American option pricing has been an active research area in financial engineering over the past few decades. Since no analytic closed-form solution exists, various numerical approaches have been developed. Among all proposed methods, the least square Monte Carlo(LSMC) approach is the most successful and popular. The LSMC utilizes linear regression for the estimation of the continuation values for the option. However, the accuracy of the LSMC is dependent on the chosen basis functions, where no objective strategy exists for the basis function selection process. Recently, it has been proposed to use kernel ridge regression (KRR) with a bundling technique for high-dimensional American option pricing to avoid selecting the basis functions.
The Heston model is an example of the stochastic volatility model, where only limited literature can be found concerning multi-dimensional American option pricing under this model. In this thesis, we reproduce the proposed KRR-based methods. Additionally, we extend the KRR-based methods for high-dimensional American options under the Heston model. We also involve the LSMC method for evaluating the pricing efficiency and accuracy of KRR-based methods. As no reliable benchmarks exist for the valuation of American options under the high-dimensional Heston model, we implement the primal-dual approach and use its result as the reference price. Nevertheless, the obtained benchmarks are biased. Therefore, we can only conclude that the KRR-based methods apply to American option pricing under the high-dimensional Heston model. Moreover, the KRR-based techniques are computationally more efficient than the LSMC method.

Magnetic anomaly detection (MAD) is a widely used method for detecting ferromagnetic targets, particularly hidden objects. A common way to localize a magnetic target is to look at the maximum of the magnitude of the magneticeld gradient. In this thesis, we consider a threeaxis total eld gradiometer which measures the gradient of the magnitude of the magnetic eld along three orthogonal axes. Based on the measured data from the three-axis total eld gradiometer, we decompose the inversion problem into two linear systems. By solving those two systems, we get an approximation for the location and moment that we are looking for. For the considered gradiometer, we can improve the parameters if the detailed geometry of the gradiometer is taken into account.