High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

Secure and scalable deduplication of horizontally partitioned health data for privacy-preserving distributed statistical computation

Background Techniques have been developed to compute statistics on distributed datasets without revealing private information except the statistical results. However, duplicate records in a distributed dataset may lead to incorrect statistical results. Therefore, to increase the accuracy of the statistical analysis of a distributed dataset, secure deduplication is an important preprocessing step. Methods We designed a secure protocol for the deduplication of horizontally partitioned datasets with deterministic record linkage algorithms. We provided a formal security analysis of the protocol in the presence of semi-honest adversaries. The protocol was implemented and deployed across three microbiology laboratories located in Norway, and we ran experiments on the datasets in which the number of records for each laboratory varied. Experiments were also performed on simulated microbiology datasets and data custodians connected through a local area network. Results The security analysis demonstrated that the protocol protects the privacy of individuals and data custodians under a semi-honest adversarial model. More precisely, the protocol remains secure with the collusion of up to N − 2 corrupt data custodians. The total runtime for the protocol scales linearly with the addition of data custodians and records. One million simulated records distributed across 20 data custodians were deduplicated within 45 s. The experimental results showed that the protocol is more efficient and scalable than previous protocols for the same problem. Conclusions The proposed deduplication protocol is efficient and scalable for practical uses while protecting the privacy of patients and data custodians

Numerical methods for option pricing under the CGMY process

In this thesis European options are priced using the CGMY process to model the underlying assets. Four different methods are implemented and investigated, including a fractional partial differential equation solver, a partial integro differential equation solver, a stochastic differential equation solver and a cosine expansion method. We also derive the forward Kolmogorov fractional partial differential equation and partial integro differential equation

Numerical methods for option pricing under the CGMY process

In this thesis European options are priced using the CGMY process to model the underlying assets. Four different methods are implemented and investigated, including a fractional partial differential equation solver, a partial integro differential equation solver, a stochastic differential equation solver and a cosine expansion method. We also derive the forward Kolmogorov fractional partial differential equation and partial integro differential equation

Numerical methods for option pricing under the CGMY process

In this thesis European options are priced using the CGMY process to model the underlying assets. Four different methods are implemented and investigated, including a fractional partial differential equation solver, a partial integro differential equation solver, a stochastic differential equation solver and a cosine expansion method. We also derive the forward Kolmogorov fractional partial differential equation and partial integro differential equation