Performance comparisons of preconditioned iterative methods for problems arising in PDE-constrained optimization

The governing dynamics of simple and complex processes, whether physical, biological, social, economic, engineering, or even rather a mere figment of imagination, can be studied via numerical simulations of mathematical models. These models in many cases can be thought to consist of one, or frequently, several coupled partial differential equations (PDEs). In many applications, the aim of such simulations is not only to study the behavior of the underlying processes, but also to optimize or control those in some optimal way. These are referred to as optimal control problems constrained by PDEs and are stated in the form of a constrained minimization problem. The general framework under which such problems are studied is referred to as PDE-constrained optimization. In this thesis, we aim to solve three benchmark optimal control problems, namely, the optimal control of the Poisson equation, the optimal control of the convection-diffusion equation and the optimal control of the Stokes system. Numerically tackling these problems lead to a large optimality system with a saddle point structure. Systems with a saddle point structure are indefinite and in general, ill-conditioned, thus posing great challenges for iterative solvers seeking to find their solution. Preconditioning the optimality system is a possible strategy to deal with the issue. The main focus of the thesis is therefore to solve the resulting optimality systems with various preconditioners available in literature and compare their efficiency. Moreover, additional challenges arise when dealing with convection-diffusion control problems which we effectively deal by employing the local projection stabilization (LPS) scheme. Furthermore, Axelsson and Neytcheva in [40] proposed a preconditioner for efficiently solving large nonlinear coupled multi-physics problems. We successfully apply this preconditioner to the first two benchmark problems with promising results

Towards conformal methods for large-scale monitoring of district heating substations

Increasing technical complexity, design variations, and customization options of IoT units create difficulties for the construction of monitoring infrastructure. These units can be associated with different domains, such as a fleet of vehicles in the mobility domain and a fleet of heat-pumps in the heating domain. The lack of labeled datasets and well-understood prior unit and fleet behavior models exacerbates the problem. Moreover, the time-series nature of the data makes it difficult to strike a reasonable balance between precision and detection delay. The thesis aims to develop a framework for scalable and cost-efficient monitoring of industrial fleets. The investigations were conducted on real-world operational data obtained from District Heating (DH) substations to detect anomalous behavior and faults. A foundational hypothesis of the thesis is that fleet-level models can mitigate the lack of labeled datasets, improve anomaly detection performance, and achieve a scalable monitoring alternative. Our preliminary investigations found that operational heterogeneity among the substations in a DH network can cause fleet-level models to be inefficient in detecting anomalous behavior at the target units. An alternative is to rely on subfleet-level models to act as a proxy for the behavior of target units. However, the main difficulty in constructing a subfleet-level model is the selection of its members such that their behavior is stable over time and representative of the target unit. Therefore, we investigated various ways of constructing the subfleets and estimating their stability. To mitigate the lack of well-understood prior unit and fleet behavior models, we proposed constructing Unit-Level and Subfleet-Level Ensemble Models, i.e., ULEM and SLEM. Herein, each member of the respective ensemble consists of a Conformal Anomaly Detector (CAD). Each ensemble yields a nonconformity score matrix that provides information about the behavior of a target unit relative to its historical data and its subfleet, respectively. However, these ensemble models can give different information about the nature of an anomaly that may not always agree with each other. Therefore, we further synthesized this information by proposing a Combined Ensemble Model (CEM). We investigated the advantages and limitations of decisions that rely on the information obtained from ULEM, SLEM, and CEM using precision and detection delay. We observed the decisions that relied on the information obtained through CEM showed a reduction in overall false alarms compared to those obtained through ULEM or SLEM, albeit at the cost of some detection delay. Finally, we combined the components of ULEM, SLEM, and CEM into what we refer to as TRANTOR: a conformal anomaly detection based indusTRiAl fleet moNiTORing framework. The proposed framework is expected to enable fleet operators in various domains to improve their monitoring infrastructure by efficiently detecting anomalous behavior and controlling false alarms at the target units

Towards large-scale monitoring of operationally diverse thermal energy systems with data-driven techniques

The core of many typical large-scale industrial infrastructure consists of hundreds or thousands of systems that are similar in their basic design and purpose. For instance, District Heating (DH) utilities rely on a large network of substations to deliver heat to their customers. Similarly, a factory may require a large fleet of specialized robots for manufacturing a certain product. Monitoring these systems is important for maintaining the overall efficiency of industrial operations by detecting various problems due to faults and misconfiguration. However, this can be challenging since a well-understood prior model for each system is rarely available. In most cases, each system in a fleet or network is fitted with a set of sensors to measure its state at different time intervals. Typically, a data-driven model for each system can be used for their monitoring. However, not all factors that can possibly influence the operations of each system in a fleet or network has an associated sensor. Moreover, sufficient instances of normal, atypical and faulty behavior are rarely available to train such a model. These issues can impede the effectiveness of a system level data-driven model. Alternatively, it can be assumed that since all the systems in a fleet or network are working on a similar task, they should all behave in a homogeneous manner. Any system that behaves differently from the majority is then considered as an outlier. This is referred to as the global model at the fleet or network level. While the approach is simple, it is less effective in the presence of non-stationary working conditions. Hence, both system level and global modeling approaches have their limitations.  This thesis investigates system level and fleet or network level (global) models for large-scale monitoring, and proposes an alternative way which is referred to as a reference-group based approach. Herein, the operational monitoring of each system, referred to as a target, is delegated to a reference-group, which consists of systems experiencing a comparable operating regime along with the target. Thus, the definition of a normal, atypical or faulty operational behavior in a target system is described relative to its reference-group. In this sense, if the target system is not behaving operationally in consort with the systems in its reference-group, then it can be inferred that this is either due to a fault or because of some atypical operation arising at the target system due to its local peculiarities. The application area for these investigations is the large-scale operational monitoring of thermal energy systems: networks of district heating (DH) substations and fleets of heatpumps. The current findings indicate three advantages of a reference-group based approach. The first is that the reference operational behavior of any system in the fleet or network does not need to be predefined. The second is that it provides a basis for what a system’s operational behavior should have been and what it is. In this respect, each system in the reference-group provides an evidence about a particular behavior during a particular time period. This can be very useful when the description of a normal, atypical and faulty operational behavior is not available. The third is that it can detect potential atypical and faulty operational behavior quicker compared to global models of outlier detection at the fleet or network level

Performance comparisons of preconditioned iterative methods for problems arising in PDE-constrained optimization

The governing dynamics of simple and complex processes, whether physical, biological, social, economic, engineering, or even rather a mere figment of imagination, can be studied via numerical simulations of mathematical models. These models in many cases can be thought to consist of one, or frequently, several coupled partial differential equations (PDEs). In many applications, the aim of such simulations is not only to study the behavior of the underlying processes, but also to optimize or control those in some optimal way. These are referred to as optimal control problems constrained by PDEs and are stated in the form of a constrained minimization problem. The general framework under which such problems are studied is referred to as PDE-constrained optimization. In this thesis, we aim to solve three benchmark optimal control problems, namely, the optimal control of the Poisson equation, the optimal control of the convection-diffusion equation and the optimal control of the Stokes system. Numerically tackling these problems lead to a large optimality system with a saddle point structure. Systems with a saddle point structure are indefinite and in general, ill-conditioned, thus posing great challenges for iterative solvers seeking to find their solution. Preconditioning the optimality system is a possible strategy to deal with the issue. The main focus of the thesis is therefore to solve the resulting optimality systems with various preconditioners available in literature and compare their efficiency. Moreover, additional challenges arise when dealing with convection-diffusion control problems which we effectively deal by employing the local projection stabilization (LPS) scheme. Furthermore, Axelsson and Neytcheva in [40] proposed a preconditioner for efficiently solving large nonlinear coupled multi-physics problems. We successfully apply this preconditioner to the first two benchmark problems with promising results

Analysis of Empirical Investor Networks and Information Events in Stock Market

We did further analysis to understand the dynamics of information diffusion in an"Empirical Investor Network (EIN) [8]. We find that the timings of the trades are of crucial importance for central investors. We find further evidence that central traders have information advantage and information diffusion plays a central role in their profitability. We verify through our robustness tests that all results hold up when profits are calculated using actual realized returns instead of a fixed holding period

Towards conformal methods for large-scale monitoring of district heating substations

Increasing technical complexity, design variations, and customization options of IoT units create difficulties for the construction of monitoring infrastructure. These units can be associated with different domains, such as a fleet of vehicles in the mobility domain and a fleet of heat-pumps in the heating domain. The lack of labeled datasets and well-understood prior unit and fleet behavior models exacerbates the problem. Moreover, the time-series nature of the data makes it difficult to strike a reasonable balance between precision and detection delay. The thesis aims to develop a framework for scalable and cost-efficient monitoring of industrial fleets. The investigations were conducted on real-world operational data obtained from District Heating (DH) substations to detect anomalous behavior and faults. A foundational hypothesis of the thesis is that fleet-level models can mitigate the lack of labeled datasets, improve anomaly detection performance, and achieve a scalable monitoring alternative. Our preliminary investigations found that operational heterogeneity among the substations in a DH network can cause fleet-level models to be inefficient in detecting anomalous behavior at the target units. An alternative is to rely on subfleet-level models to act as a proxy for the behavior of target units. However, the main difficulty in constructing a subfleet-level model is the selection of its members such that their behavior is stable over time and representative of the target unit. Therefore, we investigated various ways of constructing the subfleets and estimating their stability. To mitigate the lack of well-understood prior unit and fleet behavior models, we proposed constructing Unit-Level and Subfleet-Level Ensemble Models, i.e., ULEM and SLEM. Herein, each member of the respective ensemble consists of a Conformal Anomaly Detector (CAD). Each ensemble yields a nonconformity score matrix that provides information about the behavior of a target unit relative to its historical data and its subfleet, respectively. However, these ensemble models can give different information about the nature of an anomaly that may not always agree with each other. Therefore, we further synthesized this information by proposing a Combined Ensemble Model (CEM). We investigated the advantages and limitations of decisions that rely on the information obtained from ULEM, SLEM, and CEM using precision and detection delay. We observed the decisions that relied on the information obtained through CEM showed a reduction in overall false alarms compared to those obtained through ULEM or SLEM, albeit at the cost of some detection delay. Finally, we combined the components of ULEM, SLEM, and CEM into what we refer to as TRANTOR: a conformal anomaly detection based indusTRiAl fleet moNiTORing framework. The proposed framework is expected to enable fleet operators in various domains to improve their monitoring infrastructure by efficiently detecting anomalous behavior and controlling false alarms at the target units

Analysis of Empirical Investor Networks and Information Events in Stock Market

We did further analysis to understand the dynamics of information diffusion in an"Empirical Investor Network (EIN) [8]. We find that the timings of the trades are of crucial importance for central investors. We find further evidence that central traders have information advantage and information diffusion plays a central role in their profitability. We verify through our robustness tests that all results hold up when profits are calculated using actual realized returns instead of a fixed holding period

Performance comparisons of preconditioned iterative methods for problems arising in PDE-constrained optimization

The governing dynamics of simple and complex processes, whether physical, biological, social, economic, engineering, or even rather a mere figment of imagination, can be studied via numerical simulations of mathematical models. These models in many cases can be thought to consist of one, or frequently, several coupled partial differential equations (PDEs). In many applications, the aim of such simulations is not only to study the behavior of the underlying processes, but also to optimize or control those in some optimal way. These are referred to as optimal control problems constrained by PDEs and are stated in the form of a constrained minimization problem. The general framework under which such problems are studied is referred to as PDE-constrained optimization. In this thesis, we aim to solve three benchmark optimal control problems, namely, the optimal control of the Poisson equation, the optimal control of the convection-diffusion equation and the optimal control of the Stokes system. Numerically tackling these problems lead to a large optimality system with a saddle point structure. Systems with a saddle point structure are indefinite and in general, ill-conditioned, thus posing great challenges for iterative solvers seeking to find their solution. Preconditioning the optimality system is a possible strategy to deal with the issue. The main focus of the thesis is therefore to solve the resulting optimality systems with various preconditioners available in literature and compare their efficiency. Moreover, additional challenges arise when dealing with convection-diffusion control problems which we effectively deal by employing the local projection stabilization (LPS) scheme. Furthermore, Axelsson and Neytcheva in [40] proposed a preconditioner for efficiently solving large nonlinear coupled multi-physics problems. We successfully apply this preconditioner to the first two benchmark problems with promising results

Analysis of Empirical Investor Networks and Information Events in Stock Market

We did further analysis to understand the dynamics of information diffusion in an"Empirical Investor Network (EIN) [8]. We find that the timings of the trades are of crucial importance for central investors. We find further evidence that central traders have information advantage and information diffusion plays a central role in their profitability. We verify through our robustness tests that all results hold up when profits are calculated using actual realized returns instead of a fixed holding period