Tim Palmer

Abstract:

Most Earth-system simulations run on conventional central processing units in 64-bit double precision floating-point numbers Float64, although the need for high-precision calculations in the presence of large uncertainties has been questioned. Fugaku, currently the world's fastest supercomputer, is based on A64FX microprocessors, which also support the 16-bit low-precision format Float16. We investigate the Float16 performance on A64FX with ShallowWaters.jl, the first fluid circulation model that runs entirely with 16-bit arithmetic. The model implements techniques that address precision and dynamic range issues in 16 bits. The precision-critical time integration is augmented to include compensated summation to minimize rounding errors. Such a compensated time integration is as precise but faster than mixed precision with 16 and 32-bit floats. As subnormals are inefficiently supported on A64FX the very limited range available in Float16 is 6 × 10−5 to 65,504. We develop the analysis-number format Sherlogs.jl to log the arithmetic results during the simulation. The equations in ShallowWaters.jl are then systematically rescaled to fit into Float16, using 97% of the available representable numbers. Consequently, we benchmark speedups of up to 3.8x on A64FX with Float16. Adding a compensated time integration, speedups reach up to 3.6x. Although ShallowWaters.jl is simplified compared to large Earth-system models, it shares essential algorithms and therefore shows that 16-bit calculations are indeed a competitive way to accelerate Earth-system simulations on available hardware.

Abstract:

This paper draws on a number of Roger Penrose's ideas - including the non-Hamiltonian phase-space flow of the Hawking Box, conformal cyclic cosmology, non-computability, and gravitationally induced quantum state reduction - in order to propose a radically unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: "quantum"process is associated with fine-scale fractal geometry, "gravitational"process with larger-scale heterogeneous geometry. With this, it becomes possible to explain the experimental violation of Bell inequalities without having to abandon key ingredients of general relativity: determinism and local causality. Ensembles in IST can be described by complex Hilbert states over a finite set C p of complex numbers, where p is a large finite integer. The quantum mechanics of finite-dimensional Hilbert spaces is emergent as a singular limit when p → ∞. A small modification to the field equations of general relativity is proposed to make it consistent with IST.