Current Research

Analytical and Computational Mechanics

I am currently working on the modeling and numerical simulation of the nonlinear dynamics of quasi-unidimensional structures, such as beams and cables, with occult boundary conditions. More specifically my current project involves the simulation of the dynamics of a cable under external forces with extrusion and moving boundary conditions. The geometrically complex nature of the dynamics necessitates the use of numerical methods that conserve geometrical invariants. The non-conservative nature of the extrusion boundary condition adds an additional layer of complexity to the problem.

This project is being in collaboration with the company D-ICE Engineering based at Nantes which is directly interested in this problem. One of the challenges in this project is to develop methods that yield performant simulations appropriate for an industrial setting. This additional constraint, along with the peculiarities of the problem, makes this a particularly interesting subject of research.


Past Research

Theory of Singularities and Real and Complex Analytic Geometry

I proved that certain numerical and algebraic invariants of analytic singularities are preserved under algebraic approximation. This work involved the adaptation of and the innovative use of tools normally associated with computational algebraic geometry. These results are related to deeper questions of the possibility of the algebraic classification of analytic singularities by Hironaka’s resolution of singularities.

I also studied the geometry of locally bounded rational functions on real algebraic varieties. This work leads naturally to possible generalizations using the concept of the real spectrum developed by M. F. Roy, M. Coste and others.

Publications:

  1. A. Patel. Algebraic approximation of Cohen-Macaulay algebras. J. Algebra 625: 66-81, 2023.

  2. A. Patel. Finite determinacy and approximation of flat maps. Proc. Amer. Math. Soc., 151(1):201–213, 2023.

  3. J. Adamus, A. Patel. Equisingular algebraic approximation of real and complex analytic germs. J. Singul. 20:289-310, 2020.

  4. V. Delage, G.Fichou, A. Patel. The geometry of locally bounded rational functions. Advances in Geometry 25.3: 409–427, 2025.

Numerical Linear Algebra

I worked on the development of a massively parallel algorithm for Incomplete LU, and Cholesky preconditioners. The fine-grained parallelism and asynchronicity of this algorithm render it highly performant on modern computer architectures, and on discrete computing devices such as GPUs. As part of this project I worked with Eric Boman and Siva Rajamanickam at Sandia National Labs to validate and compare the performance of the method on CPUs, GPUs, and other computing devices (see the cited technical report below). Our paper in the SIAM Journal on Scientific Computing (SISC) on this work has been cited more than 250 times, and marks a paradigm change in the effective application of preconditioning methods in the solution of linear and nonlinear systems of algebraic equations. This work was done at Georgia Tech in the USA.

Publications:

  1. E. Chow, A. Patel. Fine-grained parallel incomplete LU factorization. SIAM journal on Scientific Computing 37.2: C169–C193, 2015.

  2. A. Patel, E. Boman, S. Rajamanickam, E. Chow. Cross Platform Fine Grained ILU and ILDL Factorizations Using Kokkos. Center for Computing Research Summer Proceedings (2015), A.M. Bradley and M.L. Parks, eds., Technical Report SAND2016-0830R, Sandia National Laboratories, pp. 159-177, 2015

High Performance Computing

I have worked on the development and implementation of new parallel algorithms in scientific computing, adapted to modern supercomputers. Notably, my work on the development of a new parallel algorithm for the Hartree-Fock methods won the award for best paper in one of the top international conferences in High Performance Computing (see the citation below). This work was done at Georgia Tech in the USA.

Publications:

  1. X. Liu, A. Patel, and E. Chow. A new scalable parallel algorithm for Fock matrix construction. IEEE 28th international parallel and distributed processing symposium (IPDPS), 2014.