arXiv subjects

Cole Brower

Publications and source records attributed to Cole Brower.

6 recordsLinked to original sources

arXiv2026

Exceeding the Numerical and Performance Characteristics of IEEE-754 SGEMM with BFloat16 Tensor Cores on GPUs for Scientific Computing

Largely due to their increased native capacity for numerical intensity and power efficiency, reduced-precision floating-point computing resources, primarily used in artificial intelligence (AI) applications, have expanded at a greater rate than their higher-precision relatives. This has led to various efforts focused upon leveraging plentiful reduced-precision hardware to mimic higher-precision mathematical calculations. This paper studies a specific use case, namely the use of bfloat16 (BF16) Tensor Cores found on modern GPUs in service of single precision (FP32) matrix multiply operations. Given that BF16 and FP32 share the same dynamic range, the option to accumulate BF16 operations into FP32 accumulators (at full-speed), and additional BF16 arithmetic characteristics specific to the Blackwell GPU architecture, such as integrated scaling hardware, such emulation is highly motivated. This paper examines the performance, efficiency, power, and numerical characteristics of FP32 matrix multiplication via BF16-based emulation and demonstrates how it exceeds numerical and performance characteristics of native FP32 for scientific applications. We also discuss a full library-ready implementation that correctly deals with denormals.

cs.DC↗

arXiv2026

Hunting for quantum advantage in electronic structure calculations is a highly non-trivial task

In light of major developments over the past decades in both quantum computing and simulations on classical hardware, it is a serious challenge to identify a real-world problem where quantum advantage is expected to appear. In quantum chemistry, electronic structure calculations of strongly correlated, i.e. multi-reference problems, are often argued to fall into such category because of their intractability with standard methods based on mean-field theory. Therefore, providing state-of-the-art benchmark data by classical algorithms is necessary to make a decisive conclusion when such competing development directions are compared. We report cutting-edge performance results together with high accuracy ground state energy for the Fe$_4$S$_4$ molecular cluster on a CAS(54,36) model space, a problem that has been included quite recently among the list of systems in the {\it Quantum Advantage Tracker} webpage maintained by IBM and RIKEN. Pushing the limits even further, we also present CAS-SCF based orbital optimizations for unprecedented CAS sizes of up to 89 electrons in 102 orbitals [CAS(89,102)] for the Fe$_5$S$_{12}$H$_4^{5-}$ molecular system comprising twenty five open shell orbitals in its sextet ground state and an active spaces size of 331 electrons in 451 orbitals. We have achieved our results via mixed-precision spin-adapted \textit{ab initio} Density Matrix Renormalization Group (DMRG) electronic structure calculations interfaced with the ORCA program package and utilizing the NVIDIA Blackwell graphics processing unit (GPU) platform. We argue that DMRG benchmark data should be taken as a classical reference when quantum advantage is reported. In addition, full exploitation of classical hardware should also be considered since even the most advanced DMRG implementations are still in a premature stage regarding utilization of all the benefits of GPU technology.

physics.chem-ph↗

arXiv2025

Guaranteed DGEMM Accuracy While Using Reduced Precision Tensor Cores Through Extensions of the Ozaki Scheme

The rapid growth of artificial intelligence (AI) has made low-precision formats such as FP16, FP8, and, most recently, block-scaled FP4 the primary focus of modern GPUs, where Tensor Cores now deliver orders-of-magnitude higher throughput than traditional FP64 pipelines. This hardware shift has sparked a new line of algorithm research: using low-precision units to emulate double-precision accuracy through schemes such as Ozaki decompositions. We advance this direction with Automatic Dynamic Precision (ADP), a fully GPU-resident framework that makes emulated FP64 matrix multiplication both efficient and reliable. At its core is the Exponent Span Capacity (ESC), a hardware-agnostic estimator that conservatively determines the decomposition parameter (also known as slices) required to achieve FP64-level accuracy. Built on ESC, ADP integrates exception handling, run time heuristics, and seamless fallback to native FP64, ensuring correctness without host-device synchronization or user intervention. Additionally, we further improve Ozaki-style decompositions with an unsigned integer slicing scheme, which increases representational efficiency and reduces computational waste. Validated against recently proposed BLAS grading tests, ADP consistently preserves FP64 fidelity on challenging inputs while incurring less than 10% run time overhead. In a 55-bit mantissa setting, our approach achieves up to 2.3x and 13.2x speedups over native FP64 GEMM on NVIDIA Blackwell GB200 and the RTX Pro 6000 Blackwell Server Edition, respectively. Our results demonstrate that low-precision accelerators can serve as a practical, production-ready foundation for high-fidelity and high-performance scientific computing workloads.

cs.DC↗

arXiv2025

Mixed-precision ab initio tensor network state methods adapted for NVIDIA Blackwell technology via emulated FP64 arithmetic

We report cutting-edge performance results via mixed-precision spin adapted ab initio Density Matrix Renormalization Group (DMRG) electronic structure calculations utilizing the Ozaki scheme for emulating FP64 arithmetic through the use of fixed-point compute resources. By approximating the underlying matrix and tensor algebra with operations on a modest number of fixed-point representatives (``slices''), we demonstrate on smaller benchmark systems and for the active compounds of the FeMoco and cytochrome P450 (CYP) enzymes with complete active space (CAS) sizes of up to 113 electrons in 76 orbitals [CAS(113, 76)] and 63 electrons in 58 orbitals [CAS(63, 58)], respectively, that the chemical accuracy can be reached with mixed-precision arithmetic. We also show that, due to its variational nature, DMRG provides an ideal tool to benchmark accuracy domains, as well as the performance of new hardware developments and related numerical libraries. Detailed numerical error analysis and performance assessment are also presented for subcomponents of the DMRG algebra by systematically interpolating between double- and pseudo-half-precision. Our analyis represents the first quantum chemistry evaluation of FP64 emulation for correlated calculations capable of achieving chemical accuracy and emulation based on fixed-point arithmetic, and it paves the way for the utilization of state-of-the-art Blackwell technology in tree-like tensor network state electronic structure calculations, opening new research directions in materials sciences and beyond.

physics.chem-ph↗

arXiv2024

Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone

A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing $0$ and closed under addition. Each numerical semigroup $S$ with smallest positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that integer point determines certain algebraic properties of $S$. In this paper, we introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful projection of certain faces of $\mathcal C_m$. We characterize several aspects of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are known to index the faces of $\mathcal C_m$, and our results culminate in a method of "walking" the face lattice of the Kunz cone in a manner analogous to that of a Gr\"obner walk. We apply our results in several contexts, including a wealth of computational data obtained from the aforementioned "walks" and a proof of a recent conjecture concerning which numerical semigroups achieve the highest minimal presentation cardinality when one fixes the smallest positive element and the number of generators.

math.CO↗

arXiv2021

Length density and numerical semigroups

Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative integers).

math.AC↗