Interpolation conditions on rational matrix functions expressed in terms of residues are studied. As a compact way of expressing tangential interpolation conditions of arbitrarily high multiplicity possibly from both sides simultaneously, interpolation conditions are represented in terms of residues. The minimal possible complexity, measured by the McMillan degree, of interpolants is found in terms of the controllability and the observability indices of certain pairs of matrices which are part of given data. An interpolant of such complexity is obtained in realization form. This leads to another approach to the partial realization problem. As a generalization of the well-known Lagrange interpolation problem for scalar polynomials, the problem of seeking for a matrix polynomial interpolant of low complexity is studied. The main tool is state space methods borrowed from systems theory. After adoption of state space methods, problems concerning rational matrix functions are reduced to the realm of linear algebra. / Ph. D.

The convergence properties of the permanent exponential interpolation series f(Z) = 1^Zf(0) + (2^Z - 1^Z)Δf(0) + (3^Z - 2.2^Z + 1^Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation U_n(Z) = ∑ⁿ_k=0 (-1)^k(ⁿ_k)(n - i + 1)^Z, Δ⁽ⁿ⁾ f(0) = Δ(Δ-1)…(Δ - n + 1)f(0), the series can be written more compactly as f(Z) = ∑^∞₀ U_n(Z)/n!Δ⁽ⁿ⁾ f(0). It is shown that Δ⁽ⁿ⁾ f(0) can be represented as Δ⁽ⁿ⁾ f(0) = M_n(f) = 1/2πi ∫_Γ (e^ω - 1)⁽ⁿ⁾ F(ω)dω, where F(ω) is the Borel transform of f(Z) and Γ encloses the convex hull of the singularities of F(ω). It is further shown that the series ∑^∞₀ U_n(Z)/n! (e^ω - 1)⁽ⁿ⁾ forms a uniformly convergent Gregory-Newton series, convergent to e^Zω in any bounded region in the strip |I(ω)| < π/2. The Polya representation of an entire function of exponential type is then formed, and the method of kernel expansion (R. P. Boas, and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, 1964) yields the desired result. This result is summed up in the following: Theorem Any entire function of exponential type such that the convex hull of the set of singularities of its Borel transform lies in the strip |I(ω)| < π/2. admits the convergent exponential interpolation series expansion f(Z) = ∑^∞_n=0 U_n(Z)/n!Δ⁽ⁿ⁾ f(0) for all Z. / M.S.

Throughout a ground vehicle development program, it is necessary to possess the loads the vehicle will experience. Unfortunately, actual loads are only available at the conclusion of the program, when the vehicle has been built and design changes are costly. The design engineer is challenged with using predicted loads early in the design process, when changes are relatively easy and inexpensive to make. It is advantageous, therefore, to accurately predict these loads early in the program, thus improving the vehicle design and, ultimately, saving time and money. The prediction of these loads depends on the fidelity of the vehicle models and their excitation. The focus of this thesis is the development of techniques for using 3D terrain surface measurements for vehicular simulations. Contributions are made to vehicle model parameter identification, terrain filtering, and application-dependent interpolation methods for 3D terrain surfaces. Modeling and simulation are used to improve and shorten a vehicle's development cycle, thus, saving time and money. An important aspect in developing a vehicle model is to identify the parameters. Some parameters are easily measured with readily available tools; however, other parameters require dismantling the vehicle or using expensive test equipment. Initial estimates of these difficult or costly to obtain parameters are made based on similar vehicle models or standard practices. In this work, a parameter identification method is presented to obtain a better estimate of these inaccessible parameters using measured terrain excitations. By knowing the excitations to the physical vehicle, the simulated response can be compared to measured response, and then the vehicle model's parameters can be optimized such that the error between the responses is minimized. Through this process, better estimates of the vehicle's parameter are obtained, which demonstrates that measured terrain can improve vehicle development by increasing the accuracy of parameter estimates. The principal excitation to any ground vehicle is the terrain, and by obtaining more accurate representations of the terrain, vehicular simulation techniques are advanced. Many simple vehicle models use a point contact tire model, which performs poorly when short wavelength irregularities are present because the model neglects the tire's mechanical filtering properties. Therefore, a filter is used to emulate a tire's mechanical filtering mechanism and create an effective terrain profile. In this work, terrain filters are evaluated to quantify their effect on the sprung mass response of the dynamic simulation of a seven degree of freedom vehicle model. In any vehicular simulation, there is a balance between analytical expense and simulation realism. This balance often limits simulations to 2D terrain profile excitations, but as computing power increases the computational expense decreases. Thus, 3D terrain excitations for vehicular simulation are a tool for advancing simulation realism that is becoming less computationally expensive. Three dimensional terrain surfaces are measured with a non-uniform spacing in the horizontal plane; therefore, application-dependent gridding methods are developed in this work to interpolate 3D terrain surface to uniform grid spacing. The uniform grid spacing allows 3D terrain surfaces to be used more efficiently in any vehicular simulation when compared to non-uniform spacing. / Master of Science

Dynamical systems are mathematical models characterized by a set of differential or difference equations. Due to the increasing demand for more accuracy, the number of equations involved may reach the order of thousands and even millions. With so many equations, it often becomes computationally cumbersome to work with these large-scale dynamical systems. Model reduction aims to replace the original system with a reduced system of significantly smaller dimension which will still describe the important dynamics of the large-scale model. Interpolation is one method used to obtain the reduced order model. This requires that the reduced order model interpolates the full order model at selected interpolation points. Reduced order models are obtained through the Krylov reduction process, which involves solving a sequence of linear systems. The Iterative Rational Krylov Algorithm (IRKA) iterates this Krylov reduction process to obtain an optimal Η₂ reduced model. Especially in the large-scale setting, these linear systems often require employing inexact solves. The aim of this thesis is to investigate the impact of inexact solves on interpolatory model reduction. We considered preconditioning the linear systems, varying the stopping tolerances, employing GMRES and BiCG as the inexact solvers, and using different initial shift selections. For just one step of Krylov reduction, we verified theoretical properties of the interpolation error. Also, we found a linear improvement in the subspace angles between the inexact and exact subspaces provided that a good shift selection was used. For a poor shift selection, these angles often remained of the same order regardless of how accurately the linear systems were solved. These patterns were reflected in Η₂ and Η∞ errors between the inexact and exact subspaces, since these errors improved linearly with a good shift selection and were typically of the same order with a poor shift. We found that the shift selection also influenced the overall model reduction error between the full model and inexact model as these error norms were often several orders larger when a poor shift selection was used. For a given shift selection, the overall model reduction error typically remained of the same order for tolerances smaller than 1 x 10^-3, which suggests that larger tolerances for the inexact solver may be used without necessarily augmenting the model reduction error. With preconditioned linear systems as well as BiCG, we found smaller errors between the inexact and exact models while the order of the overall model reduction error remained the same. With IRKA, we observed similar patterns as with just one step of Krylov reduction. However, we also found additional benefits associated with using an initial guess in the inexact solve and by varying the tolerance of the inexact solve. / Master of Science

Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples. / Doctor of Philosophy / Simulation of mathematical models plays an important role in the development of science. There is a wide range of models and approaches that depend on the information available and the goal of the problem. In this dissertation we focus on three problems whose solution depends on parameters and for which we have either data resulting from simulations of the model or a explicit structure that describes the model. First, for the case when only data are available, we develop an algorithm that builds a data-driven approximation that is then easy to reevaluate. Second, we embed our algorithm in an already developed framework for the solution of a specific kind of model structure: nonlinear eigenvalue problems. Third, given a model with a specific nonlinear structure, we develop a method to build a model with the same structure, smaller dimension (for faster computation), and that provides an accurate approximation of the original model.

Dynamical systems and their numerical simulation are very important for investigating physical and technical problems. The more accuracy is desired, the more equations are needed to reach the desired level of accuracy. This leads to large-scale dynamical systems. The problem is that computations become infeasible due to the limitations on time and/or memory in large-scale settings. Another important issue is numerical ill-conditioning. These are the main reasons for the need of model reduction, i.e. replacing the original system by a reduced system of much smaller dimension. Then one uses the reduced models in order to simulate or control processes. The main goal of this thesis is to investigate an interpolation-based approach to the weighted-H2 model reduction problem. Nonetheless, first we will discuss the regular (unweighted) H2 model reduction problem. We will re-visit the interpolation conditions for H2-optimality, also known as Meier-Luenberger conditions, and discuss how to obtain an optimal reduced order system via projection. After having introduced the H2-norm and the unweighted-H2 model reduction problem, we will introduce the weighted-H2 model reduction problem. We will first derive a new error expression for the weighted-H2 model reduction problem. This error expression illustrates the significance of interpolation at the mirror images of the reduced system poles and the original system poles, as in the unweighted case. However, in the weighted case this expression yields that interpolation at the mirror images of the poles of the weighting system is also significant. Finally, based on the new weighted-H2 error expression, we will propose an iteratively corrected interpolation-based algorithm for the weighted-H2 model reduction problem. Moreover we will present new optimality conditions for the weighted-H2 approximation. These conditions occur as structured orthogonality conditions similar to those for the unweighted case which were derived by Antoulas, Beattie and Gugercin. We present several numerical examples to illustrate the effectiveness of the proposed approach and compare it with the frequency-weighted balanced truncation method. We observe that, for virtually all of our numerical examples, the proposed method outperforms the frequency-weighted balanced truncation method. / Master of Science

The goal of this thesis is to build an interactive tool for analyzing the quality of a digital image and predicting the scale at which it may be published. Since edges are present almost everywhere in most digital images, we use a mathematical edge model as the basis of analysis. In particular, we are interested in the luminance and chromaticity behavior at edge boundaries. We use this model to develop PhETA &mdash; Photograph Edge Transition Analyzer &mdash; an interactive tool that allows novice users to view and understand the results gained from this analysis in a clear and simple manner. / Master of Science

Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, into powerful mathematical tools, for example in signal analysis and geometric modelling. Both wavelet and subdivision analysis are based on the concept of a matrix–refinable function, i.e. a finitely supported matrix function which is self-replicating in the sense that it can be expressed as a linear combination of the integer shifts of its own dilation with factor 2: F = TAF = å k∈Z F(2 ・ −k)Ak. The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute the so-called matrix- mask sequence. Wavelets are in fact constructed as a specific linear combination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2; 9], whereas the convergence of the associated matrix- subdivision scheme c0 = c, cr+1 = SAcr, r ∈ Z+, SA : c = (ck : k ∈ Z) 7→ SAc = å ℓ∈Z Ak−2ℓ cℓ : k ∈ Z ! , subject to the necessary condition that rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å j∈Z Aǫ+2j, ǫ ∈ {0, 1}, ( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinable with respect to the mask coefficients defining the refinement equation and the subdivision scheme. Throughout this thesis, we investigate in time–domain for a given matrix mask sequence, the related issues of the existence of a matrix–refinable function and the convergence of the corresponding matrix– cascade algorithm, and finally we apply some results to the particular research area of Hermite interpolatory subdivision schemes. The dissertation is organized as follows: In order to provide a certain flexibility or freedom over the project, we established in Chapter 1 the equivalence relation between the matrix cascade algorithm and the matrix subdivision scheme, subject to a well defined class of initial iterates. Despite the general noncommutativity of matrices, we make use in the full rank case Qǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some useful tools, yielding a convergence result which comes as close to the scalar case as possible: we obtained a concrete sufficient condition on the mask sequence based on the matrix version of the generating function introduced in [3, page 22] for existence and convergence. Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29], our result on scalar case provided a progress for general mask sequences. We then applied to obtain a new one-parameter family of refinable functions which includes the cardinal splines as a special case, as well as corresponding convergent subdivision schemes. With the view to broaden the class of convergent matrix-masks, we replaced in chapter 3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theory of rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvement with a generalization of [13, Theorem 13, p.8] in to the context of rank subdivision schemes. In Chapter 4, we translated the concrete convergence criteria of the general theory from Theorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], into the context of rank, factorization and spectral radius (cf. [26]), and presented a careful analysis of the relationship between the two concepts. We then proceed with generalizations and improvements: we classified the matrix cascade algorithms in term of rank = 1, 2, . . . , d, and provided a complete characterization of each class with the use of a more general r−norming factor namely τ(r)-norming factor. On the other hand, we presented numerical methods to determine, if possible, the convergence of each class of matrix cascade algorithms. In both the scalar and matrix cases above, we also obtained explicitly the geometric constant appearing in the estimate for the geometric convergence of thematrix-cascade algorithm iterates to the matrix- refinable function. This same geometric convergence rate therefore also holds true for the corresponding matrix–cascade algorithm. Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 to the particular research area of Hermite interpolatory subdivision schemes: we provided a new convergence criterium, and end up with new convergence ranges of the parameters’ values of the famous Hermite interpolatory subdivision scheme with two parameters, due to Merrien [23]. / AFRIKAANSE OPSOMMING :(Wiskundige simbole het verander) Golfie en subdivisietegnieke het oor die afgelope twee dekades ontwikkel in kragtige wiskundige gereedskap, byvoorbeeld in seinanalise en geometriesemodellering. Beide golfie en subdivisie analise is gebaseer op die konsep van ’n matriks-verfynbare funksie; oftewel ’n eindig-ondersteunde matriksfunksie F wat selfreproduserend is in die sin dat dit uitgedruk kan word as ’n lineêre kombinasie van die heelgetalskuiwe van F se eie dilasie met faktor 2: F = Σ F(2 · −α)A(α), met A(α), α ∈ Z, wat aandui die sogenaamde matriks-masker ry. Golfies kan dan gekonstrueer word as ’n spesifieke lineêre kombinasie van die funksie ry {F(2 · −α) : α ∈ Z} (sien [2; 9]), terwyl die konvergensie van die ooreenstemmende matriks-subdivisie skema cº = c, cr+1 =(Σ β∈Z A(α − 2β) cr(β) : α ∈ Z ! , r ∈ Z+, onderhewig aan die nodige voorwaarde dat rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å α∈Z A(ǫ + 2α), ǫ ∈ {0, 1}, (sien [27]) die bestaan impliseer van ’n eindig-ondersteunde matriksfunksie F wat verfynbaar ismet betrekking tot diemaskerko¨effisi¨entewat die subdivisieskema definieer, en in terme waarvan die limietfunksie F van die subdivisieskema uitgedruk kan word as F = å α∈Z F(· − α)c(α). Ons hoofdoel hier is om , in die tydgebied, en vir ’n gegewematriks-masker ry, die verwante kwessies van die bestaan van ’nmatriks-verfynbare funksie en die konvergensie van die ooreenstemmende matriks-kaskade algoritme, en matriks-subdivisieskema, te ondersoek, en om uiteindelik sommige van ons resultate toe te pas op die spesifieke kwessie van die konvergensie van Hermite interpolerende subdivisieskemas. Summary v Eerstens, in Hoofstuk 1, ondersoek ons die verwantskap tussen matriks-kaskade algoritmes en matriks-subdivisie skemas, met verwysing na ’n goedgedefinieerde klas van begin-iterate. Vervolgens beskou ons die volle rang geval Qǫ = I, ǫ ∈ {0, 1}, om, in Hoofstuk 2, nuttige gereedskap te ontwikkel, en wat daarby ’n konvergensie resultaat met ’n sterk konneksie ten opsigte van die skalaar-geval oplewer. Met die doelstelling om ons klas van konvergente matriks-maskers te verbreed, vervang ons, in Hoofstuk 3, die volle rang voorwaarde met die rang een voorwaarde Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, en verkry ons dan ’n verbetering op ’n konvergensieresultaat in die artikel [14] deur Dubuc en Merrien, deur gebruik te maak van die teorie van rang subdivisieskemas van Micchelli en Sauer [26; 27], waarna ons die resultaat [14, Stelling 13, page 8] na die konteks van rang subdivisieskemas veralgemeen. InHoofstuk 4 herlei ons die konkrete konvergensie kriteria van Stelling 3.2, soos gebaseer op die r-normerende faktor gedefinieer in [14, Definisie 6, page 6] , na die konteks van rang, faktorisering en spektraalradius (sien [27]), en gee ons ’n streng analise van die verwantskap tussen die twee konsepte. Verder stel ons dan bekend ’n nuwe klassifikasie van matriks-kaskade algoritmes ten opsigte van rang, en verskaf ons ’n volledige karakterisering van elke klasmet behulp van ’nmeer algemene r-normerende faktor, nl. die τ(r)-normerende faktor. Daarby gee ons doeltreffende numeriesemetodes vir die implementering van ons teoretiese resultate. Ons verkry ook eksplisiet die geometriese konstante wat voorkom in die afskatting van die geometriese konvergensie van die matriks-kaskade algoritme iterate na die matriks-verfynbare funksie. Ten slotte, in Hoofstuk 5, pas ons die teorie en algoritmes ontwikkel in Hoofstuk 4 toe om die konvergensie van Hermite-interpolerende subdivisieskemas te analiseer. Spesifiek lei ons ’n nuwe konvergensie kriterium af, wat ons dan toepas om nuwe konvergensie gebiede vir die parameter waardes te verkry vir die beroemde Hermite interpolerende subdivisieskema met twee parameters, soos toegeskryf aan Merrien [24].

In this paper, there are three chapters. The first chapter discusses interpolation. Here a theorem about the uniqueness of the solution to the general interpolation problem is proven. Then the problem of how to represent this unique solution is discussed. Finally, the error involved in the interpolation and the convergence of the interpolation process is developed. In the second chapter a theorem about the uniform approximation to continuous functions is proven. Then the best approximation and the least squares approximation (a special case of best approximation) is discussed. In the third chapter orthogonal polynomials as discussed as well as bounded linear functionals in Hilbert spaces, interpolation and approximation and approximation in Hilbert space.

Cette thèse a été réalisée dans le cadre d'un projet industriel (BMCI), dont l'objectif est d'augmenter la disponibilité des équipements sur les navires. Dans cette thèse, nous proposons une approche qui met à contribution deux approches différentes, à savoir une approche à base de données pour la détection des défauts et une approche à base de connaissances d'experts pour l'isolation des défauts. Cette approche se veut générique et applicable à différents sous-systèmes du moteur ainsi qu'à divers moteurs et offre une ouverture pour une éventuelle application sur d'autres équipements. De plus, elle est tolérante vis-à-vis des éventuels changements au niveau de l'instrumentation disponible. Cette approche a été testée sur la détection et l'isolation des défauts les plus fréquents et aux conséquences graves auxquels les moteurs Diesel sont sujets. Tous les sous-systèmes du moteurs Diesel sont inclus et l'approche de diagnostic prend en considération les interactions existantes entre les sous-systèmes. L'approche de diagnostic a été testée sur un banc d'essai et sur le navire militaire Adroit de DCNS. Les défauts réalisés sur divers circuits du banc moteur et les défauts apparus en fonctionnement sur certains moteurs de l'Adroit, ont été majoritairement détectés et isolés avec succès. De plus, pour pallier à l'incertitude et au caractère flou des relations expertes utilisées dans la procédure d'isolation, une validation des relations de cause à effet a été réalisée, dans le cadre de cette thèse, par la réalisation d'un modèle analytique de simulation de défauts. / This thesis is carried out within an industrial framework (BMCI) which aims to enhance the availability of equipments on board ships. In this work, a data-based method for fault detection is combined with a knowledge-based method for fault isolation. The presented approach is generic and characterized by the ability to be applied to all the Diesel engine subsystems, to different kind of Diesel engines and can also be extended to other equipments. Moreover, this approach is tolerant regarding differences in instrumentation. This approach is tested upon the detection and isolation of the most hazardous and frequent faults which subject Diesel engines. This approach intends to make diagnosis upon the entire Diesel engine including all the subsystems and the existing interactions between the subsystems. The proposed approach is tested upon a test bench and upon the Diesel engines of the DCNS military vessel textquotedblleft Adroit". Most of the introduced faults on the test bench and the appeared faults on the Adroit engines have been successfully detected and isolated. In addition, to deal with uncertainties and fuzziness of the causal relationships given by maintenance experts, a model is developed. This model aims to validate these causal relationships used in the isolation part of the diagnosis approach.