This thesis concerns the identification of dynamic models in systems biology. and is structured into two parts. Both parts concern building dynamic models from observed data, but are quite different in perspective, rationale and mathematics. The first part considers the development of novel identification techniques that are particularly tailored to (molecular) biology and considers two approaches. The first approach reformulates the parameter estimation problem as a feasibility problem. This reformulation allows the invalidation of models by analysing entire parameter regions. The second approach utilises nonlinear observers and a transformation of the model equations into parameter free coordinates. The parameter free coordinates allow the design of a globally convergent observer, which in turn estimates the parameter values, and further, allows to identify modelling errors or unknown inputs/influences. Both approaches are bottom up approaches that require a mechanistic understanding of the underlying processes (in terms of a biochemical reaction network) leading to complex nonlinear models. The second part is an example of what can be done with classical, well developed tools from systems identification when applied to hitherto unattended problems.In particular, part two of my thesis develops a modelling framework for rat movements in an experimental setup that it widely used to study learning and memory.The approach is a top down approach that is data driven resulting in simple linear models.

The problem of detecting a vector of symbols, drawn from a finite alphabet and transmitted over a multiple-input multiple-output (MIMO) channel with Gaussian noise, is of central importance in digital communications and is encountered in several different applications. Examples include, but are not limited to; detection of symbols spatially multiplexed over a multiple-antenna channel and the multiuser detection problem in a code division multiple access (CDMA) system. Two algorithms previously proposed in the literature are considered and analyzed. Both algorithms have their origin in other fields of science but have gained mainstream recognition as efficient algorithms for the detection problem considered herein. Specifically, we consider the sphere decoder and semidefinite relaxation detector. By incorporating assumptions applicable in the communications context the performance of the two algorithms is addressed. The first algorithm, the sphere decoder, offers optimal performance in terms of its error probability. Further, the algorithm has proved extremely efficient in terms of computational complexity for moderately sized problems at high signal to noise ratio (SNR). Although it is recognized that the algorithm has an exponential worst case complexity, there has been a widespread belief that the algorithm has a polynomial average complexity at high SNR. A contribution made herein is to show that this is incorrect and that the average complexity, as the worst case complexity, is exponential in the number of symbols detected. Instead, another explanation of the observed efficiency of the algorithm is offered by deriving the exponential rate of growth and showing that this rate, although strictly positive for finite SNR, is small in the high SNR regime. The second algorithm, the semidefinite relaxation (SDR) detector, offers polynomial complexity at the expense of suboptimal performance in terms of error probability. Nevertheless, previous numerical observations suggest that error probability of the SDR algorithm is close to that of the optimal detector. Herein, the near optimality is of the SDR algorithm is given a precise meaning by studying the diversity of the SDR algorithm when applied to the (real valued) i.i.d.~Rayleigh fading channel and it is shown that the SDR algorithm achieves the same diversity order as the optimal detector. Further, criteria under which the SDR estimates coincide with the optimal estimates are derived and discussed. / Ett grundläggande problem som påträffats inom digital kommunikation är detektering av en symbolvektor, tillhörande ett ändligt symbolalfabet, som sänts över en MIMO (från engelskans multiple-input multiple-output) kanal med Gausiskt brus. Detta problem påträffas bland annat då symboler sänts över en trådlös kanal med flera antenner hos mottagaren och sändaren samt då flera användare i ett CDMA system simultant skall avkodas. In denna avhandling behandlas två mottagaralgoritmer konstruerade för detta ändamål. Algoritmerna har sin bakgrund i andra forskningsområden men kan i nuläget sägas vara mycket välkända inom kommunikationsområdet. De benämns vanligtvis som sfäravkodaren (eng. sphere decoder) samt den semidefinita relaxeringsdetektorn (eng. semidefinite relaxation detector). Algoritmerna analyseras i denna avhandling matematiskt genom att införa förenklande antaganden som är relevanta och applicerbara för de kommunikationsproblem som är av intesse. Den första algoritmen, sfäravkodaren, löser dessa detektionsproblem på ett optimalt sätt i betydelsen att den minimerar sannolikheten för att detektorn fattar ett felaktigt beslut rörande det sända meddelandet (symbolvektorn). Också vad gäller algoritmens komplexitet har simuleringar visat att den är oväntat låg, åtminstone vid höga signalbrusförhållanden (SNR). Trots att det är allmänt känt att algoritmen i sämsta fall har exponentiell komplexitet så har detta lett till den allmänt spridda uppfattningen att medelkomplexiteten (eller den förväntade komplexiteten) endast är polynomisk vid höga signalbrusförhållanden. Ett av huvudbidragen i denna avhandling är att visa att denna uppfattning är felaktig och att också medelkomplexiteten växer exponentiellt i antalet symboler som simultant detekteras. Ytterligare ett bidrag ligger i att ge en alternativ förklaring till den observerat låga medelkomplexiteten. Det visas att den exponentiella hastighet med vilken komplexiteten växer beror på signalbrusförhållande, och att den är låg för höga SNR. Den andra algoritmen, den semidefinita relaxeringsdetektorn, erbjuder polynomisk komplexitet vid en något högre felsannolikhet. Intressant nog har dock felsannolikheten tidigare, genom simuleringar, visat sig vara endast marginellt högre än felsannolikheten hos den optimala mottagaren. Bidraget som relaterar till den semidefinita relaxeringsmottagaren ligger i att både förklara och i att ge en specifik kvatifierbar mening åt uttalandet att felsannolikheten endast är marginellt högre. I syfte att åstadkomma detta studeras diversitetsordningen för detektorn, och det bevisas att diversitetsordningen för den semidefinita relaxeringsdetektorn är densamma som för den optimala mottagaren. Utöver detta karakteriseras också de krav som måste uppfyllas för att den detektorn skall finna den optimala lösningen. / QC 20100901

Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of different methods developed to find an approximate solution to the subproblem. We study the well-known method of More and Sorensen and two recent methods for large sparse subproblems: the so-called Lanczos method of Gould et al. and the Rendland Wolkowicz algorithm. The common ground to explore these methods will be semidefinite programming. This approach has been used by Rendl and Wolkowicz to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that the Rendl and Wolkowicz algorithm is more robust.

Harnessing the power of DNA microarray technology requires the existence of analysis methods that accurately interpret microarray data. Current literature abounds with algorithms meant for the investigation of microarray data. However, there is need for an efficient approach that combines different techniques of microarray data analysis and provides a viable solution to dimensionality reduction of microarray data. Reducing the high dimensionality of microarray data is one approach in striving to better understand the information contained within the data. We propose a novel approach for dimensionality reduction of microarray data that effectively combines different techniques in the study of DNA microarrays. Our method, <strong><em>KAS</em></strong> (<em>kernel alignment with semidefinite embedding</em>), aids the visualization of microarray data in two dimensions and shows improvement over existing dimensionality reduction methods such as PCA, LLE and Isomap.

Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of different methods developed to find an approximate solution to the subproblem. We study the well-known method of More and Sorensen and two recent methods for large sparse subproblems: the so-called Lanczos method of Gould et al. and the Rendland Wolkowicz algorithm. The common ground to explore these methods will be semidefinite programming. This approach has been used by Rendl and Wolkowicz to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that the Rendl and Wolkowicz algorithm is more robust.

Harnessing the power of DNA microarray technology requires the existence of analysis methods that accurately interpret microarray data. Current literature abounds with algorithms meant for the investigation of microarray data. However, there is need for an efficient approach that combines different techniques of microarray data analysis and provides a viable solution to dimensionality reduction of microarray data. Reducing the high dimensionality of microarray data is one approach in striving to better understand the information contained within the data. We propose a novel approach for dimensionality reduction of microarray data that effectively combines different techniques in the study of DNA microarrays. Our method, <strong><em>KAS</em></strong> (<em>kernel alignment with semidefinite embedding</em>), aids the visualization of microarray data in two dimensions and shows improvement over existing dimensionality reduction methods such as PCA, LLE and Isomap.

In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, thus guaranteeing stability.

In this thesis, we consider joint beamforming, power, and channel allocation in a multi-user and multi-channel underlay cognitive radio network (CRN). In this system, beamforming is implemented at each SU-TX to minimize the co-channel interference. The formulated joint optimization problem is a non-convex, mixed integer nonlinear programming (MINLP) problem. We propose a solution which consists of two stages. At first, given a channel allocation, a feasible solutions for power and beamforming vectors are derived by converting the problem into a convex form with an introduced optimal auxiliary variable and semidefinite relaxation (SDR) approach. Next, two explicit searching algorithms, i.e., genetic algorithm (GA) and simulated annealing (SA)-based algorithm are proposed to determine optimal channel allocations. Simulation results show that beamforming, power and channel allocation with SA (BPCA-SA) algorithm achieves a close optimal sum-rate with a lower computational complexity compared with beamforming, power and channel allocation with GA (BPCA-GA) algorithm. Furthermore, our proposed allocation scheme shows significant improvement than zero-forcing beamforming (ZFBF).

This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.

This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.