|Title:||The BSDE solvers for high-dimensional PDEs and BSDEs|
|Advisors:||Lin, Yanping (AMA)|
Qiao, Zhonghua (AMA)
|Subject:||Stochastic differential equations -- Numerical solutions|
Differential equations, Parabolic -- Numerical solutions
Hong Kong Polytechnic University -- Dissertations
|Department:||Department of Applied Mathematics|
|Pages:||xxvi, 111 pages : color illustrations|
|Abstract:||Conventional numerical methods for high-dimensional parabolic partial differential equations (PDEs) suffer from the notorious "curse of dimensionality". Inspired by the FCNN-based deep BSDE solver in E et al. (2017) and Han et al. (2018), this thesis presents a CNN3H-based deep BSDE solver and a CNN2H-based BSDE solver by converting the fully connected neural networks (FCNNs) to the convolutional neural networks with 3 hidden layers (CNN3Hs) or with 2 hidden layers (CNN2Hs), and a linear BSDE solver by replacing the FCNNs with linear combinations. We also employ the connection between PDEs and backward stochastic differential equations (BSDEs), i.e. the Feynman-Kac formula. Owing to fewer parameters, the proposed BSDE solvers demonstrate higher efficiency than the FCNN-based deep BSDE solver without sacrificing accuracy when solving some 100-dimensional and 1000-dimensional PDEs.|
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