A plethora of complex dynamical systems from disordered media to biological systems exhibit mathematical characteristics (e.g., long-range dependence, self-similar and power law magnitude increments) that are well-fitted by fractional partial differential equations (PDEs). For instance, some biological systems displaying an anomalous diffusion behavior, which is characterized by a non-linear mean-square displacement relation, can be mathematically described by fractional PDEs. In general, the PDEs represent various physical laws or rules governing complex dynamical systems. Since prior knowledge about the mathematical equations describing complex dynamical systems in biology, healthcare, disaster mitigation, transportation, or environmental sciences may not be available, we aim to provide algorithmic strategies to discover the integer or fractional PDEs and their parameters from system's evolution data. Toward deciphering non-trivial mechanisms driving a complex system, we propose a data-driven approach that estimates the parameters of a fractional PDE model. We study the space-time fractional diffusion model that describes a complex stochastic process, where the magnitude and the time increments are stable processes. Starting from limited time-series data recorded while the system is evolving, we develop a fractional-order moments-based approach to determine the parameters of a generalized fractional PDE. We formulate two optimization problems to allow us to estimate the argumentsmore »
Solving PDEs in space-time: 4D tree-based adaptivity, mesh-free and matrix-free approaches
Numerically solving partial differential equations (PDEs) remains a compelling application of supercomputing resources. The next generation of computing resources - exhibiting increased parallelism and deep memory hierarchies - provide an opportunity to rethink how to solve PDEs, especially time dependent PDEs. Here, we consider time as an additional dimension and simultaneously solve for the unknown in large blocks of time (i.e. in 4D space-time), instead of the standard approach of sequential time-stepping. We discretize the 4D space-time domain using a mesh-free kD tree construction that enables good parallel performance as well as on-the-fly construction of adaptive 4D meshes. To best use the 4D space-time mesh adaptivity, we invoke concepts from PDE analysis to establish rigorous a posteriori error estimates for a general class of PDEs. We solve canonical linear as well as non-linear PDEs (heat diffusion, advection-diffusion, and Allen-Cahn) in space-time, and illustrate the following advantages: (a) sustained scaling behavior across a larger processor count compared to sequential time-stepping approaches, (b) the ability to capture "localized" behavior in space and time using the adaptive space-time mesh, and (c) removal of any time-stepping constraints like the Courant-Friedrichs-Lewy (CFL) condition, as well as the ability to utilize spatially varying time-steps. We believe more »
- Publication Date:
- NSF-PAR ID:
- 10161729
- Journal Name:
- SC '19: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis
- Page Range or eLocation-ID:
- 1 to 61
- Sponsoring Org:
- National Science Foundation
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