<p>This thesis is concerned with the solution of the blind source separation problem (BSS). The BSS problem occurs frequently in various scientific and technical applications. In essence, it consists in separating meaningful underlying components out of a mixture of a multitude of superimposed signals.</p> <P> In the recent research literature there are two related approaches to the BSS problem: The first is known as Independent Component Analysis (ICA), where the goal is to transform the data such that the components become as independent as possible. The second is based on the notion of diagonality of certain characteristic matrices derived from the data. Here the goal is to transform the matrices such that they become as diagonal as possible. In this thesis we study the latter method of approximate joint diagonalization (AJD) to achieve a solution of the BSS problem. After an introduction to the general setting, the thesis provides an overview on particular choices for the set of target matrices that can be used for BSS by joint diagonalization.</p> <P> As the main contribution of the thesis, new algorithms for approximate joint diagonalization of several matrices with non-orthogonal transformations are developed.</p> <P> These newly developed algorithms will be tested on synthetic benchmark datasets and compared to other previous diagonalization algorithms.</p> <P> Applications of the BSS methods to biomedical signal processing are discussed and exemplified with real-life data sets of multi-channel biomagnetic recordings.</p> / <p>Diese Arbeit befasst sich mit der Lösung des Problems der blinden Signalquellentrennung (BSS). Das BSS Problem tritt häufig in vielen wissenschaftlichen und technischen Anwendungen auf. Im Kern besteht das Problem darin, aus einem Gemisch von überlagerten Signalen die zugrundeliegenden Quellsignale zu extrahieren.</p> <P> In wissenschaftlichen Publikationen zu diesem Thema werden hauptsächlich zwei Lösungsansätze verfolgt:</p> <P> Ein Ansatz ist die sogenannte "Analyse der unabhängigen Komponenten", die zum Ziel hat, eine lineare Transformation <B>V</B> der Daten <B>X</B> zu finden, sodass die Komponenten U_n der transformierten Daten <B>U</B> = <B> V X</B> (die sogenannten "independent components") so unabhängig wie möglich sind. Ein anderer Ansatz beruht auf einer simultanen Diagonalisierung mehrerer spezieller Matrizen, die aus den Daten gebildet werden. Diese Möglichkeit der Lösung des Problems der blinden Signalquellentrennung bildet den Schwerpunkt dieser Arbeit.</p> <P> Als Hauptbeitrag der vorliegenden Arbeit präsentieren wir neue Algorithmen zur simultanen Diagonalisierung mehrerer Matrizen mit Hilfe einer nicht-orthogonalen Transformation.</p> <P> Die neu entwickelten Algorithmen werden anhand von numerischen Simulationen getestet und mit bereits bestehenden Diagonalisierungsalgorithmen verglichen. Es zeigt sich, dass unser neues Verfahren sehr effizient und leistungsfähig ist. Schließlich werden Anwendungen der BSS Methoden auf Probleme der biomedizinischen Signalverarbeitung erläutert und anhand von realistischen biomagnetischen Messdaten wird die Nützlichkeit in der explorativen Datenanalyse unter Beweis gestellt.</p>

Charge excitations in different 3d transition metal compounds are studied. In particular, the influence of the lattice geometry on the character of these excitations is investigated. For this purpose, the momentum dependent loss function of electron energy-loss spectroscopy (EELS) as well as the optical conductivity are calculated and compared with the experimental data of NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$, and CuGeO$_{3}$ . A quarter-filled extended Hubbard model on a system of coupled ladders provides a qualitative explanation for the highly anisotropic charge excitations of NaV$_{2}$O$_{5}$ and LiV$_{2}$O$_{5}$. These ladder compounds do not only differ from the charge ordering pattern but also from the coupling between different ladders: In LiV$_{2}$O$_{5}$ one finds a strong inter-ladder hopping which is very small in NaV$_{2}$O$_{5}$. On the other hand, in NaV$_{2}$O$_{5}$ the ladders are coupled by a strong inter-ladder Coulomb interaction. The charge excitations of quasi one-dimensional cuprates reflect both the properties of the CuO$_{4}$ plaquettes and the character of the coupling between different plaquettes. Independently from the geometry of the cuprat chains, the local excitation of the copper hole onto the adjacent oxygen orbitals is always found. Further transitions with an excitation energy below the local excitation of a single plaquette result from a hole transfer to another plaquette. These excitations with hole delocalization dominate the spectra of the corner-shared Sr$_{2}$CuO$_{3}$. In contrast to this, the hole transfer leads only to a pre-peak in the spectra of the edge-shared CuGeO$_{3}$. Furthermore, it is shown that the hole transfer is determined by the geometry of the edge-shared CO. / Gegenstand dieser Arbeit ist die theoretische Analyse von Ladungsanregungen in verschiedenen niedrigdimensionalen 3d-Übergangsmetallverbindungen, wobei insbesondere der Einfluß der Gittergeometrie auf die Charakteristik der Anregungen untersucht wurde. Mit Hilfe des Lanczos-Algorithmus' wurden dazu sowohl die impulsabhängige Verlustfunktion der Elektron-Energie-Verlust-Spektroskopie (EELS) als auch die optische Leitfähigkeit für NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$ und CuGeO$_{3}$ berechnet und mit den experimentellen Ergebnissen verglichen. Unter der Verwendung eines Modells viertelgefüllter Leitern kann man die Ladungsanregungen sowohl für NaV$_{2}$O$_{5}$ als auch LiV$_{2}$O$_{5}$ sehr gut beschreiben. In diesen Materialien findet man nicht nur unterschiedliche Ladungsordnungen sondern vor allem auch verschiedene Kopplungsarten zwischen den Leitern. Während die Leitern im NaV$_{2}$O$_{5}$ durch die Coulomb-Wechselwirkung miteinander gekoppelt sind, existiert im LiV$_{2}$O$_{5}$ ein Austausch aufgrund einer starken Hybridisierung zwischen den Leitern. Die Ladungsanregungen von quasi eindimensionalen Kupratketten spiegeln sowohl die Plaketteneigenschaften als auch die Plakettenkopplung wider. Unabhängig von der Geometrie der Ketten findet man stets die lokale Anregung des Kupferloches auf die umliegenden Sauerstofforbitale. Aus einem möglichen Lochtransfer zu benachbarten Plaketten resultieren außerdem noch Anregungen, die energetisch unterhalb der Plakettenanregung liegen und unmittelbar von der Kettengeometrie abhängen. Während im eckenvernetzten Sr$_{2}$CuO$_{3}$ diese Anregungen die Spektren dominieren, spielt der Lochtransfer im kantenvernetzten CuGeO$_{3}$ nur eine untergeordnete Rolle.

Charge excitations in different 3d transition metal compounds are studied. In particular, the influence of the lattice geometry on the character of these excitations is investigated. For this purpose, the momentum dependent loss function of electron energy-loss spectroscopy (EELS) as well as the optical conductivity are calculated and compared with the experimental data of NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$, and CuGeO$_{3}$ . A quarter-filled extended Hubbard model on a system of coupled ladders provides a qualitative explanation for the highly anisotropic charge excitations of NaV$_{2}$O$_{5}$ and LiV$_{2}$O$_{5}$. These ladder compounds do not only differ from the charge ordering pattern but also from the coupling between different ladders: In LiV$_{2}$O$_{5}$ one finds a strong inter-ladder hopping which is very small in NaV$_{2}$O$_{5}$. On the other hand, in NaV$_{2}$O$_{5}$ the ladders are coupled by a strong inter-ladder Coulomb interaction. The charge excitations of quasi one-dimensional cuprates reflect both the properties of the CuO$_{4}$ plaquettes and the character of the coupling between different plaquettes. Independently from the geometry of the cuprat chains, the local excitation of the copper hole onto the adjacent oxygen orbitals is always found. Further transitions with an excitation energy below the local excitation of a single plaquette result from a hole transfer to another plaquette. These excitations with hole delocalization dominate the spectra of the corner-shared Sr$_{2}$CuO$_{3}$. In contrast to this, the hole transfer leads only to a pre-peak in the spectra of the edge-shared CuGeO$_{3}$. Furthermore, it is shown that the hole transfer is determined by the geometry of the edge-shared CO. / Gegenstand dieser Arbeit ist die theoretische Analyse von Ladungsanregungen in verschiedenen niedrigdimensionalen 3d-Übergangsmetallverbindungen, wobei insbesondere der Einfluß der Gittergeometrie auf die Charakteristik der Anregungen untersucht wurde. Mit Hilfe des Lanczos-Algorithmus' wurden dazu sowohl die impulsabhängige Verlustfunktion der Elektron-Energie-Verlust-Spektroskopie (EELS) als auch die optische Leitfähigkeit für NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$ und CuGeO$_{3}$ berechnet und mit den experimentellen Ergebnissen verglichen. Unter der Verwendung eines Modells viertelgefüllter Leitern kann man die Ladungsanregungen sowohl für NaV$_{2}$O$_{5}$ als auch LiV$_{2}$O$_{5}$ sehr gut beschreiben. In diesen Materialien findet man nicht nur unterschiedliche Ladungsordnungen sondern vor allem auch verschiedene Kopplungsarten zwischen den Leitern. Während die Leitern im NaV$_{2}$O$_{5}$ durch die Coulomb-Wechselwirkung miteinander gekoppelt sind, existiert im LiV$_{2}$O$_{5}$ ein Austausch aufgrund einer starken Hybridisierung zwischen den Leitern. Die Ladungsanregungen von quasi eindimensionalen Kupratketten spiegeln sowohl die Plaketteneigenschaften als auch die Plakettenkopplung wider. Unabhängig von der Geometrie der Ketten findet man stets die lokale Anregung des Kupferloches auf die umliegenden Sauerstofforbitale. Aus einem möglichen Lochtransfer zu benachbarten Plaketten resultieren außerdem noch Anregungen, die energetisch unterhalb der Plakettenanregung liegen und unmittelbar von der Kettengeometrie abhängen. Während im eckenvernetzten Sr$_{2}$CuO$_{3}$ diese Anregungen die Spektren dominieren, spielt der Lochtransfer im kantenvernetzten CuGeO$_{3}$ nur eine untergeordnete Rolle.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.

In the vast realm of inorganic materials, the Cu2+-containing cuprates form one of the richest classes. Due to the combined effect of crystal-field, covalency and strong correlations, all undoped cuprates are magnetic insulators with well-localized spins S=1/2, whereas the charge and orbital degrees of freedom are frozen out. The combination of the spin-only nature of their magnetism with the unique structural diversity renders cuprates as excellent model systems. The experimental studies, boosted by the discovery of high-temperature superconductivity in doped La2CuO4, revealed a fascinating variety of magnetic behaviors observed in cuprates. A digest of prominent examples should include the spin-Peierls transition in CuGeO3, the Bose-Einstein condensation of magnons in BaCuSi2O6, and the quantum critical behavior of Li2ZrCuO4. The magnetism of cuprates originates from short-range (typically, well below 1 nm) exchange interactions between pairs of spins Si and Sj, localized on Cu atoms i and j. Especially in low-dimensional compounds, these interactions are strongly anisotropic: even for similar interatomic distances |Rij|, the respective magnetic couplings Jij can vary by several orders of magnitude. On the other hand, there is an empirical evidence for the isotropic nature of this interaction in the spin space: different components of Si are coupled equally strong. Thus, the magnetism of cuprates is mostly described by a Heisenberg model, comprised of Jij(Si*Sj) terms. Although the applicability of this approach to cuprates is settled, the model parameters Jij are specific to a certain material, or more precisely, to a particular arrangement of the constituent atoms, i.e. the crystal structure. Typically, among the infinite number of Jij terms, only several are physically relevant. These leading exchange couplings constitute the (minimal) microscopic magnetic model. Already at the early stages of real material studies, it became gradually evident that the assignment of model parameters is a highly nontrivial task. In general, the problem can be solved experimentally, using elaborate measurements, such as inelastic neutron scattering on large single crystals, yielding the magnetic excitation spectrum. The measured dispersion is fitted using theoretical models, and in this way, the model parameters are refined. Despite excellent accuracy of this method, the measurements require high-quality samples and can be carried out only at special large-scale facilities. Therefore, less demanding (especially, regarding the sample requirements), yet reliable and accurate procedures are desirable. An alternative way to conjecture a magnetic model is the empirical approach, which typically relies on the Goodenough-Kanamori rules. This approach links the magnetic exchange couplings to the relevant structural parameters, such as bond angles. Despite the unbeatable performance of this approach, it is not universally applicable. Moreover, in certain cases the resulting tentative models are erroneous. The recent developments of computational facilities and techniques, especially for strongly correlated systems, turned density-functional theory (DFT) band structure calculations into an appealing alternative, complementary to the experiment. At present, the state-of-the-art computational methods yield accurate numerical estimates for the leading microscopic exchange couplings Jij (error bars typically do not exceed 10-15%). Although this computational approach is often regarded as ab initio, the actual procedure is not parameter-free. Moreover, the numerical results are dependent on the parameterization of the exchange and correlation potential, the type of the double-counting correction, the Hubbard repulsion U etc., thus an accurate choice of these crucial parameters is a prerequisite. In this work, the optimal parameters for cuprates are carefully evaluated based on extensive band structure calculations and subsequent model simulations. Considering the diversity of crystal structures, and consequently, magnetic behaviors, the evaluation of a microscopic model should be carried out in a systematic way. To this end, a multi-step computational approach is developed. The starting point of this procedure is a consideration of the experimental structural data, used as an input for DFT calculations. Next, a minimal DFT-based microscopic magnetic model is evaluated. This part of the study comprises band structure calculations, the analysis of the relevant bands, supercell calculations, and finally, the evaluation of a microscopic magnetic model. The ground state and the magnetic excitation spectrum of the evaluated model are analyzed using various simulation techniques, such as quantum Monte Carlo, exact diagonalization and density-matrix renormalization groups, while the choice of a particular technique is governed by the dimensionality of the model, and the presence or absence of magnetic frustration. To illustrate the performance of the approach and tune the free parameters, the computational scheme is applied to cuprates featuring rather simple, yet diverse magnetic behaviors: spin chains in CuSe2O5, [NO]Cu(NO3)3, and CaCu2(SeO3)2Cl2; quasi-two-dimensional lattices with dimer-like couplings in alpha-Cu2P2O7 and CdCu2(BO3)2, as well as the 3D magnetic model with pronounced 1D correlations in Cu6Si6O18*6H2O. Finally, the approach is applied to spin liquid candidates --- intricate materials featuring kagome-lattice arrangement of the constituent spins. Based on the DFT calculations, microscopic magnetic models are evaluated for herbertsmithite Cu3(Zn0.85Cu0.15)(OH)6Cl2, kapellasite Cu3Zn(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2, as well as for volborthite Cu3[V2O7](OH)2*2H2O. The results of the DFT calculations and model simulations are compared to and challenged with the available experimental data. The advantages of the developed approach should be briefly discussed. First, it allows to distinguish between different microscopic models that yield similar macroscopic behavior. One of the most remarkable example is volborthite Cu3[V2O7](OH)2*2H2O, initially described as an anisotropic kagome lattice. The DFT calculations reveal that this compound features strongly coupled frustrated spin chains, thus a completely different type of magnetic frustration is realized. Second, the developed approach is capable of providing accurate estimates for the leading magnetic couplings, and consequently, reliably parameterize the microscopic Hamiltonian. Dioptase Cu6Si6O18*6H2O is an instructive example showing that the microscopic theoretical approach eliminates possible ambiguity and reliably yields the correct parameterization. Third, DFT calculations yield even better accuracy for the ratios of magnetic exchange couplings. This holds also for small interchain or interplane couplings that can be substantially smaller than the leading exchange. Hence, band structure calculations provide a unique possibility to address the interchain or interplane coupling regime, essential for the magnetic ground state, but hardly perceptible in the experiment due to the different energy scales. Finally, an important advantage specific to magnetically frustrated systems should be mentioned. Numerous theoretical and numerical studies evidence that low-dimensionality and frustration effects are typically entwined, and their disentanglement in the experiment is at best challenging. In contrast, the computational procedure allows to distinguish between these two effects, as demonstrated by studying the long-range magnetic ordering transition in quasi-1D spin chain systems. The computational approach presented in the thesis is a powerful tool that can be directly applied to numerous S=1/2 Heisenberg materials. Moreover, with minor modifications, it can be largely extended to other metallates with higher value of spin. Besides the excellent performance of the computational approach, its relevance should be underscored: for all the systems investigated in this work, the DFT-based studies not only reproduced the experimental data, but instead delivered new valuable information on the magnetic properties for each particular compound. Beyond any doubt, further computational studies will yield new surprising results for known as well as for new, yet unexplored compounds. Such "surprising" outcomes can involve the ferromagnetic nature of the couplings that were previously considered antiferromagnetic, unexpected long-range couplings, or the subtle balance of antiferromagnetic and ferromagnetic contributions that "switches off" the respective magnetic exchange. In this way, dozens of potentially interesting systems can acquire quantitative microscopic magnetic models. The results of this work evidence that elaborate experimental methods and the DFT-based modeling are of comparable reliability and complement each other. In this way, the advantageous combination of theory and experiment can largely advance the research in the field of low-dimensional quantum magnetism. For practical applications, the excellent predictive power of the computational approach can largely alleviate designing materials with specific properties.:List of Figures List of Tables List of Abbreviations 1. Introduction 2. Magnetism of cuprates 3. Experimental methods 4. DFT-based microscopic modeling 5. Simulations of a magnetic model 6. Model spin systems: challenging the computational approach 7. Kagome lattice compounds 8. Summary and outlook Appendix Bibliography List of publications Acknowledgments