Randomized Sketching and Tensor-Format Solvers for High-Dimensional Models

Modern science and engineering increasingly rely on numerical simulations of systems with large numbers of interacting variables: new quantum materials with intricate atomic geometries, hot plasmas inside fusion reactors, time-resolved medical image, and many more applications. Storage and manipulation of such high-dimensional information are often impossible using conventional methods, because the required memory and runtime grow exponentially with the number of variables. This project develops a new set of mathematical tools that represent this information in compact, structured form and use carefully designed random projections to extract answers far more efficiently than was previously possible, while rigorously ensuring a near-zero probability of failure. These techniques directly address scalability bottlenecks in modern AI and machine learning pipelines, where high-dimensional tensor operations are increasingly central to large-scale model training and inference. The resulting methods accelerate discovery in areas central to national priorities, including quantum science, where they enable improved quantum chemistry calculations, and the design of next-generation quantum moire materials for electronics; fusion energy, where they support fast digital twins of plasma turbulence in tokamak reactors; and medical imaging, where they speed up reconstruction of dynamic scans from limited data. The project also trains undergraduate and graduate students at Virginia Tech in modern computational mathematics, develops new courses that integrate tensor-based algorithms with high-performance scientific computing, and supports K-12 outreach through the Blacksburg Math Circle using visually engaging topics such as the geometry of moire patterns. All algorithms and software are released openly so that researchers across many disciplines can adopt, extend, and benefit from the advances. This project develops a new class of randomized sketching operators, the block-structured TTStack sketches, that combine the storage efficiency of tensor-train representations with rigorous embedding guarantees of Johnson-Lindenstrauss type. These sketches are designed to overcome two well-documented limitations of existing tensor-network sketching: the exponential scaling in tensor dimension that afflicts Khatri-Rao embeddings, and the absence of provable subspace embedding guarantees for current Gaussian tensor-train sketches. The research pursues four coordinated objectives. First, it establishes oblivious subspace embedding and distortion bounds for TTStack sketches and characterizes the trade-offs among sketch size, accuracy, and block structure. Second, it develops scalable randomized algorithms that operate directly in tensor formats, including sketch-and-compress routines for rank-increasing operations, randomized Krylov solvers such as TT-GMRES and randomized Lanczos for linear and eigenvalue problems, and streaming tensor-train approximation methods that overcome current dimensional limitations. Third, it builds theoretical foundations for non-Hermitian iterative eigensolvers in tensor format, enabling reliable ground-state calculations for transcorrelated quantum chemistry Hamiltonians where standard variational methods fail. Fourth, it validates the methods on demanding benchmarks drawn from quantum many-body physics, gyrokinetic plasma turbulence compression, dynamic medical imaging reconstruction, and atomic-scale modeling of multilayer moire quantum materials. High-performance open-source Julia implementations, designed for algorithmic experimentation and reproducibility, are released to the community to support broad adoption and to seed further interdisciplinary advances in computational science. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2608677 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Paul Cazeaux | Institution: Virginia Polytechnic Institute and State University, BLACKSBURG, VA | Award Amount: $249,581 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2608677 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2608677.html