Sivaram Ambikasaran

We present a fast iterative solver for scattering problems in 2D,where a penetrable object with compact support is considered. By representing the scattered field as a volume potential in terms of the Green's function, we arrive at the Lippmann- Schwinger equation in integral form, which is then discretized using an appropriate quadrature technique. The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel [1], and the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation (NCA) [2]. The advantage of the NCA described in [2] is that the search space of so-called far-field pivots is smaller than that of the existing NCAs [3, 4]. Another significant contribution of this work is the use of HODLR based direct solver [5] as a preconditioner to further accelerate the iterative solver. In one of our numerical experiments, the iterative solver does not converge without a preconditioner. We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not. Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver, DAFMMbased fast iterative solver, and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem. To the best of our knowledge, this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions. In the spirit of reproducible computational science, the implementation of the algorithms developed in this article.

Ignition delay time (IDT) is an important global combustion property that affects the thermal efficiency of the engine and emissions (particularly NO (Formula presented.)). IDT is generally measured by performing experiments using Shock-tube and Rapid Compression Machine (RCM). The numerical calculation of IDT is a computationally expensive and time-consuming process. Arrhenius type empirical correlations offer an inexpensive alternative to obtain IDT. However, such correlations have limitations as these typically involve ad-hoc parameters and are valid only for a specific fuel and particular range of temperature/pressure conditions. This study aims to formulate a data-driven scientific way to obtain IDT for new fuels without performing detailed numerical calculations or relying on ad-hoc empirical correlations. We propose a rigorous, well-validated data-driven study to obtain IDT for new fuels using a regression-based clustering algorithm. In our model, we bring in the fuel structure through the overall activation energy ((Formula presented.)) by expressing it in terms of the different bonds present in the molecule. Gaussian Mixture Model (GMM) is used for the formation of clusters, and regression is applied over each cluster to generate models. The proposed algorithm is used on the ignition delay dataset of straight-chain alkanes (C (Formula presented.) H (Formula presented.) for n = 4 to 16). The high level of accuracy achieved demonstrates the applicability of the proposed algorithm over IDT data. The algorithm and framework discussed in this article are implemented in python and made available at https://doi.org/10.5281/zenodo.5774617.

Using accurate multi-component diffusion treatment in numerical combustion studies remains formidable due to the computational cost associated with solving for diffusion velocities. To obtain the diffusion velocities, for low density gases, one needs to solve the Stefan-Maxwell equations along with the zero diffusion flux criteria, which scales as O(N3), when solved exactly. In this paper, we propose an accurate, fast, direct and robust algorithm to compute multi-component diffusion velocities. We also take into account the Soret effect, while computing the multi-component diffusion velocities. To our knowledge, this is the first provably accurate algorithm (the solution can be obtained up to an arbitrary degree of precision) scaling at a computational complexity of O(N) in finite precision. The key idea involves leveraging the fact that the matrix of the reciprocal of the binary diffusivities, V, is low rank, with its rank being independent of the number of species involved. The low rank representation of matrix V is computed in a fast manner at a computational complexity of O(N) and the Sherman-Morrison-Woodbury formula is used to solve for the diffusion velocities at a computational complexity of O(N). Rigorous proofs and numerical benchmarks illustrate the low rank property of the matrix V and scaling of the algorithm.