Abstract
The sparse grid combination technique is an efficient method to reduce the curse of dimensionality for high-dimensional problems, since it uses only selected grids for spatial discretization. To further reduce the computational complexity in the temporal dimension, we choose the Parareal algorithm, a parallel-in-time algorithm. For the coarse and fine solvers in time, we use an efficient implementation of the Alternating Direction Implicit (ADI) method, which is an unusual choice due to the larger computational cost compared to the usual choice of one-step or Runge-Kutta methods. In this paper we combine both approaches and therefore obtain a even more efficient computational method for parallelism. The application problem is to determine a fair price of a Put option using the Heston model with correlation. We analyze this model as an example to illustrate this advantageous combination of the sparse grid with the Parareal algorithm. Finally, we present further ideas to improve this advantageous combination of methods.
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Clevenhaus, A., Ehrhardt, M., Günther, M. (2022). The Parareal Algorithm and the Sparse Grid Combination Technique in the Application of the Heston Model. In: Ehrhardt, M., Günther, M. (eds) Progress in Industrial Mathematics at ECMI 2021. ECMI 2021. Mathematics in Industry(), vol 39. Springer, Cham. https://doi.org/10.1007/978-3-031-11818-0_62
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DOI: https://doi.org/10.1007/978-3-031-11818-0_62
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