We present an arbitrary updated Lagrangian Material Point Method (A-ULMPM) to alleviate issues, such as the cell-crossing instability and numerical fracture, that plague state of the art Eulerian formulations of MPM, while still allowing for large de
formations that arise in fluid simulations. Our proposed framework spans MPM discretizations from total Lagrangian formulations to Eulerian formulations. We design an easy-to-implement physics-based criterion that allows A-ULMPM to update the reference configuration adaptively for measuring physical states including stress, strain, interpolation kernels and their derivatives. For better efficiency and conservation of angular momentum, we further integrate the APIC[Jiang et al. 2015] and MLS-MPM[Hu et al. 2018] formulations in A-ULMPM by augmenting the accuracy of velocity rasterization using both the local velocity and its first-order derivatives. Our theoretical derivations use a nodal discretized Lagrangian, instead of the weak form discretization in MLS-MPM[Hu et al. 2018], and naturally lead to a modified MLS-MPM in A-ULMPM, which can recover MLS-MPM using a completely Eulerian formulation. A-ULMPM does not require significant changes to traditional Eulerian formulations of MPM, and is computationally more efficient since it only updates interpolation kernels and their derivatives when large topology changes occur. We present end-to-end 3D simulations of stretching and twisting hyperelastic solids, splashing liquids, and multi-material interactions with large deformations to demonstrate the efficacy of our novel A-ULMPM framework.
Transition probability densities are fundamental to option pricing. Advancing recent work in deep learning, we develop novel transition density function generators through solving backward Kolmogorov equations in parametric space for cumulative proba
bility functions, using neural networks to obtain accurate approximations of transition probability densities, creating ultra-fast transition density function generators offline that can be trained for any underlying. These are single solve , so they do not require recalculation when parameters are changed (e.g. recalibration of volatility) and are portable to other option pricing setups as well as to less powerful computers, where they can be accessed as quickly as closed-form solutions. We demonstrate the range of application for one-dimensional cases, exemplified by the Black-Scholes-Merton model, two-dimensional cases, exemplified by the Heston process, and finally for a modified Heston model with time-dependent parameters that has no closed-form solution.
