Tag
Numerical Analysis
Numerical analysis studies the design, convergence, and stability of algorithms that approximate continuous mathematical objects with finite, computable structures. Classical methods (finite differences, finite elements, spectral methods) each preserve fragments of the underlying geometry by accident. Discrete exterior calculus (DEC) preserves it by construction, discretising differential forms on simplicial complexes so that topological identities like Stokes' theorem hold exactly at the discrete level. My work uses DEC through the cartan library for applications in fluid dynamics, electromagnetics, and quantum mechanics.
Blog
August 9, 2026
Semantic Segmentation: Resolving a Domain and Typing its Pieces
Every segmentation method resolves a domain into pieces and then assigns each piece a type. Reading the field through those two operations puts pixels, point clouds, meshes and shape spaces on one axis, recovers the primal-dual mesh pair on which discrete exterior calculus is built, and gives a precise reason why a message-passing network may place its dual vertices where a discrete Hodge star may not.
July 11, 2026
Quaternions Are the Rotors of Space
Hamilton's quaternions reconstructed on their own terms and then recognised as the even subalgebra of the geometric algebra of space: the imaginaries i, j, k are the basis bivectors, a unit quaternion is a rotor, and the half-angle, the two-to-one cover, and the absence of gimbal lock become plain facts about rotors. A companion to the classical-mechanics article, closing with the native rotation expressions this settles in code.
July 5, 2026
Mathematical Pharmacology
Pharmacokinetics, pharmacodynamics, and quantitative systems pharmacology reconstructed as one state-space object: controllability, observability, BIBO stability, and Metzler positivity built once at arbitrary compartment count, specialised to the reversible/dissipative (GENERIC) structure of compartmental kinetics, generalised to tensor QSP networks, and closed by a computable answer to when adding a compartment buys real structure versus an unidentifiable direction.
July 3, 2026
Classical Mechanics from Zero, in Two Languages
Classical mechanics constructed from nothing but an inertial frame, in matrix and linear algebra and in geometric algebra side by side: rotations and rigid-body dynamics without Euler angles, one vector derivative replacing grad, div, and curl, and the Lagrangian and Hamiltonian formalisms in both languages.
April 15, 2026
From RVE to Mesh: A Pipeline for Heterogeneous Continua
A single pipeline from microstructure to discrete solver: mean-field homogenisation on a representative volume element produces an SPD permeability tensor field, which induces a Riemannian metric, whose Hodge star discretises the Laplace-Beltrami operator, and whose scalar curvature drives adaptive remeshing.
April 4, 2026
Numerical Analysis via Discrete Exterior Calculus
A self-contained reconstruction of numerical analysis through discrete exterior calculus: simplicial complexes, cochains, the discrete Hodge star, and the Hodge Laplacian, applied to quantum mechanics, computational electromagnetics, and fluid dynamics.
Engineering
ferrum-gpu
Pure-Rust GPU compute substrate with Python bindings. FFT kernels compile from Rust source straight to PTX via cuda-oxide (no CUDA C in the build) and run on NVIDIA GPUs today; cross-vendor support through spirv-oxide and Vulkan is the v0.2 roadmap. A `no_std` `Backend` trait, typed `Device<B>` and `Buffer<T, B>` facades, and 1D/2D radix-2 Stockham C2C kernels cross-validated against `numpy.fft` across 29 GPU integration tests within 1e-3 to 1e-4 relative error. Published on PyPI.
gpufft
Cross-vendor GPU FFT for Rust, backed by VkFFT on Vulkan and cuFFT on CUDA. A single trait surface runs identically on NVIDIA, AMD, and Intel; buffers and plans are typed at the backend-and-scalar level, so mixing a Vulkan buffer with a CUDA plan, or an `f32` plan with a `Complex64` buffer, is a compile error. `plan_c2c`, `plan_r2c`, and `plan_c2r` at `f32` and `f64` over 1D, 2D, and 3D. Ships a dual-backend manylinux Python wheel cross-validated against `numpy`. Sibling to ferrum-gpu, sharing its FFT API.
Elworthy
JIT compiler that specialises Bismut-Elworthy-Li formulas into SIMD kernels for unbiased Monte Carlo Greeks on non-stationary SDEs. Symbolic AST, Cranelift lowering (scalar and 2-lane F64X2), multi-dimensional Heston driver, pathwise and likelihood-ratio Malliavin parameter Greeks (machine-checked with SymPy). European call price and BEL delta cross-validated against Black-Scholes closed form and the independent blackscholes crate; both agree within four Monte Carlo standard errors. About 22x over a tree-walking interpreter on GBM paths.
Kloeden
Hand-written SIMD C++ vs Rust (LLVM + Cranelift) benchmark companion to pathwise and elworthy. Same Brownian-increment fixture across four impls; single-thread pinned-core throughput on scalar Euler / Milstein / Taylor 1.5 on GBM, plus a digital-delta correctness table showing naive pathwise silently returns 0 in both languages while the Bismut-Elworthy-Li constant-flow weight matches analytic within 4 Monte Carlo standard errors (bitwise-identical between hand-rolled C++, hand-rolled Rust, and elworthy_rt::from_paths). Named after Peter Kloeden.