Tesi sul tema "Asymptotic equivalents"

In this thesis, the asymptotic behavior and oscillation of solutions of dynamic equations on time scales are studied. In the first part of the thesis, asymptotic equivalence and asymptotic equilibrium of dynamic systems are investigated. Sufficient conditions are established for the asymptotic equivalence of linear systems and linear and quasilinear systems, respectively, and for the asymptotic equilibrium of quasilinear systems by unifying and extending some known results for differential systems and difference systems to dynamic systems on arbitrary time scales. In particular, for the asymptotic equivalence of differential systems, the well-known theorems of Levinson and Yakubovich are improved and the well-known theorem of Wintner for the asymptotic equilibrium of linear differential systems is generalized to arbitrary time scales. Some of our results for asymptotic equilibrium are new even for difference systems. In the second part, the oscillation of solutions of a particular class of second order nonlinear delay dynamic equations and, more generally, two-dimensional nonlinear dynamic systems, including delay-dynamic systems, are discussed. Necessary and sufficient conditions are derived for the oscillation of solutions of nonlinear delay dynamic equations by extending some continuous results. Specifically, the classical theorems of Atkinson and Belohorec are generalized. Sufficient conditions are established for the oscillation of solutions of nonlinear dynamic systems by unifying and extending the corresponding continuous and discrete results. Particularly, the oscillation criteria of Atkinson, Belohorec, Waltman, and Hooker and Patula are generalized.

In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional a-posteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.

Les mesures de résistivité en forage sont communément utilisées pour obtenirune meilleure caractérisation du sous-sol. L’utilisation d’un tube métallique pourcouvrir le puits complique énormément les simulations numériques pour lepotentiel électrique à cause de la faible épaisseur du tube et de sa conductivitéélevée par rapport à celle des formations du sous-sol. Dans ce travail, motivé pardes configurations réalistes, le tube est modélisé par une couche mincecylindrique d’épaisseur uniforme et la résistivité du tube est proportionnelle aucube de son épaisseur.Dans cette thèse, on se concentre sur ce problème pour obtenir des Conditionsde transmission (ITCs) approchées pour le potentiel électrique à travers le tubemétallique. Pour ce faire, on considère dans une première approche, un modèle2D en coordonnées cartésiennes, puis on résout le problème 3D axisymétriquequi est considéré dans la majorité des simulations de mesures de résistivité enforage à travers un tube. On considère d’abord le cas statique (fréquence nulle),puis on obtient des ITCs pour des fréquences non-nulles, lesquelles sontimportantes pour comprendre certains phénomènes physiques, comme les effetsDelaware et Groningen. Ensuite, on analyse les modèles en prouvant desrésultats de stabilité et convergence, et on évalue la performance numérique deces modèles en utilisant la méthode des éléments finis. Enfin, on construit dessolutions semi-analytiques pour ces modèles, lesquelles nous fournissent unemanière plus efficace d’évaluer nos modèles approchés par rapport aux solutionsnumériques (éléments finis)
Through-casing borehole resistivity measurements are commonly acquired inorder to characterize the Earth’s subsurface. The use of a casing surrounding theborehole highly complicates numerical simulations of the electric potential due toits thinness and a large contrast between casing conductivity and surroundingrock formation conductivity. In this work, we model the casing as a thin cylindricallayer of uniform thickness. Motivated by realistic scenarios, we realize that theconductivity of the case is typically proportional to its thickness to the power ofminus three.In this Ph.D. Dissertation, we focus on the above problem to derive ImpedanceTransmission Conditions (ITCs) in order to replace the metallic casing. To do so,we start by considering a 2D model in Cartesian coordinates that serves as aninitial approximation to solve the more realistic 3D axi-symmetric model (usingcylindrical coordinates) considered in most realistic through casing boreholesimulations. We start by considering the static (zero frequency) case, and we alsoderive ITCs for nonzero frequencies, which are important to understand certainphysical phenomena occurring in through casing borehole measurements,namely, the so called Delaware and Groningen effects. Then, we analyze thesemodels by proving stability and convergence results, and we asses the numericalperformance of these models by employing a Finite Element Method. Finally, wederive semi-analytical solutions for such models, which provide a more efficientway of evaluating our approximate models as in comparison with full numericalsolutions
Las medidas de resistividad en perforaciones a traves de tubos se utilizan demanera común para obtener una mejor caracterización del subsuelo de la tierra.El uso de un tubo que cubre el pozo complica enormemente las simulacionesnuméricas debido a su finura y al gran contraste entre la conductividad del tuboy la de las formaciones rocosas. En este trabajo, modelizamos el tubo como unamembrana cilíndrica fina de grosor uniforme. Basándonos en configuracionesrealistas, consideramos que la conductividad del tubo es proporcional a su grosora la potencia de menos tres.En esta tesis doctoral, nos concentramos en el problema anterior para obtenercodiciones de transmisión de impedancia (ITCs) que sirvan para reemplazar eltubo metálico. Para ello, empezamos por considerar un modelo 2D encoordenadas cartesianas, que sirve como una primera aproximación pararesolver el problema 3D con simetría axial (empleando coordenadas cilíndricas)considerado en la mayoría de las simulaciones realistas de perforaciones contubos. Empezamos por considerar el caso estático (frecuencia nula), y más tardeobtenemos ITCs para frecuencias no nulas, las cuales son importantes paraentender ciertos fenómenos físicos que ocurren al obtener medidas deresistividad en pozos a través de tubos, como por ejemplo, los efectos deDelaware y Groningen. Después, analizamos estos modelos demostrandoresultados de estabilidad y convergencia, y evaluamos el rendimiento numéricode estos modelos empleando el método de elementos finitos. Por último,obetnemos soluciones semi-analíticas para dichos modelos, las cualesproporcionan una manera más eficiente de evaluar las soluciones a nuestrosmodelos aproximados en comparación con soluciones puramente numéricas

Diese Arbeit untersucht Inferenz für Streuungsparameter bedingter Gaußprozesse anhand diskreter verrauschter Beobachtungen in einem Hochfrequenz-Setting. Unser Ziel dabei ist es, eine asymptotische Charakterisierung von effizienter Schätzung in einem allgemeine Gaußschen Rahmen zu finden. Für ein parametrisches Fundamentalmodell wird ein Hájek-Le Cam-Faltungssatz hergeleitet, welcher eine exakte asymptotische untere Schranke für Schätzmethoden liefert. Dazu passende obere Schranken werden konstruiert und die Bedeutung des Satzes wird verdeutlicht anhand zahlreicher Beispiele wie der (fraktionellen) Brownschen Bewegung, dem Ornstein-Uhlenbeck-Prozess oder integrierten Prozessen. Die Herleitung der Effizienzresultate basiert auf asymptotischen Äquivalenzen und kann für verschiedene Verallgemeinerungen des parametrischen Fundamentalmodells verwendet werden. Als eine solche Erweiterung betrachten wir das Schätzen der quadrierten Kovariation eines stetigen Martingals anhand verrauschter asynchroner Beobachtungen, welches ein fundamentales Schätzproblem in der Öknometrie ist. Für dieses Modell erhalten wir einen semi-parametrischen Faltungssatz, welcher bisherige Resultate im Sinne von Multidimensionalität, Asynchronität und Annahmen verallgemeinert. Basierend auf den vorhergehenden Herleitungen entwickeln wir einen statistischen Test für den Hurst-Parameter einer fraktionellen Brownschen Bewegung. Ein Score- und ein Likelihood-Quotienten-Test werden implementiert sowie analysiert und erste empirische Eindrücke vermittelt.
This work studies inference on scaling parameters of a conditionally Gaussian process under discrete noisy observations in a high-frequency regime. Our aim is to find an asymptotic characterisation of efficient estimation for a general Gaussian framework. For a parametric basic case model a Hájek-Le Cam convolution theorem is derived, yielding an exact asymptotic lower bound for estimators. Matching upper bounds are constructed and the importance of the theorem is illustrated by various examples of interest such as the (fractional) Brownian motion, the Ornstein-Uhlenbeck process or integrated processes. The derivation of the efficiency result is based on asymptotic equivalences and can be employed for several generalisations of the parametric basic case model. As such an extension we consider estimation of the quadratic covariation of a continuous martingale from noisy asynchronous observations, which is a fundamental estimation problem in econometrics. For this model, a semi-parametric convolution theorem is obtained which generalises existing results in terms of multidimensionality, asynchronicity and assumptions. Based on the previous derivations, we develop statistical tests on the Hurst parameter of a fractional Brownian motion. A score test and a likelihood ratio type test are implemented as well as analysed and first empirical impressions are given.

The statistical practice of equating is needed when scores on different versions of the same standardized test are to be compared. This thesis constitutes four contributions to the observed-score equating framework kernel equating. Paper I introduces the open source R package kequate which enables the equating of observed scores using the kernel method of test equating in all common equating designs. The package is designed for ease of use and integrates well with other packages. The equating methods non-equivalent groups with covariates and item response theory observed-score kernel equating are currently not available in any other software package. In paper II an alternative bandwidth selection method for the kernel method of test equating is proposed. The new method is designed for usage with non-smooth data such as when using the observed data directly, without pre-smoothing. In previously used bandwidth selection methods, the variability from the bandwidth selection was disregarded when calculating the asymptotic standard errors. Here, the bandwidth selection is accounted for and updated asymptotic standard error derivations are provided. Item response theory observed-score kernel equating for the non-equivalent groups with anchor test design is introduced in paper III. Multivariate observed-score kernel equating functions are defined and their asymptotic covariance matrices are derived. An empirical example in the form of a standardized achievement test is used and the item response theory methods are compared to previously used log-linear methods. In paper IV, Wald tests for equating differences in item response theory observed-score kernel equating are conducted using the results from paper III. Simulations are performed to evaluate the empirical significance level and power under different settings, showing that the Wald test is more powerful than the Hommel multiple hypothesis testing method. Data from a psychometric licensure test and a standardized achievement test are used to exemplify the hypothesis testing procedure. The results show that using the Wald test can provide different conclusions to using the Hommel procedure.

Le fonctionnement des algorithmes d’apprentissage automatique repose grandement sur la structure des données qu’ils doivent utiliser. La majorité des travaux de recherche en apprentissage automatique se concentre sur l’étude de données homogènes, souvent modélisées par des variables aléatoires indépendantes et identiquement distribuées. Pourtant, les données apparaissant en pratique sont souvent hétérogènes. Nous proposons dans cette thèse de considérer des données hétérogènes en les dotant d’un profil de variance. Cette notion, issue de la théorie des matrices aléatoires, nous permet notamment d’étudier des données issues de modèles de mélanges. Nous nous intéressons plus particulièrement à la problématique de la régression ridge à travers deux modèles : la régression ridge linéaire (linear ridge model) et la régression ridge à caractéristiques aléatoires (random feature ridge model). Nous étudions dans cette thèse la performance de ces deux modèles dans le cadre de la grande dimension, c’est-à-dire lorsque la taille de l’échantillon d’entraînement et la dimension des données tendent vers l’infini avec des vitesses comparables. Dans cet objectif, nous proposons des équivalents asymptotiques de l’erreur d’entraînement et de l’erreur de test relatives aux modèles d’intérêt. L’obtention de ces équivalents repose grandement sur l’étude spectrale issue de la théorie des matrices aléatoires, des probabilités libres et de la théorie des trafics. En effet, la mesure de la performance de nombreux modèles d’apprentissage dépend de la distribution des valeurs propres de matrices aléatoires. De plus, ces résultats nous ont permis d’observer des phénomènes spécifiques à la grande dimension, comme le phénomène de la double descente. Notre étude théorique s’accompagne d’expériences numériques illustrant la précision des équivalents asymptotiques que nous fournissons
The functioning of machine learning algorithms relies heavily on the structure of the data they are given to study. Most research work in machine learning focuses on the study of homogeneous data, often modeled by independent and identically distributed random variables. However, data encountered in practice are often heterogeneous. In this thesis, we propose to consider heterogeneous data by endowing them with a variance profile. This notion, derived from random matrix theory, allows us in particular to study data arising from mixture models. We are particularly interested in the problem of ridge regression through two models: the linear ridge model and the random feature ridge model. In this thesis, we study the performance of these two models in the high-dimensional regime, i.e., when the size of the training sample and the dimension of the data tend to infinity at comparable rates. To this end, we propose asymptotic equivalents for the training error and the test error associated with the models of interest. The derivation of these equivalents relies heavily on spectral analysis from random matrix theory, free probability theory, and traffic theory. Indeed, the performance measurement of many learning models depends on the distribution of the eigenvalues of random matrices. Moreover, these results enabled us to observe phenomena specific to the high-dimensional regime, such as the double descent phenomenon. Our theoretical study is accompanied by numerical experiments illustrating the accuracy of the asymptotic equivalents we provide

The thesis is in two parts. The first part is the paper “The Distribution of k-free integers” that my advisor, Dr. Roger Baker, and I submitted in February 2009. The reader will note that I have inserted additional commentary and explanations which appear in smaller text. Dr. Baker and I improved the asymptotic formula for the number of k-free integers less than x by taking advantage of exponential sum techniques developed since the 1980's. Both of us made substantial contributions to the paper. I discovered the exponent in the error term for the cases k=3,4, and worked the case k=3 completely. Dr. Baker corrected my work for k=4 and proved the result for k=5. He then generalized our work into the paper as it now stands. We also discussed and both contributed to parts of section 3 on bounds for exponential sums. The second part represents my own work guided by my advisor. I study the zeros of derivatives of Dirichlet L-functions. The first theorem gives an analog for a result of Speiser on the zeros of ζ'(s). He proved that RH is equivalent to the hypothesis that ζ'(s) has no zeros with real part strictly between 0 and ½. The last two theorems discuss zero-free regions to the left and right for L^{(k)}(s,χ).

Dans cette thèse, nous nous intéressons à la modélisation mathématique des hétérogénéités de longueurs caractéristiques beaucoup plus petites que la longueur d'ondes. La thèse consiste en deux parties. La partie théorique est dédiée à l'obtention d'un développement asymptotique raccordé: la solution est décrite à l'aide d'un développement de champ proche au voisinage de l'obstacle et par un développement de champ lointain hors de ce voisinage. Le développement de champ lointain met en jeu des solutions singulières de l'équation des ondes tandis que le champ proche lui est régi par un modèle quasi-statique. Ces deux développements sont alors raccordés dans une zone intermédiaire dite de raccord. Nous obtenons alors des estimations d'erreurs permettant de rendre rigoureux ce développement asymptotique formel. La deuxième partie est numérique. Elle décrit à la fois la méthode de Galerkine discontinue, une méthode de raffinement de maillage espace-temps et propose une discrétisation des modèles asymptotiques obtenues précédemment. Elle est illustrée par un certain nombre de tests numériques
In this thesis, we focus our attention on the modeling of heterogeneities which are smaller than the wavelength. The document is decomposed into two parts : a theoretical one and a numerical one. In the first part, we derive a matched asymptotic expansion composed of a far-field expansion and a near-field expansion. The terms of the far-field expansion are singular solutions of the wave equation whereas the terms of the near-field expansion satisfy quasistatic problems. These expansions are matched in an intermediate region. We justify mathematically this theory by proving error estimates. In the second part, we describe the Discontinuous Galerkin method, a local time stepping method and the implementation of the matched asymptotic method. Numerical simulations illustrate these results

This thesis presents a mathematical model of the nanoporous anode within a dyesensitised solar cell (DSC). The main purpose of this work is to investigate interfacial charge transfer and charge transport within the porous anode of the DSC under both illuminated and non-illuminated conditions. Within the porous anode we consider many of the charge transfer reactions associated with the electrolyte species, adsorbed dye molecules and semiconductor electrons at the semiconductor-dye- electrolyte interface. Each reaction at this interface is modelled explicitly via an electrochemical equation, resulting in an interfacial model that consists of a coupled system of non-linear algebraic equations. We develop a general model framework for charge transfer at the semiconductor-dye-electrolyte interface and simplify this framework to produce a model based on the available interfacial kinetic data. We account for the charge transport mechanisms within the porous semiconductor and the electrolyte filled pores that constitute the anode of the DSC, through a one- dimensional model developed under steady-state conditions. The governing transport equations account for the diffusion and migration of charge species within the porous anode. The transport model consists of a coupled system of non-linear differential equations, and is coupled to the interfacial model via reaction terms within the mass-flux balance equations. An equivalent circuit model is developed to account for those components of the DSC not explicitly included in the mathematical model of the anode. To obtain solutions for our DSC mathematical model we develop code in FORTRAN for the numerical simulation of the governing equations. We additionally employ regular perturbation analysis to obtain analytic approximations to the solutions of the interfacial charge transfer model. These approximations facilitate a reduction in computation time for the coupled mathematical model with no significant loss of accuracy. To obtain predictions of the current generated by the cell we source kinetic and transport parameter values from the literature and from experimental measurements associated with the DSC commissioned for this study. The model solutions we obtain with these values correspond very favourably with experimental data measured from standard DSC configurations consisting of titanium dioxide porous films with iodide/triiodide redox couples within the electrolyte. The mathematical model within this thesis enables thorough investigation of the interfacial reactions and charge transport within the DSC.We investigate the effects of modified cell configurations on the efficiency of the cell by varying associated parameter values in our model. We find, given our model and the DSC configuration investigated, that the efficiency of the DSC is improved with increasing electron diffusion, decreasing internal resistances and with decreasing dark current. We conclude that transport within the electrolyte, as described by the model, appears to have no limiting effect on the current predicted by the model until large positive voltages. Additionally, we observe that the ultrafast injection from the excited dye molecules limits the interfacial reactions that affect the DSC current.

This thesis presents a mathematical model of the nanoporous anode within a dyesensitised solar cell (DSC). The main purpose of this work is to investigate interfacial charge transfer and charge transport within the porous anode of the DSC under both illuminated and non-illuminated conditions. Within the porous anode we consider many of the charge transfer reactions associated with the electrolyte species, adsorbed dye molecules and semiconductor electrons at the semiconductor-dye- electrolyte interface. Each reaction at this interface is modelled explicitly via an electrochemical equation, resulting in an interfacial model that consists of a coupled system of non-linear algebraic equations. We develop a general model framework for charge transfer at the semiconductor-dye-electrolyte interface and simplify this framework to produce a model based on the available interfacial kinetic data. We account for the charge transport mechanisms within the porous semiconductor and the electrolyte filled pores that constitute the anode of the DSC, through a one- dimensional model developed under steady-state conditions. The governing transport equations account for the diffusion and migration of charge species within the porous anode. The transport model consists of a coupled system of non-linear differential equations, and is coupled to the interfacial model via reaction terms within the mass-flux balance equations. An equivalent circuit model is developed to account for those components of the DSC not explicitly included in the mathematical model of the anode. To obtain solutions for our DSC mathematical model we develop code in FORTRAN for the numerical simulation of the governing equations. We additionally employ regular perturbation analysis to obtain analytic approximations to the solutions of the interfacial charge transfer model. These approximations facilitate a reduction in computation time for the coupled mathematical model with no significant loss of accuracy. To obtain predictions of the current generated by the cell we source kinetic and transport parameter values from the literature and from experimental measurements associated with the DSC commissioned for this study. The model solutions we obtain with these values correspond very favourably with experimental data measured from standard DSC configurations consisting of titanium dioxide porous films with iodide/triiodide redox couples within the electrolyte. The mathematical model within this thesis enables thorough investigation of the interfacial reactions and charge transport within the DSC.We investigate the effects of modified cell configurations on the efficiency of the cell by varying associated parameter values in our model. We find, given our model and the DSC configuration investigated, that the efficiency of the DSC is improved with increasing electron diffusion, decreasing internal resistances and with decreasing dark current. We conclude that transport within the electrolyte, as described by the model, appears to have no limiting effect on the current predicted by the model until large positive voltages. Additionally, we observe that the ultrafast injection from the excited dye molecules limits the interfacial reactions that affect the DSC current.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface. We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem.

We present several results for growth functions of ideals of different com- binatorial structures. An ideal is a set downward closed under a containment relation, like the relation of subpartition for partitions, or the relation of induced subgraph for graphs etc. Its growth function (GF) counts elements of given size. For partition ideals we establish an asymptotics for GF of ideals that do not use parts from a finite set S and use this to construct ideal with highly oscillating GF. Then we present application characterising GF of particular partition ideals. We generalize ideals of ordered graphs to ordered uniform hypergraphs and show two dichotomies for their GF. The first result is a constant to linear jump for k-uniform hypergraphs. The second result establishes the polynomial to exponential jump for 3-uniform hypergraphs. That is, there are no ordered hypergraph ideals with GF strictly inside the constant-linear and polynomial- exponential range. We obtain in both dichotomies tight upper bounds. Finally, in a quite general setting we present several methods how to generate for various combinatorial structures pairs of sets defining two ideals with iden- tical GF. We call these pairs Wilf equivalent pairs and use the automorphism method and the replacement method to obtain such pairs. 1

碩士
國立交通大學
控制工程系
85
In this thesis, the partial differential equations for computing heat transferin PCB's and IC packages are approximated by using the finite difference method. We have proposed efficient equivalent circuit models to simulate the heat transfer mechanism of PCB's and IC's. We have also considered the complicate wiring models of PCB's in our simulation. In the modeling of IC's, we also pro-posed the asymptotic equivalent method to transfer the traditional three dimen-sion equivalent circuits of IC's into two dimension asymptotic equivalent cir- cuits. With the usage of this method, the simulation computation time and com-puter memory size can be reduced greatly. The asymptotic equivalent network ofIC's are used to build the library models designed in SPICE. According to ourconsideration of the accuracy, we can determine the order of simplified modelsi.e. different asymptotic equivalent models.

The submitted thesis studies penalty function methods for stochastic programming problems. The main objective of the paper is to examine penalty function methods for deterministic nonlinear programming, in particular exact penalty function methods, in order to enhance penalty function methods for stochastic programming. For this purpose, the equivalence of the original de- terministic nonlinear and the corresponding penalty function problem using arbi- trary vector norm as the penalty function is shown for convex and invex functions occurring in the problems, respectively. The obtained theorems are consequently applied to multiple chance constrained problems under finite discrete probability distribution to show the asymptotic equivalence of the probabilistic and the cor- responding penalty function problems. The practical use of the newly obtained methods is demonstrated on a numerical study, in which a comparison with other approaches is provided as well. 1