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Explore advanced randomization techniques for tackling high-dimensional computational problems in this 53-minute conference talk from JuliaCon Local Paris 2025. Discover how randomization serves as a powerful dimensionality reduction technique that enables solving large-scale problems while leveraging optimized kernels and mixed precision capabilities. Learn about recent progress in applying randomization to linear systems of equations and eigenvalue problems, including sketching techniques that embed large-dimensional subspaces while preserving geometrical properties and their parallel implementations. Examine randomized versions of vector orthogonalization processes and their integration into the Arnoldi iteration method. Understand how these innovations lead to Krylov subspace methods that maintain the numerical stability of classic approaches while reducing communication overhead and improving efficiency on modern massively parallel computers. Gain insights into the practical implementation of these methods through a Julia library, making these advanced techniques accessible for real-world applications in high-performance computing environments.
