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.

Quasi-keyhole, including plasma keyhole and double-sided welding, is a novel approach proposed to operate the keyhole arc welding process. It can result in a high quality weld, but also raise higher demand of the operator. A computer control system to detect the keyhole and control the arc current can improve the performance of the welding process. To this effect, developing automatic pipe welding, instead of manual welding, is a hot research topic in the welding field. The objective of this research is to design an automatic quasi-keyhole pipe welding system that can monitor the keyhole and control its establishment time to track the reference trajectory as the dynamic behavior of welding processes changes. For this reason, an automatic plasma welding system is proposed, in which an additional electrode is added on the back side of the workpiece to detect the keyhole, as well as to provide the double-side arc in the double-sided arc welding mode. In the automatic pipe welding system the arc current can be controlled by the computer controller. Based on the designed automatic plasma pipe welding system, two kinds of model predictive controller − linear and bilinear − are developed, and an optimal algorithm is designed to optimize the keyhole weld process. The result of the proposed approach has been verified by using both linear and bilinear model structures in the quasi-keyhole plasma welding (QKPW) process experiments, both in normal plasma keyhole and double-sided arc welding modes.

The application of polymers in robust engineering designs is on the rise due to their excellent mechanical properties such as high fracture toughness, specific strength, durability, as well as, thermal and chemical resistances. Implementation of some advanced polymeric solids is limited due to the lack of available mechanical properties. In order for these materials to endure strenuous engineering designs it is vital to investigate their response in multiple loading rates and conditions. In this thesis, the mechanical response of polyethermide (PEI) is characterized under quasi-static, high strain rate, and multiple impact conditions. Standard tension, torsion, and compression experiments are performed in order to distinguish the multi-regime response of PEI. The effects of physical ageing and rejuvenation on the quasi-static mechanical response are investigated. The strain softening regime resulting from strain localization is eliminated by thermal and mechanical rejuvenation, and the advantages of these processes are discussed. The dynamic fracture toughness of the material in response to notched impact via Charpy impact test is evaluated. The high strain-rate response of PEI to uniaxial compression is evaluated at rates exceeding 104/s via miniaturized Split Hopkinson Pressure Bar (MSHPB), and compared to the quasi-static case to determine strain-rate sensitivity. The elastic response of the aged material to multiple loading conditions are correlated using the Ramberg-Osgood equation, while the elastoplastic response of rejuvenated PEI is correlated using a both the Ramberg-Osgood equation and a novel model. The strain-rate sensitivity of the strength is found to be nominally bilinear and transition strains are modeled using the Ree-Erying formulation. Finally, multiple impact experiments are performed on PEI using the MSHPB and a model is proposed to quantify damage as a result of collision.

由於時間序列在不同範疇的廣泛應用，許多實證結果已明白指出時間序列 資料普遍地存在非線性(nonlinearity)，使得非線型方法在最近幾年受到 極大的重視。然而，對於某些特定的非線型模式，縱然現在已有學者提出 模式選取之檢定方法，但是它們的模式階數確認問題至今卻仍無法有效率 地解決，更遑論得到最佳的模式配適與預測結果了。所以，我們試圖利用 一已於其他科學領域成功應用之新技術──神經網路，來解決非線型時間 序列之預測問題，而我們之所以利用神經網路的原因是其多層前輸網路是 泛函數的近似器(functional approximator)，對任意函數均有極佳之逼 近能力，使我們免除對時間序列資料之屬性(線性或非線性)作事先檢定或 假設的必要。在本篇論文中，我們首先建構15組雙線型時間序列資料，然 後對於這些數據分別以神經網路與自我迴歸整合移動平均(ARIMA) 模式配 適。藉著比較兩者間的配適與預測結果，我們發現神經網路對於預測非線 型時間序列是較具有穩健性。最後，我們以台幣對美元之即期匯率作為我 們的實證資料，結果亦證實了神經網路對於預測一般經濟時間序列亦較具 穩健性。 / With rapid development at the study of time series, the nonlinear approaches have attracted great attention in recent years. However, there are no efficient processes for the problem of identification to many specifically nonlinear models . Even if many testing methods have been proposed, we still can not find the best fitted model and obtain the best forecast performance. Hence, we try to solve the forecast problems by a new technique -- neurocomputing, which has been successfully applied in many scientific fields. The reason why we apply the neural networks is that the multilayer feedforward networks are functional approximators for the unknown function. In this paper, we will first construct several sets of bilinear time series and then fit these series by neural networks and ARIMA models. In this simulation study, we have found that the neural networks perform the robust forecast for some nonlinear time series. Finally, forecasting performance with favorable models will also be compared through the empirical realization of Taiwan.

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

La présente thèse vise à explorer différents types de modèles log-bilinéaires dans le domaine des sciences actuarielles. Le point de départ consiste en le modèle de Lee-Carter, utilisé pour les problèmes de projection de la mortalité. Différentes variantes sont développées, et notamment le modèle de Poisson log-bilinéaire. L'introduction de variables explicatives est également analysée. Enfin, une tentative de d'exportation de ces modèles au cas des triangles IBNR est effectuée.

Résumé : Dans ce projet, un décodage neuronal bayésien est effectué sur le colliculus inférieur du cochon d'Inde. Premièrement, On lit les potentiels évoqués grâce aux électrodes et ensuite on en déduit les potentiels d'actions à l'aide de technique de classification des décharges des neurones. Ensuite, un modèle linéaire généralisé (GLM) est entraîné en associant un stimulus acoustique en même temps que les mesures de potentiel qui sont effectuées. Enfin, nous faisons le décodage neuronal de l'activité des neurones en utilisant une méthode d'estimation statistique par maximum à posteriori afin de reconstituer la représentation spectro-temporelle du signal acoustique qui correspond au stimulus acoustique. Dans ce projet, nous étudions l'impact de différents modèles de codage neuronal ainsi que de différentes représentations spectro-temporelles (qu'elles sont supposé représenter le stimulus acoustique équivalent) sur la précision du décodage bayésien de l'activité neuronale enregistrée par le système auditif central. En fait, le modèle va associer une représentation spectro-temporelle équivalente au stimulus acoustique à partir des mesures faites dans le cerveau. Deux modèles de codage sont comparés: un GLM et un modèle bilinéaire généralisé (GBM), chacun avec trois différentes représentations spectro-temporelles des stimuli d'entrée soit un spectrogramme ainsi que deux représentations bio-inspirées: un banc de filtres gammatones et un spikegramme. Les paramètres des GLM et GBM, soit le champ récepteur spectro-temporel, le filtre post décharge et l'entrée non linéaire (seulement pour le GBM) sont adaptés en utilisant un algorithme d'optimisation par maximum de vraisemblance (ML). Le rapport signal sur bruit entre la représentation reconstruite et la représentation originale est utilisé pour évaluer le décodage, c'est-à-dire la précision de la reconstruction. Nous montrons expérimentalement que la précision de la reconstruction est meilleure avec une représentation par spikegramme qu'avec une représentation par spectrogramme et, en outre, que l'utilisation d'un GBM au lieu d'un GLM augmente la précision de la reconstruction. En fait, nos résultats montrent que le rapport signal à bruit de la reconstruction d'un spikegramme avec le modèle GBM est supérieur de 3.3 dB au rapport signal à bruit de la reconstruction d'un spectrogramme avec le modèle GLM. / Abstract : In this project, Bayesian neural decoding is performed on the neural activity recorded from the inferior colliculus of the guinea pig following the presentation of a vocalization. In particular, we study the impact of different encoding models on the accuracy of reconstruction of different spectro-temporal representations of the input stimulus. First voltages recorded from the inferior colliculus of the guinea pig are read and the spike trains are obtained. Then, we fit an encoding model to the stimulus and associated spike trains. Finally, we do neural decoding on the pairs of stimuli and neural activities using the maximum a posteriori optimization method to obtain the reconstructed spectro-temporal representation of the signal. Two encoding models, a generalized linear model (GLM) and a generalized bilinear model (GBM), are compared along with three different spectro-temporal representations of the input stimuli: a spectrogram and two bio-inspired representations, i.e. a gammatone filter bank (GFB) and a spikegram. The parameters of the GLM and GBM including spectro-temporal receptive field, post spike filter and input non linearity (only for the GBM) are fitted using the maximum likelihood optimization (ML) algorithm. Signal to noise ratios between the reconstructed and original representations are used to evaluate the decoding, or reconstruction accuracy. We experimentally show that the reconstruction accuracy is better with the spikegram representation than with the spectrogram and GFB representation. Furthermore, using a GBM instead of a GLM significantly increases the reconstruction accuracy. In fact, our results show that the spikegram reconstruction accuracy with a GBM fitting yields an SNR that is 3.3 dB better than when using the standard decoding approach of reconstructing a spectrogram with GLM fitting.