<p> The thesis presents an efficient self-adjoint approach to the S-parameter sensitivity analysis based on full-wave electromagnetic (EM) time-domain simulations with two commonly used numerical techniques: the finite-difference time-domain (FDTD) method and the transmission-line matrix (TLM) method. Without any additional simulations, we extract the response gradient with respect to all the design variables making use of the full-wave solution already generated by the system analysis. It allows the computation of the S-parameter derivatives as an independent post-process with negligible overhead. The sole requirement is the ability of the solver to export the field solution at user-defined points. Most in-house and commercial solvers have this ability, which makes our approach readily applicable to practical design problems.</p> <p> In the TLM-based self-adjoint techniques, we propose an algorithm to convert the electrical and magnetic field solutions into TLM voltages. The TLM-based discrete adjoint variable method (AVM) is originally developed to use incident and reflected voltages as the state variables. Our conversion algorithm makes the TLM-AVM method applicable to all time-domain commercial solvers, FDTD simulators included, with comparable accuracy and less memory overhead. Our approach is illustrated through waveguide examples using a TLM-based commercial simulator.</p> <p> Currently, our TLM-based self-adjoint approach is limited to loss-free homogeneous problems. However, our FDTD-based self-adjoint approach is valid for lossy inhomogeneous cases as well. The FDTD-based self-adjoint technique needs only the E-field values as the state variables. In order to make it also applicable to a TLM-based solver, whose mesh grid is displaced from the FDTD grid, we interpolate the E-field solution from the TLM mesh to that on the FDTD mesh. Our FDTD-based approach is validated through the response derivatives computation with respect to both shape and constitutive parameters in waveguide and antenna structures. The response derivatives can be used not only to guide a gradient-based optimizer, but also to provide a sufficient good initial guess for the solution of nonlinear inverse problems.</p> <p> Suggestions for further research are provided.</p> / Thesis / Master of Applied Science (MASc)

[ES] Uno de los objetivos más importantes en el análisis de la seguridad en el campo de la ingeniería nuclear es el cálculo, rápido y preciso, de la evolución de la potencia dentro del núcleo del reactor. La distribución de los neutrones se puede describir a través de la ecuación de transporte de Boltzmann. La solución de esta ecuación no puede obtenerse de manera sencilla para reactores realistas, y es por ello que se tienen que considerar aproximaciones numéricas. En primer lugar, esta tesis se centra en obtener la solución para varios problemas estáticos asociados con la ecuación de difusión neutrónica: los modos lambda, los modos gamma y los modos alpha. Para la discretización espacial se ha utilizado un método de elementos finitos de alto orden. Diversas características de cada problema espectral se analizan y se comparan en diferentes reactores. Después, se investigan varios métodos de cálculo para problemas de autovalores y estrategias para calcular los problemas algebraicos obtenidos a partir de la discretización espacial. La mayoría de los trabajos destinados a la resolución de la ecuación de difusión neutrónica están diseñados para la aproximación de dos grupos de energía, sin considerar dispersión de neutrones del grupo térmico al grupo rápido. La principal ventaja de la metodología que se propone es que no depende de la geometría del reactor, del tipo de problema de autovalores ni del número de grupos de energía del problema. Tras esto, se obtiene la solución de las ecuaciones estacionarias de armónicos esféricos. La implementación de estas ecuaciones tiene dos principales diferencias respecto a la ecuación de difusión neutrónica. Primero, la discretización espacial se realiza a nivel de pin. Por tanto, se estudian diferentes tipos de mallas. Segundo, el número de grupos de energía es, generalmente, mayor que dos. De este modo, se desarrollan estrategias a bloques para optimizar el cálculo de los problemas algebraicos asociados. Finalmente, se implementa un método modal actualizado para integrar la ecuación de difusión neutrónica dependiente del tiempo. Se presentan y comparan los métodos modales basados en desarrollos en función de los diferentes modos espaciales para varios tipos de transitorios. Además, también se desarrolla un control de paso de tiempo adaptativo, que evita la actualización de los modos de una manera fija y adapta el paso de tiempo en función de varias estimaciones del error. / [CA] Un dels objectius més importants per a l'anàlisi de la seguretat en el camp de l'enginyeria nuclear és el càlcul, ràpid i precís, de l'evolució de la potència dins del nucli d'un reactor. La distribució dels neutrons pot modelar-se mitjançant l'equació del transport de Boltzmann. La solució d'aquesta equació per a un reactor realístic no pot obtenir's de manera senzilla. És per això que han de considerar-se aproximacions numèriques. En primer lloc, la tesi se centra en l'obtenció de la solució per a diversos problemes estàtics associats amb l'equació de difusió neutrònica: els modes lambda, els modes gamma i els modes alpha. Per a la discretització espacial s'ha utilitzat un mètode d'elements finits d'alt ordre. Algunes de les característiques dels problemes espectrals s'analitzaran i es compararan per a diferents reactors. Tanmateix, diversos solucionadors de problemes d'autovalors i estratègies es desenvolupen per a calcular els problemes obtinguts de la discretització espacial. La majoria dels treballs per a resoldre l'equació de difusió neutrònica estan dissenyats per a l'aproximació de dos grups d'energia i sense considerar dispersió de neutrons del grup tèrmic al grup ràpid. El principal avantatge de la metodologia exposada és que no depèn de la geometria del reactor, del tipus de problema d'autovalors ni del nombre de grups d'energia del problema. Seguidament, s'obté la solució de les equacions estacionàries d'harmònics esfèrics. La implementació d'aquestes equacions té dues principals diferències respecte a l'equació de difusió. Primer, la discretització espacial es realitza a nivell de pin a partir de l'estudi de diferents malles. Segon, el nombre de grups d'energia és, generalment, major que dos. D'aquesta forma, es desenvolupen estratègies a blocs per a optimitzar el càlcul dels problemes algebraics associats. Finalment, s'implementa un mètode modal amb actualitzacions dels modes per a integrar l'equació de difusió neutrònica dependent del temps. Es presenten i es comparen els mètodes modals basats en l'expansió dels diferents modes espacials per a diversos tipus de transitoris. A més a més, un control de pas de temps adaptatiu es desenvolupa, evitant l'actualització dels modes d'una manera fixa i adaptant el pas de temps en funció de vàries estimacions de l'error. / [EN] One of the most important targets in nuclear safety analyses is the fast and accurate computation of the power evolution inside of the reactor core. The distribution of neutrons can be described by the neutron transport Boltzmann equation. The solution of this equation for realistic nuclear reactors is not straightforward, and therefore, numerical approximations must be considered. First, the thesis is focused on the attainment of the solution for several steady-state problems associated with neutron diffusion problem: the $\lambda$-modes, the $\gamma$-modes and the $\alpha$-modes problems. A high order finite element method is used for the spatial discretization. Several characteristics of each type of spectral problem are compared and analyzed on different reactors. Thereafter, several eigenvalue solvers and strategies are investigated to compute efficiently the algebraic eigenvalue problems obtained from the discretization. Most works devoted to solve the neutron diffusion equation are made for the approximation of two energy groups and without considering up-scattering. The main property of the proposed methodologies is that they depend on neither the reactor geometry, the type of eigenvalue problem nor the number of energy groups. After that, the solution of the steady-state simplified spherical harmonics equations is obtained. The implementation of these equations has two main differences with respect to the neutron diffusion. First, the spatial discretization is made at level of pin. Thus, different meshes are studied. Second, the number of energy groups is commonly bigger than two. Therefore, block strategies are developed to optimize the computation of the algebraic eigenvalue problems associated. Finally, an updated modal method is implemented to integrate the time-dependent neutron diffusion equation. Modal methods based on the expansion of the different spatial modes are presented and compared in several types of transients. Moreover, an adaptive time-step control is developed that avoids setting the time-step with a fixed value and it is adapted according to several error estimations. / Carreño Sánchez, AM. (2020). Integration methods for the time dependent neutron diffusion equation and other approximations of the neutron transport equation [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/144771

With the rapid increase in available compute power and memory, and bolstered by the advent of efficient formulations and algorithms, the role of 3D full-wave computational methods for accurate modelling of complex electromagnetic (EM) structures has gained in significance. The range of problems includes Radar Cross Section (RCS) computation, analysis and design of antennas and passive microwave circuits, bio-medical non-invasive detection and therapeutics, energy harvesting etc. Further, with the rapid advances in technology trends like System-in-Package (SiP) and System-on-Chip (SoC), the fidelity of chip-to-chip communication and package-board electrical performance parameters like signal integrity (SI), power integrity (PI), electromagnetic interference (EMI) are becoming increasingly critical. Rising pin-counts to satisfy functionality requirements and decreasing layer-counts to maintain cost-effectiveness necessitates 3D full wave electromagnetic solution for accurate system modelling. Method of Moments (MoM) is one such widely used computational technique to solve a 3D electromagnetic problem with full-wave accuracy. Due to lesser number of mesh elements or discretization on the geometry, MoM has an advantage of a smaller matrix size. However, due to Green's Function interactions, the MoM matrix is dense and its solution presents a time and memory challenge. The thesis focuses on formulation and development of novel techniques that aid in fast MoM based electromagnetic solutions. With the recent paradigm shift in computer hardware architectures transitioning from single-core microprocessors to multi-core systems, it is of prime importance to parallelize the serial electromagnetic formulations in order to leverage maximum computational benefits. Therefore, the thesis explores the possibilities to expedite an electromagnetic simulation by scalable parallelization of near-linear complexity algorithms like Fast Multipole Method (FMM) on a multi-core platform. Secondly, with the best of parallelization strategies in place and near-linear complexity algorithms in use, the solution time of a complex EM problem can still be exceedingly large due to over-meshing of the geometry to achieve a desired level of accuracy. Hence, the thesis focuses on judicious placement of mesh elements on the geometry to capture the physics of the problem without compromising on accuracy- a technique called Adaptive Mesh Refinement. This facilitates a reduction in the number of solution variables or degrees of freedom in the system and hence the solution time. For multi-scale structures as encountered in chip-package-board systems, the MoM formulation breaks down for parts of the geometry having dimensions much smaller as compared to the operating wavelength. This phenomenon is popularly known as low-frequency breakdown or low-frequency instability. It results in an ill-conditioned MoM system matrix, and hence higher iteration count to converge when solved using an iterative solver framework. This consequently increases the solution time of simulation. The thesis thus proposes novel formulations to improve the spectral properties of the system matrix for real-world complex conductor and dielectric structures and hence form well-conditioned systems. This reduces the iteration count considerably for convergence and thus results in faster solution. Finally, minor changes in the geometrical design layouts can adversely affect the time-to-market of a commodity or a product. This is because the intermediate design variants, in spite of having similarities between them are treated as separate entities and therefore have to follow the conventional model-mesh-solve workflow for their analysis. This is a missed opportunity especially for design variant problems involving near-identical characteristics when the information from the previous design variant could have been used to expedite the simulation of the present design iteration. A similar problem occurs in the broadband simulation of an electromagnetic structure. The solution at a particular frequency can be expedited manifold if the matrix information from a frequency in its neighbourhood is used, provided the electrical characteristics remain nearly similar. The thesis introduces methods to re-use the subspace or Eigen-space information of a matrix from a previous design or frequency to solve the next incremental problem faster.

Micromagnetics is a continuum mechanics theory of magnetic materials widely used in industry and academia. In this thesis we describe a complete numerical method, with a number of novel components, for the computational solution of dynamic micromagnetic problems by solving the Landau-Lifshitz-Gilbert (LLG) equation. In particular we focus on the use of the implicit midpoint rule (IMR), a time integration scheme which conserves several important properties of the LLG equation. We use the finite element method for spatial discretisation, and use nodal quadrature schemes to retain the conservation properties of IMR despite the weak-form approach. We introduce a novel, generally-applicable adaptive time step selection algorithm for the IMR. The resulting scheme selects error-appropriate time steps for a variety of problems, including the semi-discretised LLG equation. We also show that it retains the conservation properties of the fixed step IMR for the LLG equation. We demonstrate how hybrid FEM/BEM magnetostatic calculations can be coupled to the LLG equation in a monolithic manner. This allows the coupled solver to maintain all properties of the standard time integration scheme, in particular stability properties and the energy conservation property of IMR. We also develop a preconditioned Krylov solver for the coupled system which can efficiently solve the monolithic system provided that an effective preconditioner for the LLG sub-problem is available. Finally we investigate the effect of the spatial discretisation on the comparative effectiveness of implicit and explicit time integration schemes (i.e. the stiffness). We find that explicit methods are more efficient for simple problems, but for the fine spatial discretisations required in a number of more complex cases implicit schemes become orders of magnitude more efficient.

[pt] Companhias de petróleo investindo no desenvolvimento de campos de hidrocarboneto dependem de estudos de reservatórios para realizarem previsões de produção e quantificarem os riscos associados à economicidade dos projetos. Neste sentido, a área de modelagem de reservatórios é de suma importância, sendo responsável por prever o desempenho futuro do reservatório sob diversas condições operacionais. Considerando que a solução dos sistemas de equações construídos a cada passo de tempo de uma simulação, durante o ciclo de linearização, é a parte que apresenta a maior demanda computacional, esta dissertação foca na análise de diferentes técnicas de solvers numéricos que podem ser aplicadas a simuladores, para mensurar seus desempenhos. Os solvers numéricos mais adequados para a solução de grandes sistemas de equações, tais como os encontrados em simulações de reservatórios, são os denominados solvers iterativos, que gradativamente aproximam a solução de um dado problema por meio da combinação de um método iterativo e um precondicionador. Os métodos iterativos avaliados nesta pesquisa foram o Gradiente Biconjugado Estabilizado (BiCGSTAB), Mínimos Resíduos Generalizado (GMRES) e Minimização Ortogonal (ORTHOMIN). Além disso, três técnicas de precondicionamento foram implementadas para auxiliar os métodos iterativos, sendo estas a Decomposição LU Incompleta (ILU), Fatoração Aninhada (NF) e Pressão Residual Restrita (CPR). A combinação destes diferentes métodos iterativos e precondicionadores permite a avaliação de diversas configurações distintas de solvers, em termos de seus desempenhos em um simulador. Os testes numéricos conduzidos neste trabalho utilizaram um novo simulador de reservatórios que está sendo desenvolvido pela Pontifícia Universidade Católica (PUC-Rio) em conjunto com a Petrobras. O objetivo dos testes foi analisar a robustez e eficiência de cada um dos solvers quanto à sua capacidade de resolver as equações de escoamento multifásico no meio poroso, visando assim auxiliar na seleção do solver mais adequado para o simulador. / [en] Petroleum companies investing in the development of hydrocarbon fields rely upon a variety of reservoir studies to perform production forecasts and quantify the risks associated with the economics of their projects. Integral to these studies is the discipline of reservoir modeling, responsible for predicting future reservoir performance under various operational conditions. Considering that the most time-demanding aspect of reservoir simulations is the solution of the systems of equations that arise within the linearization cycles at each time-step, this research focuses on analyzing different numerical solver techniques to be applied to a simulator, in order to assess their performance. The numerical solvers most suited for the solution of very large systems of equations, such as those encountered in reservoir simulations, are the so-called iterative solvers, which gradually approach the solution to a problem by combining an iterative strategy with a preconditioning method. The iterative methods examined in this research were the Stabilized Biconjugate Gradient (BiCGSTAB), the Generalized Minimum Residual (GMRES), and the Orthogonal Minimization (ORTHOMIN) methods. Furthermore, three preconditioning techniques were implemented to aid the iterative methods, namely the Incomplete LU Factorization (ILU), the Nested Factorization (NF), and the Constrained Pressure Residual (CPR) methods. The combination of these different iterative methods and preconditioners enables the appraisal of several distinct solver configurations, in terms of their performance in a simulator. The numerical tests conducted in this work made use of a new reservoir simulator currently under development at Pontifical Catholic University of Rio de Janeiro (PUC-Rio), as part of a joint project with Petrobras. The objective of these tests was to assess the robustness and efficiency of each solver in the solution of the multiphase flow equations in porous media, and support the selection of the solver most suited for the simulator.

This research focuses on implementing and improving a parallel solution framework for the linear static analysis of large structural models on PC clusters. The framework consists of two separate programs where the first one is responsible from preparing data for the parallel solution that involves partitioning, workload balancing, and equation numbering. The second program is a fully parallel nite element program that utilizes substructure based solution approach with direct solvers. The first step of data preparation is partitioning the structure into substructures. After creating the initial substructures, the estimated imbalance of the substructures is adjusted by iteratively transferring nodes from the slower substructures to the faster ones. Once the final substructures are created, the solution phase is initiated. Each processor assembles its substructure&#039 / s stiffness matrix and condenses it to the interfaces. The interface equations are then solved in parallel with a block-cyclic dense matrix solver. After computing the interface unknowns, each processor calculates the internal displacements and element stresses or forces. Comparative tests were done to demonstrate the performance of the solution framework.

Made available in DSpace on 2014-12-17T15:47:48Z (GMT). No. of bitstreams: 1 DiegoCBO.pdf: 564820 bytes, checksum: eedd81c1881d60fea03c3dcdd8556734 (MD5) Previous issue date: 2007-11-07 / O m?todo de combina??o de Nelson-Oppen permite que v?rios procedimentos de decis?o, cada um projetado para uma teoria espec?&#64257;ca, possam ser combinados para inferir sobre teorias mais abrangentes, atrav?s do princ?pio de propaga??o de igualdades. Provadores de teorema baseados neste modelo s?o bene&#64257;ciados por sua caracter?stica modular e podem evoluir mais facilmente, incrementalmente. Di&#64256;erence logic ? uma subteoria da aritm?tica linear. Ela ? formada por constraints do tipo x &#8722; y &#8804; c, onde x e y s?o vari?veis e c ? uma constante. Di&#64256;erence logic ? muito comum em v?rios problemas, como circuitos digitais, agendamento, sistemas temporais, etc. e se apresenta predominante em v?rios outros casos. Di&#64256;erence logic ainda se caracteriza por ser modelada usando teoria dos grafos. Isto permite que v?rios algoritmos e&#64257;cientes e conhecidos da teoria de grafos possam ser utilizados. Um procedimento de decis?o para di&#64256;erence logic ? capaz de induzir sobre milhares de constraints. Um procedimento de decis?o para a teoria de di&#64256;erence logic tem como objetivo principal informar se um conjunto de constraints de di&#64256;erence logic ? satisfat?vel (as vari?veis podem assumir valores que tornam o conjunto consistente) ou n?o. Al?m disso, para funcionar em um modelo de combina??o baseado em Nelson-Oppen, o procedimento de decis?o precisa ter outras funcionalidades, como gera??o de igualdade de vari?veis, prova de inconsist?ncia, premissas, etc. Este trabalho apresenta um procedimento de decis?o para a teoria de di&#64256;erence logic dentro de uma arquitetura baseada no m?todo de combina??o de Nelson-Oppen. O trabalho foi realizado integrando-se ao provador haRVey, de onde foi poss?vel observar o seu funcionamento. Detalhes de implementa??o e testes experimentais s?o relatados

Dans le contexte de l'analyse numérique en calcul de structures, la génération de maillages conformes sur des modèles à géométrie complexe conduit à des tailles de modèles importantes, et amène à imaginer de nouvelles approches éléments finis. Le temps de génération d'un maillage est directement lié à la complexité de la géométrie, augmentant ainsi considérablement le temps de calcul global. Les processeurs graphiques (GPU) offrent de nouvelles opportunités pour le calcul en temps réel. L'architecture grille des GPU a été utilisée afin d'implémenter une méthode éléments finis sur maillage cartésien. Ce maillage est particulièrement adapté à la parallélisation souhaitée par les processeurs graphiques et permet un gain de temps important par rapport à un maillage conforme à la géométrie. Les formulations de la méthode des éléments finis ainsi que de la méthode des éléments finis étendue ont été reprises afin d'être adaptées à notre méthode. La méthode des éléments finis étendus permet de prendre en compte la géométrie et les interfaces à travers un choix adéquat de fonctions d'enrichissement. Cette méthode discrétise par exemple sans mailler explicitement les fissures, et évite surtout de remailler au cours de leur propagation. Des adaptations de cette méthode sont faites afin de ne pas avoir besoin d'un maillage conforme à la géométrie. La géométrie est définie implicitement par une fonction surfaces de niveau, ce qui permet une bonne approximation de la géométrie et des conditions aux limites sans pour autant s'appuyer sur un maillage conforme. La géométrie est représentée par une fonction surfaces de niveau que nous appelons la densité. La densité est supérieure à 0.5 à l'intérieur du domaine de calcul et inférieure à 0.5 à l'extérieur. Cette fonction densité, définie par ses valeurs aux points noeuds du maillage, est interpolée à l'intérieur de chaque élément. Une méthode d'intégration adaptée à cette représentation géométrique est proposée. En effet, certains éléments sont coupés par la fonction surfaces de niveau et l'intégration de la matrice de raideur ne doit se faire que sur la partie pleine de l'élément. La méthode de quadrature de Gauss qui permet d'intégrer des polynômes de manière exacte n'est plus adaptée. Nous proposons d'utiliser une méthode de quadrature avec des points d'intégration répartis sur une grille régulière et dense. L'intégration peut s'avérer coûteuse en temps de calcul, c'est pour cette raison que nous proposons une technique d'apprentissage donnant la matrice élémentaire de rigidité en fonction des valeurs de la fonction surfaces de niveau aux sommets de l'élément considéré. Cette méthode d'apprentissage permet de grandes améliorations du temps de calcul des matrices élémentaires. Les résultats obtenus après analyse par la méthode des éléments finis standard ou par la méthode des éléments finis sur maillage cartésien ont une taille qui peut croître énormément selon la complexité des modèles, ainsi que la précision des schémas de résolution. Dans un contexte de programmation sur processeurs graphiques, où la mémoire est limitée, il est intéressant d'arriver à compresser ces données. Nous nous sommes intéressés à la compression des modèles et des résultats éléments finis par la transformée en ondelettes. La compression mise en place aidera aussi pour les problèmes de stockage en réduisant la taille des fichiers générés, et pour la visualisation des données. / Generating a conformal mesh on complex geometries leads to important model size of structural finite element simulations. The meshing time is directly linked to the geometry complexity and can contribute significantly to the total turnaround time. Graphics processing units (GPUs) are highly parallel programmable processors, delivering real performance gains on computationally complex, large problems. GPUs are used to implement a new finite element method on a Cartesian mesh. A Cartesian mesh is well adapted to the parallelism needed by GPUs and reduces the meshing time to almost zero. The novel method relies on the finite element method and the extended finite element formulation. The extended finite element method was introduced in the field of fracture mechanics. It consists in enriching the basis functions to take care of the geometry and the interface. This method doesn't need a conformal mesh to represent cracks and avoids refining during their propagation. Our method is based on the extended finite element method, with a geometry implicitly defined, wich allows for a good approximation of the geometry and boundary conditions without a conformal mesh.To represent the model on a Cartesian grid, we use a level set representing a density. This density is greater than 0.5 inside the domain and less than 0.5 outside. It takes 0.5 on the boundary. A new integration technique is proposed, adapted to the geometrical representation. For the element cut by the levet set, only the part full of material has to be integrated. The Gauss quadrature is no longer adapted. We introduce a quadrature method with integration points on a cartesian dense grid.In order to reduce the computational effort, a learning approach is then considered to form the elementary stiffness matrices as function of density values on the vertices of the elements. This learning method reduces the stiffness matrices time computation. Results obtained after analysis by finite element method or the novel finite element method can have important storage size, dependant of the model complexity and the resolution scheme exactitude. Due to the limited direct memory of graphics processing units, the data results are compressed. We compress the model and the element finite results with a wavelet transform. The compression will help for storage issue and also for data visualization.

In this thesis, we construct a new optimal contractivity-preserving (CP) explicit, 2-step, 6-stage, 6-derivative, Hermite--Birkhoff--Obrechkoff method of order 13, denoted by HBO(13) with nonnegative coefficients, for solving nonstiff first-order initial value problems y'=f(t,y), y(t_0)=y_0. This new method is the combination of a CP 2-step, 6-derivative, Hermite--Obrechkoff of order 9, denoted by HO(9), and a 6-stage Runge-Kutta method of order 5, denoted by RK(6,5). The new HBO(13) method has order 13. We compare this new method, programmed in Matlab, to Adams-Bashforth-Moulton method of order 13 in PECE mode, denoted by ABM(13), by testing them on several frequently used test problems, and show that HBO(13) is more efficient with respect to the CPU time, the global error at the endpoint of integration and the relative energy error. We show that the new HBO(13) method has a larger scaled interval of absolute stability than ABM(13) in PECE mode.

We establish multiresolution norm equivalences in weighted spaces <i>L²_w</i>((0,1)) with possibly singular weight functions <i>w(x)</i>&ge;0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function <i>w(x)</i> within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning <i>p</i>-Version FEM and wavelet discretizations of degenerate elliptic problems.