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Modern hardware tells a curious story: the gap between peak floating-point throughput and sustainable memory bandwidth keeps widening, while the AI revolution has pushed GPU vendors toward ever lower precision, FP16, FP8, and now FP4. For scientific simulations, this raises a natural question: can we borrow the same tricks without giving up the stability and convergence we cannot negotiate away?

This talk explores structure-aware low precision for discontinuous Galerkin methods. In DG discretizations, the solution is represented locally by polynomial expansions. For smooth solutions, the coefficients of these expansions often decay with polynomial degree: some coefficients carry large-scale features of the solution, while others encode finer details. Standard floating-point formats ignore this hierarchy, assigning the same number of bits to every coefficient. Not all bits are created equal, and standard formats spend them as if they were.

This motivates Adaptive Spectral Block Floating Point, a degree-aware number format that allocates precision according to modal importance. The format generalizes to arbitrary polynomial degree and bit budgets, and extends to tensor-product bases in two and three dimensions. Numerical experiments show that ASBFP can preserve expected DG convergence rates while substantially reducing memory footprint, with configurations ranging from aggressive compression to FP64-like accuracy.

The main message is that mixed precision for scientific computing need not be a blunt FP64-to-FP32 replacement. When the discretization has mathematical structure, precision can be allocated where it matters most.