{"response":{"award":[{"abstractText":"Numerical linear algebra (NLA) is foundational to mathematical and statistical computing, scientific simulation, data analysis, and artificial intelligence. Yet new NLA algorithms are still designed largely by hand, and claims about their correctness, reliability, and efficiency often rely on arguments that cannot be checked automatically. AI methods, including reinforcement learning (RL) and large language models (LLMs), can explore algorithm designs efficiently, but they may produce opaque or inconsistent results without dependable guarantees. This project will create a trustworthy framework in which AI helps discover new NLA algorithms, statistically guided methods identify promising designs, and formal proof software verifies their mathematical correctness and certifies their computational and communication costs. By making computational tools faster, statistically guided, and mathematically sound, the project will serve the national interest and support the mission of advancing national health, prosperity, and security. The project’s broader impacts will include training undergraduate and graduate students and strengthening regional research capacity by anchoring training activities in New Hampshire. The investigators will release openly available software and educational materials, host public workshops, and broaden participation.\r\n\r\nThe technical goal is to develop Linear Algebra Algorithm Morphing (LAAM), a statistically guided framework for discovering, analyzing, and certifying new NLA algorithms. Rather than treating an algorithm as unstructured code or a fixed vector of tuning parameters, LAAM will represent each candidate as a structured mathematical derivation that records algebraic transformations, nested-loop organization, compositional primitives, admissible parameters, and symbolic arithmetic and communication costs. This structural representation creates a new mathematical and statistical optimization problem: the search space is mixed discrete-continuous, variable-dimensional, highly redundant, and organized by equivalence among derivations. The project will therefore develop new Bayesian optimization and RL methods, including novel uncertainty-aware surrogate models and equivalence-aware graph search, to identify promising and mathematically distinct candidates while minimizing expensive proof and evaluation calls. Lean will certify deterministic correctness and cost claims, thereby converting AI-generated candidates into reusable mathematical objects rather than opaque algorithmic programs. Because LLM-generated algorithms and proofs vary across prompts, the LAAM will also develop sequentially valid inference to separate genuine algorithmic improvements from model-induced randomness. These statistical advances will provide a principled foundation for reproducible AI-assisted mathematical discovery. LAAM will compose randomized sketches, iterative updates, preconditioners, and orthogonalization procedures across matrix families varying in conditioning, sparsity, rank, leverage structure, and spectral decay. The resulting theory and algorithms will not only advance NLA itself by establishing new correctness, stability, and complexity results, but also advancing statistics through faster and more reliable methods for principal component analysis, least squares, regression, low-rank approximation, and matrix completion. Finally, a verified mathematical library will attach machine-readable accuracy, precondition, cost, and confidence contracts to certified routines and record certificate-backed provenance.\r\n\r\nThis 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.","activeAwd":"true","agency":"NSF","awardAgencyCode":"4900","awardee":"WILLIAM MARSH RICE UNIVERSITY","awardeeAddress":"6100 MAIN ST","awardeeCity":"HOUSTON","awardeeCountryCode":"US","awardeeDistrict":"07","awardeeDistrictCode":"TX07","awardeeName":"William Marsh Rice University","awardeePhone":"7133484820","awardeeStateCode":"TX","awardeeZipCode":"77005","cfdaNumber":"47.049","date":"08/17/2026","dirAbbr":"MPS","divAbbr":"DMS","estimatedTotalAmt":"150000","expDate":"08/31/2029","fundAgencyCode":"4900","fundProgramName":"MSPA-INTERDISCIPLINARY","fundsObligated":["FY 2026 = $150,000.00"],"fundsObligatedAmt":"150000","histAwd":"false","id":"2616828","initAmendmentDate":"08/17/2026","latestAmendmentDate":"08/17/2026","managingPec":"745400","orgCodeDir":"03000000","orgCodeDiv":"03040000","orgLongName":"Directorate for Mathematical and Physical Sciences","orgLongName2":"Division Of Mathematical Sciences","orgUrl":"http://www.nsf.gov/div/index.jsp?div=dms","parentUeiNumber":"","pdPIName":"HENGRUI LUO","perfAddress":"6100 MAIN ST","perfCity":"HOUSTON","perfCountryCode":"US","perfDistrict":"07","perfDistrictCode":"TX07","perfLocation":"William Marsh Rice University","perfStateCode":"TX","perfZipCode":"77005","pi":["HENGRUI LUO hl180@rice.edu"],"piEmail":"hl180@rice.edu","piFirstName":"HENGRUI","piId":"270096981","piLastName":"LUO","poEmail":"jzhu@nsf.gov","poName":"Jun Zhu","poPhone":"7032924551","primaryProgram":["01002627DB NSF RESEARCH & RELATED ACTIVIT"],"progEleCode":"745400","program":"Artificial Intelligence (AI), Machine Learning Theory","progRefCode":"075Z, 079Z","publicAccessMandate":"1","startDate":"09/15/2026","title":"Collaborative Research: AIMing: Statistically Guided Discovery for Certified and Efficient Randomized Numerical Linear Algebra Algorithms","transType":"Standard Grant","ueiNumber":"K51LECU1G8N3"}],"metadata":{"offset":0,"rpp":25,"totalCount":1}}}