Section: Research Program

Efficient and certified approximation methods

Safe numerical approximations

The last twenty years have seen the advent of computer-aided proofs in mathematics and this trend is getting more and more important. They request: fast and stable numerical computations; numerical results with a guarantee on the error; formal proofs of these computations or computations with a proof assistant. One of our main long-term objectives is to develop a platform where one can study a computational problem on all (or any) of these three levels of rigor. At this stage, most of the necessary routines are not easily available (or do not even exist) and one needs to develop ad hoc tools to complete the proof. We plan to provide more and more algorithms and routines to address such questions. Possible applications lie in the study of mathematical conjectures where exact mathematical results are required (e.g., stability of dynamical systems); or in more applied questions, such as the automatic generation of efficient and reliable numerical software for function evaluation. On a complementary viewpoint, numerical safety is also critical in robust space mission design, where guidance and control algorithms become more complex in the context of increased satellite autonomy. We will pursue our collaboration with specialists of that area whose questions bring us interesting focus on relevant issues.

Floating-point computing

Floating-point arithmetic is currently undergoing a major evolution, in particular with the recent advent of a greater diversity of available precisions on a same system (from 8 to 128 bits) and of coarser-grained floating-point hardware instructions. This new arithmetic landscape raises important issues at the various levels of computing, that we will address along the following three directions.

Floating-point algorithms, properties, and standardization

One of our targets is the design of building blocks of computing (e.g., algorithms for the basic operations and functions, and algorithms for complex or double-word arithmetic). Establishing properties of these building blocks (e.g., the absence of “spurious” underflows/overflows) is also important. The IEEE 754 standard on floating-point arithmetic (whose next version, a rather minor revision, will be released soon) will have to undergo a major revision within a few years: first because advances in technology or new needs make some of its features obsolete, and because new features need standardization. We aim at playing a leading role in the preparation of the next standard.

Error bounds

We will pursue our studies in rounding error analysis, in particular for the “low precision–high dimension” regime, where traditional analyses become ineffective and where improved bounds are thus most needed. For this, the structure of both the data and the errors themselves will have to be exploited. We will also investigate the impact of mixed-precision and coarser-grained instructions (such as small matrix products) on accuracy analyses.

High performance kernels

Most directions in the team are concerned with optimized and high performance implementations. We will pursue our efforts concerning the implementation of well optimized floating-point kernels, with an emphasis on numerical quality, and taking into account the current evolution in computer architectures (the increasing width of SIMD registers, and the availability of low precision formats). We will focus on computing kernels used within other axes in the team such as, for example, extended precision linear algebra routines within the FPLLL and HPLLL libraries.