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Numerical exceptions, which may be caused by overflow, operations like division by 0 or sqrt(−1), or convergence failures, are unavoidable in many cases, in particular when software is used on unforeseen and difficult inputs. As more aspects of society become automated e.g., self-driving cars, health monitors, and cyber-physical systems more generally, it is becoming increasingly important to design software that is resilient to exceptions, and that responds to them in a consistent way. Consistency is needed to allow users to build higher-level software that is also resilient and consistent (and so on recursively). In this paper we explore the design space of consistent exception handling for the widely used BLAS and LAPACK linear algebra libraries, pointing out a variety of instances of inconsistent exception handling in the current versions, and propose a new design that balances consistency, complexity, ease of use, and performance. Some compromises are needed, because there are preexisting inconsistencies that are outside our control, including in or between existing vendor BLAS implementations, different programming languages, and even compilers for the same programming language. And user requests from our surveys are quite diverse. We also propose our design as a possible model for other numerical software, and welcome comments on our design choices. 

We propose a new algorithm to improve the strong scalability of right-looking sparse LU factorization on distributed memory systems. Our 3D sparse LU algorithm uses a three-dimensional PI process grid, aggressively exploits elimination tree parallelism and trades off increased memory for reduced per-process communication. We also analyze the asymptotic improvements for planar graphs (e.g., from 2D grid or mesh domains) and certain non-planar graphs (specifically for 3D grids and meshes). For planar graphs with n vertices, our algorithm reduces communication volume asymptotically in n by a factor of O(sqrt(logn)) and latency by a factor of O(logn). For non-planar cases, our algorithm can reduce the per-process communication volume by 3× and latency by O(n^1/3) times. In all cases, the memory needed to achieve these gains is a constant factor. We implemented our algorithm by extending the 2D data structure used in SuperLU_DIST. Our new 3D code achieves speedups up to 27× for planar graphs and up to 3.3× for non-planar graphs over the baseline 2D SuperLU_DIST when run on 24,000 cores of a Cray XC30.