``Research is what I am doing when I don’t know what I am doing.” – Wernher von Braun
My research and teaching interests include computational and applied mathematics. In particular, my current research focuses on constructing highly accurate fast algorithms leveraging approximation theory and numerical linear algebra.
The SAFRAN (Stable Accurate Fast Robust Algorithms & Numerics) group was born in April 2016 at IISc and moved to IITM in September 2017. Students interested to be part of the group can apply for the research program at IITM Madras. Interested candidates are strongly encouraged to apply to either the Mathematics Department or the Interdisciplinary program. More details on the research at Mathematics Department and the Interdisciplinary program can be found here.
Members:
Current
 Sivaram Ambikasaran (Convenor)
 Nachiketa Mishra  (Post Doctoral Fellow, ICTS)
 Kandappan  (Ph.D. Student, IISc)
 Manisha  (Ph.D. Student, ICTS)
 Vaishnavi  (Ph.D. Student, IISc)
 Abhay Gupta  (M.Tech. Research Student, IISc)
Alumni
 Richa Naik (Intern, IITM)
 Deeksha Koul (Final Year Project student, DAICT)
 Karan Raj Singh (Project Assistant, BITS)
 Chaitanya Tappu (S.N.Bhat Intern, IISc)
 Ankit Kumar (S.N.Bhat Intern, NIT Rourkela)
Publications
 Daniel ForemanMackey, Eric Agol, Ruth Angus, Sivaram Ambikasaran, “Fast and scalable Gaussian process modeling with applications to astronomical time series”, Submitted to AAS journals.
 Sivaram Ambikasaran, Krithika Narayanaswamy, “An accurate, fast, mathematically robust, universal, noniterative algorithm for computing multicomponent diffusion velocities”, Proceedings of Combustion Institute.
 Sivaram Ambikasaran, Carlos Borges, LiseMarie ImbertGerard, Leslie Greengard, “Fast, adaptive, high order discretization of the LippmannSchwinger equation in two dimension”, SIAM Journal of Scientific Computing
 Sivaram Ambikasaran, “Generalized Rybicki Press algorithm”, Numerical Linear Algebra with Applications.
 Sivaram Ambikasaran, Michael O’Neil, Karan Raj Singh, “Fast symmetric factorization of hierarchical matrices with applications”
 Amirhossein Aminfar, Sivaram Ambikasaran, Eric Darve, ``A fast block lowrank dense solver with applications to finiteelement matrices”, Journal of Computational Physics
 Sivaram Ambikasaran, Daniel ForemanMackey, Leslie Greengard, David W Hogg, Michael O’Neil, ``Fast Direct Methods for Gaussian Processes”, Transactions on Pattern Analysis and Machine Intelligence
 Jun Lai, Sivaram Ambikasaran, Leslie F Greengard, ``A fast direct solver for high frequency scattering from a large cavity in two dimensions”, SIAM Journal of Scientific Computing
 Sivaram Ambikasaran, Eric Darve, “The Inverse Fast Multipole Method”
 Judith Y Li, Sivaram Ambikasaran, Eric Darve, Peter K Kitanidis, “A Kalman filter powered by ℋ^{2}matrices for quasicontinuous data assimilation problems”, Water Resources Research.
 Sivaram Ambikasaran, “Fast Algorithms for Dense Numerical Linear Algebra and Applications, Stanford Thesis”, Stanford Thesis
 Sivaram Ambikasaran, Arvind Krishna Saibaba, Eric Darve, Peter K Kitanidis, “Fast Algorithms for Bayesian Inversion”, The IMA Volumes in Mathematics and its Applications
 Arvind Krishna Saibaba, Sivaram Ambikasaran, Judith Y Li, Peter K Kitanidis, Eric Darve, “Application of hierarchical matrices in geostatistics”, Oil & Gas Science and Technology  Revue d’IFP Energies Nouvelles.
 Sivaram Ambikasaran, Judith Y Li, Peter K Kitanidis, Eric Darve, “Largescale stochastic linear inversion using hierarchical matrices”, Computational Geosciences.
 Sivaram Ambikasaran, and Eric Darve, “An 𝒪(NlogN) fast direct solver for partially hierarchical semiseparable matrices”, Journal of Scientific Computing.
 K. Bhaskar, Sivaram Ambikasaran, ``Untruncated infinite series superposition method for accurate flexural analysis of isotropic/orthotropic rectangular plates with arbitrary edge conditions”, Composite Structures.
Sample Research

Fast Linear Algebra Algorithms
The major focus of the group is to reduce the computational complexity of linear algebra algorithms at the same time maintaining high accuracy.

Dense linear algebra
Typically, linear algebra algorithms for dense matrix vector products scale as 𝒪(N^{2}) and for solving dense linear systems scale as 𝒪(N^{3}), where the underlying matrix is of size N × N. This is excruciatingly slow for largescale practical problems, making it unattractive for largescale simulation needed for optimization and design. However, note that since most of the computation is done in finite precision (say machine precision of 16 digits), we can devise approximate but arbitrarily accurate numerical algorithms, that scales favorably and guaranteeing high precision.

Sparse linear algebra
To solve partial differential equations, discretizing the PDE using finite difference or finite element method is preferred in practice. Such discretizations result in a sparse linear system, which needs to be solved. Solving the corresponding linear system of size N × N using nested dissection or multifrontal strategies requires a computational cost of 𝒪(N^{(d+1)/2}), where d is the underlying dimension. Our focus is on reducing these computational costs to 𝒪(N) guaranteeing high precision.


Design Centric PDE Solvers
One major application focus of our group is on designing accurate, fast ODE solvers for a wide range of applications, for instance, electromagnetic scattering from penetrable and impenetrable objects, advance stealth applications, material homogenization, imaging, etc. Traditional solvers have been mainly based on iterative techniques, which has its own set of disadvantages. For instance, the number of iterations to converge may be extremely large making it impractical. Another disadvantage, especially for design, is that any modifciation to the geometry or incoming field typically results in restarting the entire iterative solver, which is definitely not desirable. To circumvent these disadvantages, our focus is to design high accurate and fast direct solvers for such systems.

Computational Statistics
One of the core issues in computational statistics and large scale data analysis is efficient manipulation of dense covariance matrices, which describe the interdependence between several random variables or random processes. This is especially the case in spatial statistics and time series. In the previously mentioned contexts, the correlation matrices have a highly exploitable structure.

Blackbox tools, reproducible and open source computational science
There are several implementations of the fast multipole method and fast direct solvers, but very few provide standalone implementations or the basic ``building blocks” that can be easily reused. One of our focus is to develop such standalone implementations, which will be of use in many other applications. Another endeavor is to have the tools for computational science available to the public as open source standalone numerical libraries. To this end, we maintain a GitHub repository(https://github.com/sivaramambikasaran) and a GitLab repository (https://gitlab.com/groups/SAFRAN), where all the numerical packages, which include fast multipole method, fast direct solvers, quadratures, etc., developed by us are posted and made available to the public.
List of possible research topics
Below are some of the possible research projects, one can pursue as a member of our group. The list is only indicative and by no means exhaustive. Look here for more details on the research program at IIT Madras. Students are also welcome to discuss other projects that interest them to pursue research under my guidance. Choose Mathematics Department and Interdisciplinary program, if you wish to be considered for the group.
 HODLR: Inhouse fast linear algebra package for hierarchical matrices
 Direct solvers for high frequency scattering
 Linear complexity solvers for finite element matrices
 Fast algorithms for integral equations based inverse obstacle scattering
 Fast algorithms for integral equations based inverse medium scattering
 Linear complexity algorithms for computational statistics
 Algorithms for solving elliptic PDE’s with rapidly oscillating coefficient with applications to material homogenisation
 Matrix Analysis: Pivoting algorithms and analysis of growth factor
 Accurate multicomponent diffusion in counterflow diffusion flame
 Compact chemical mechanism
 Accurate simulation of cool flames
Mathematical Packages
A significant part of the group’s effort is to develop inhouse mathematical packages and make them available to the scientific community. These are made available in the hope that they will be useful, but without any warranty. All the codes can be redistributed and/or modified under the terms of the appropriate license.
 BBFMM2D: Black Box Fast Multipole Method in two dimensions
 FLIPACK: Fast Linear Inversion PACKage
 HODLR: Fast Direct Solver for hierarchical offdiagonal lowrank matrices
 ESS: Extended SemiSeparable and the Generalized Rybicki Press algorithms
 George: Fast and flexible Python library for Gaussian Process Regression
 Celerite: A scalable method for Gaussian Process regression.
Funding
Present
 INSPIRE faculty award by Department of Science & Technology
 Young Scientist Research Award by Department of Atomic Energy
Past
 Startup grant by Indian Institute of Science
 Simons Foundation Fellowship by ICTS
 Infosys Foundation Funding by ICTS
 AIG