreadern.
Apexit Plus zeigte in Kombination mit CHX (0,73 MPa) und NaOCl (0,62 MPa) die
höchsten Haftwerte. Nach einer SpĂŒlung mit EDTA zeigten sich insgesamt die geringsten
Haftwerte (0,21 MPa). Es ergab sich ein signifikanter Einfluss der SpĂŒlungen
auf die Haftwerte (Kruskal-Wallis Test p= 0,034).
Unter den Bedingungen dieser Studie profitierte der Sealer Apexit Plus nicht von der
Schmierschichtentfernung durch EDTA. Die SpĂŒlung fĂŒhrte zu geringen Haftwerten.
Der calciumsalicylatbasierte Sealer verhielt sich somit gegensÀtzlich zu den meisten
anderen Wurzelkanalsealern, die in der Regel zu besseren Haftwerten nach Schmierschichtentfernung
gelangen.
Basierend auf den Ergebnissen dieser Studie mit Apexit Plus ist eine alleinige SpĂŒlung
mit EDTA nicht empfehlenswert. Ob Apexit Plus eine Verbesserung der Haftkraft
durch eine zusĂ€tzliche SpĂŒlung mit CHX erlangen kann, war nicht Ziel dieser Arbeit
und ist anderweitig zu erforschen
Eliminating Gibbs phenomena: A non-linear PetrovâGalerkin method for the convectionâdiffusionâreaction equation
Comparative analysis of machine learning algorithms for multi-syndrome classification of neurodegenerative syndromes
Importance: The entry of artificial intelligence into medicine is pending. Several methods have been used for the predictions of structured neuroimaging data, yet nobody compared them in this context.
Objective: Multi-class prediction is key for building computational aid systems for differential diagnosis. We compared support vector machine, random forest, gradient boosting, and deep feed-forward neural networks for the classification of different neurodegenerative syndromes based on structural magnetic resonance imaging.
Design, setting, and participants: Atlas-based volumetry was performed on multi-centric T1-weighted MRI data from 940 subjects, i.e., 124 healthy controls and 816 patients with ten different neurodegenerative diseases, leading to a multi-diagnostic multi-class classification task with eleven different classes.
Interventions: N.A.
Main outcomes and measures: Cohenâs kappa, accuracy, and F1-score to assess model performance.
Results: Overall, the neural network produced both the best performance measures and the most robust results. The smaller classes however were better classified by either the ensemble learning methods or the support vector machine, while performance measures for small classes were comparatively low, as expected. Diseases with regionally specific and pronounced atrophy patterns were generally better classified than diseases with widespread and rather weak atrophy.
Conclusions and relevance: Our study furthermore underlines the necessity of larger data sets but also calls for a careful consideration of different machine learning methods that can handle the type of data and the classification task best