Adaptive Meshfree Methods for Partial Differential Equations

There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending on the governing equations, the domains, and the boundary conditions. The MFS is used as the main meshfree method to solve the Laplace equation in this dissertation, and we propose adaptive algorithms in different versions based on the residual type of an error estimator in 2D and 3D domains. Popular techniques for handling parameters and different approaches are considered in each example to obtain satisfactory results. Dirichlet boundary conditions are carefully chosen to validate the efficiency of the adaptive method. The RBF collocation method and the Method of Approximate Particular Solutions (MAPS) are used for solving the Poisson equation. Due to the type of the PDE, different strategies for constructing the adaptive method had to be followed, and proper error estimators are considered for this part. This results in having a new point of view when observing the numerical results. Methodologies of meshfree methods that are employed in this dissertation are introduced, and numerical examples are presented with various boundary conditions to show how the adaptive method performs. We can observe the benefit of using the adaptive method and the improved error estimators provide better results in the experiments

A Comparison of Two Boundary Methods For Biharmonic Boundary Value Problems

The purpose of this thesis is to solve biharmonic boundary value problems using two different boundary methods and compare their performances. The two boundary methods used are the method of fundamental solutions (MFS) and the method of approximate fundamental solutions (MAFS). The Delta-shaped basis function with the Abel regularization technique is used in the construction of the approximate fundamental solutions in MAFS. The MFS produces more accurate results but needs known fundamental solutions for the differential operator. The MAFS can provide comparable results, and is applicable to more general differential operators. The numerical results using both methods are presented

Relational versus structural embeddedness: the roles of uncertainty in information technology outsourcing

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In response to uncertainty imposed on ITO business environments, it is reported that relational and structural embeddedness play an important role in safeguarding against opportunistic behaviour and improving long-term performance. A firm can outsource its IT services to a partner who is believed to be reliable and competent among existing parties for whom it has the outsourcing histories in the perspective of relational embeddedness. In contrast, from the viewpoint of structural embeddedness, a firm can collect information on multiple alternative candidates through the observation of their network linkages and the information transmission via third parties although it has no outsourcing histories for them. Also, based on this information, it can outsource its IT services to a new partner who could make better performance as well as who is considered reliable. However, the building and maintenance of new outsourcing relationships require resources which could be better used for the refinement of existing outsourcing relationships. Therefore, a firm faces the tension between the two types of embeddedness. Prior studies addressing relational and structural embeddedness in the context of ITO are mainly based on relational exchange theory and social capital theory respectively. They also provide a body of empirical evidence rooted in these theories. However, each ITO research stream on relational or structural embeddedness has mainly focused on its own advantages in response to uncertainty. That is, the conditional superiority of each type of embeddedness has not been investigated in ITO studies. Furthermore, although they have been compared in other research contexts, the main research focus has been on which is preferred at the high level of uncertainty rather than which leads to better performance according to the type and level of uncertainty. Therefore, this research aims at answering the following research question in the context of ITO: which of the two types of embeddedness is more appropriate in improving long-term performance in the presence of uncertainty of which the type and level are not uniform across a wide range of outsourced IT services? In particular, the following uncertainties from two different sources are introduced for the comparison between the two types of embeddedness: the uncertainty stemming from the unpredictability of technological requirements and the uncertainty originating in the difficulty in measuring performance. In this research, they are called “technological unpredictability” and “measurement difficulty” respectively. It is widely accepted that the two uncertainties discovered from transaction cost theory and agency theory increase the possibility of opportunism and threaten performance. Therefore, the different levels of technological unpredictability and measurement difficulty can create an ideal platform to investigate the conditional superiority of relational or structural embeddedness.In order to address the research question, an ITO network is simulated. Firms in this network perform the role of a coordinator or a partner in establishing ITO consortia to respond to outsourcing opportunities with the different levels of the two uncertainties. As coordinators, firms take the partner selection and control strategy based on relational or structural embeddedness, which is called “the relational strategy” or “the structural strategy” in this research. They also compete with each other to maximise their longterm profits. As partners, firms behave cooperatively or opportunistically. Their decision-makings and payoffs are modelled through a game-theoretic method. In addition, a full factorial design of experiments is applied for efficient simulation experiments and systematic analyses. Consequently, the simulation results show that the superiority of each type of embeddedness is different according the type and level of uncertainty. The research on relational embeddedness emphasises the advantage of trust and commitment generated by the repetition or long-term maintenance of outsourcing relationships with reliable partners as shown in the literature on long-term cooperative ITO relationships. The findings in this research support this argument when measurement difficulty is at the high level and technological unpredictability is at the low level. On the other hand, the study on structural embeddedness focuses on the use of (potential) partners’ network positions and information transmitters as revealed in the literature on network-based ITO relationships. The simulation results support this claim when technological unpredictability is at the high level regardless of the level of measurement difficulty. Especially, at the high levels of both uncertainties, structural embeddedness enables better performance. This research contributes to the literature in three research areas: (1) IT outsourcing, (2) network dynamics and (3) environmental adaptation. Firstly, this research examines the conditional superiority of each type of embeddedness at the different levels of technological unpredictability and measurement difficulty. Therefore, the findings resolve the tension between the two types of embeddedness in ITO studies. Especially,this resolution can provide possible theoretical answers to why an ITO partnership based on relational embeddedness fails in spite of its popularity in the industry and academia, and in which condition structural embeddedness is preferred in ITO business environments. Secondly, the simulation results reveal that some coordinators preferring relational embeddedness consolidate their existing network ties while others favouring structural embeddedness increase the number of network ties. Therefore, this research improves an understanding of how the strength and structure of network ties at the egocentric level can be changed by the type and level of uncertainty. Thirdly, the relational and structural strategy in this research focus on the utilisation of present partners and the search for alternative partners respectively. Therefore, the concepts underlying the two types of embeddedness are in line with those underlying exploitation and exploration. The examination on the relative advantage of each type of embeddedness can extend the general argument that more resources should be invested in exploration than in exploitation to adapt to uncertain business settings

Governance of IT Service Procurement: Relationship vs Network based Approach

Relational and structural embeddedness are reported to play an important role in the context of information technology outsourcing (ITO). However, we do not fully understand which of the two types of embeddedness is more appropriate in preventing opportunistic behaviour and improving long-term performance in the presence of uncertainty which is not uniform across a wide range of outsourced IT services and products. In order to address this question, a virtual ITO network is simulated where firms take the partner selection and control strategy based on relational or structural embeddedness. They also compete with each other to maximise their long-term profits. The simulation results show that the advantage of each type of embeddedness is different according to the levels of measurement difficulty and requirement unpredictability which coexist in the ITO business environments. Therefore, this study provides a better understanding of the conditional superiority of each type of embeddedness in the precence of the two uncertainties and offers ITO managers with a guideline for a choice between relational and structural embeddedness

An Adaptive Method of Fundamental Solutions for Solving the Laplace Equation

In this paper, we propose a residual-type adaptive method of fundamental solutions (AMFS) for solving the two-dimensional Laplace equation. An error estimator is defined only on the boundary of the domain. Initial distributions of source points and collocation points are determined by using approaches proposed in Chen et al. (2006). The adding, removing, and stopping strategies are designed so that the required accuracy can be satisfied within finite steps. Numerical experiments reveal that AMFS improves the accuracy of the MFS approximation obtained from uniformly distributed sources and collocation points, which makes the MFS more practical for non-harmonic and non-smooth boundary conditions. Moreover, it is shown that the error estimator becomes equidistributed after an adaptive iteration. A detailed comparison between AMFS and MFS using uniformly distributed points is also presented for each numerical example

The Method of Transformed Angular Basis Function for Solving the Laplace Equation

In this paper, we propose a new approach to improve the method of angular basis function (MABF) proposed by Young et al. (2015) for the Laplace equation in two-dimensional settings. Instead of the fundamental solution ln r used in the traditional Method of Fundamental Solution (MFS), MABF employs a different basis function θ and produces good approximate solutions on the domains with acute, narrow regions and exterior problems (Young et al., 2015). However, the definition of θ inevitably incurs a singularity situation for many different types of domains. Therefore, the selection of source points of MABF is not as convenient as the traditional MFS. To avoid the singularity situation in implementing, we introduce a transformation so that the transformed angular basis function does not exhibit this type of singularity for commonly used distributions of source points. As a result, source points for the method of transformed angular basis function (MTABF) can then be chosen in a similar way to traditional MFS. Numerical experiments demonstrate that the proposed approach significantly simplifies the selection of source points in MABF for different types of domains, which makes MABF more applicable. Numerical results of MTABF and MFS are presented for comparison purposes

A Challenge for Emphysema Quantification Using a Deep Learning Algorithm With Low-dose Chest Computed Tomography

Purpose: We aimed to identify clinically relevant deep learning algorithms for emphysema quantification using low-dose chest computed tomography (LDCT) through an invitation-based competition. Materials and Methods: The Korean Society of Imaging Informatics in Medicine (KSIIM) organized a challenge for emphysema quantification between November 24, 2020 and January 26, 2021. Seven invited research teams participated in this challenge. In total, 558 pairs of computed tomography (CT) scans (468 pairs for the training set, and 90 pairs for the test set) from 9 hospitals were collected retrospectively or prospectively. CT acquisition followed the hospitals' protocols to reflect the real-world clinical setting. Using the training set, each team developed an algorithm that generated converted LDCT by changing the pixel values of LDCT to simulate those of standard-dose CT (SDCT). The agreement between SDCT and LDCT was evaluated using the intraclass correlation coefficient (ICC; 2-way random effects, absolute agreement, and single rater) for the percentage of low-attenuated area below -950 HU (LAA(-950 HU)), kappa value for emphysema categorization (LAA(-950 HU), <5%, 5% to 10%, and >= 10%) and cosine similarity of LAA(-950 HU). Results: The mean LAA(-950 HU) of the test set was 14.2%+/- 10.5% for SDCT, 25.4%+/- 10.2% for unconverted LDCT, and 12.9%+/- 10.4%, 11.7%+/- 10.8%, and 12.4%+/- 10.5% for converted LDCT (top 3 teams). The agreement between the SDCT and converted LDCT of the first-place team was 0.94 (95% confidence interval: 0.90, 0.97) for ICC, 0.71 (95% confidence interval: 0.58, 0.84) for categorical agreement, and 0.97 (interquartile range: 0.94 to 0.99) for cosine similarity. Conclusions: Emphysema quantification with LDCT was feasible through deep learning-based CT conversion strategies.N