The FASTMath SciDAC Institute is developing and deploying scalable mathematical algorithms and software tools for reliable simulation of complex physical phenomena and collaborating with Department of Energy (DOE) domain scientists to ensure the usefulness and applicability of our work. The focus of our work is strongly driven by the requirements of DOE application scientists who require fast, accurate, and robust forward simulation along with the ability to efficiently perform ensembles of simulations in optimization or uncertainty quantification studies.

The FASTMath Institute in SciDAC-4 brings together the successful SciDAC-3 teams of the Frameworks, Algorithms and Scalable Technologies for Mathematics (FASTMath) and Quantification of Uncertainty in Extreme Scale Simulations (QUEST) Institutes, which focused on critical capabilities needed for fast, robust forward simulations (FASTMath) and uncertainty quantification (QUEST). Our contributions in SciDAC-3 ranged from providing the foundations for next-generation application codes to developing key numerical capabilities that enabled faster time to solution, higher fidelity, and more robust simulations. In SciDAC-4, we will continue our efforts to further develop the full range of technologies to improve the reliability, accuracy, and robustness of application simulation codes.

In particular, our team will conduct research in eight foundational topical areas. We will advance numerical methods and software in each of these eight focused topical areas, prioritizing our developments based on application needs both now and as they evolve into the future. We will also more tightly integrate these methods and software together to improve overall functionality, efficiency, and performance of next-generation simulation codes.

- structured meshes
- unstructured meshes
- time integrators
- linear solvers
- eigensolvers
- numerical optimization
- data analytics
- uncertainty quantification

A number of critical Department of Energy applications, characterized by complex geometry and/or physics, have greatly benefited from the application of high-order and/or adaptive mesh methods that have been developed by the FASTMath team. Over the course of the SciDAC program, these methods and tools have evolved to the point where application scientists are able to focus their attention on the development of domain-specific discretization technologies, using FASTMath tools for controlling and adapting the meshes. Our focus in FASTMath is the continued development of structured, multi-block technologies, unstructured mesh techniques, particle-mesh methods, and time discretization methods. Of critical importance in these areas is high-order discretization techniques and adaptive mesh refinement.

Making block-structured grid adaptive mesh refinement software frameworks available for the development of massively parallel, multiphysics applications. |
Enabling the high-fidelity simulation of physical phenomena on complex geometries by developing and deploying unstructured mesh technologies for leadership-class computers. |
Developing time integration software tailored to efficient solution of problems arising in the context of simulation and optimization of multiphysics and multiscale simulations. |

Solution of systems of algebraic equations is undoubtedly one of the most common computational kernels in scientific applications of interest to the Department of Energy. Efficient, scalable, and reliable algorithms are crucial for the success of large-scale simulations. FASTMath work focuses on both iterative and direct linear solution methods, and eigensolvers. Our work in the first half of the project in these areas centered around two primary themes. First, we are developing new algorithms and using them to better solve physics problems including advanced multigrid techniques, improved preconditioners, and new eigensystem solvers. Second, we are creating efficient many-core solution methodologies using hybrid programming techniques, communication-reducing strategies, and intelligent task-mapping methods. This work has application to fusion, nuclear structure calculation, quantum chemistry, accelerator modeling, climate, and dislocation dynamics research.

Researching and developing iterative linear solvers, including multigrid and multilevel methods. |
Providing factorization-based sparse solvers and preconditioners on large-scale parallel machines. |
Developing scalable eigenvalue algorithms and solvers for problems with millions to billions of degrees of freedom. |

Numerical optimization is an outer-loop problem used in many applications to select parameters to minimize or maximize a quality of interest. SciDAC and the DOE ASCR Applied Mathematics Program have funded research on methods for solving derivative-free optimization problems, discrete optimization problems, and PDE-constrained optimization problems and for calculating sensitivities. Our focus in FASTMath is to develop methods for dynamic optimization problems with constraints that may include discrete variables and multiple objectives, methods for sensitivity analysis using surrogate models, and capabilities for computing adjoint and forward sensitivities.

Developing a framework in the Toolkit for Advance Optimization for PDE-constrained optimization with nontrivial design and state constraints that may include discrete variables (using MINOTAUR) and multiple objectives (using POUNDERS and APOSMM). |
Developing iterative sampling methods that employ sensitivity analysis and parametric surrogate models to determine the most important parameters for a given problem and use this information to reduce the numerical optimization search space. |
Developing advanced adjoint and forward sensitivity capabilities in PETSc and Albany/Trilinos to provide essential derivative information that will support important quantities of interest that arise in dynamic optimization problems and data assimilation. |

We are focused on the development of all aspects of exascale data analytic methods. Efforts include the functional representation of data, using sparse polynomial approximation, low dimensional manifolds, and high order regularizers to enable faster storage, retrieval, and analysis of large datasets. Targeted methods are being developed in sparse storage and retrieval of large data, uncertainty estimates for sparse data representation, fast estimation of data statistics, and importance ranking in streaming data.

Models of physical systems typically involve inputs/parameters that are determined from empirical measurements, and therefore exhibit a certain degree of uncertainty. Estimating the propagation of this uncertainty into computational model output predictions is crucial for purposes of model validation, hypothesis testing, model optimization, and decision support. FASTMath researchers are involved in the development of probabilistic methods and software for efficient uncertainty quantification (UQ) in computational models. These efforts are organized on a number of fronts, addressing a range of algorithmic and software developments, targeting key challenges in UQ. more>>