arXiv subjects

Ritesh Khan

Publications and source records attributed to Ritesh Khan.

4 recordsLinked to original sources

arXiv2026

Hybrid hierarchical matrices with adaptive mixed precision storage

Hierarchical matrices are data-sparse approximations of dense matrices that are widely used for fast matrix computations. Hierarchical matrices are built using a tree data structure, with low-rank blocks identified by various admissibility conditions, such as standard admissibility and weak admissibility. This paper introduces a novel hierarchical matrix framework, namely $\mathcal{H}_h$, based on a hybrid admissibility condition: we use the standard admissibility at the coarser levels (larger blocks) and the weak admissibility at the finer levels (smaller blocks). This hybrid strategy confines dense blocks only along the diagonal. We provide a criterion that ensures lower storage cost for $\mathcal{H}_h$-matrices compared to $\mathcal{H}$-matrices under the standard admissibility condition. We carry out a rounding error analysis of $\mathcal{H}_h$-matrices and show that the admissible blocks of $\mathcal{H}_h$-matrices can be represented in low precision (precision lower than the working precision) without degrading the overall approximation quality. We provide an explicit rule for dynamically selecting the precision of a given admissible block, thereby proposing an adaptive mixed precision algorithm for constructing and storing $\mathcal{H}_h$-matrices. Furthermore, we show that the use of mixed precision does not compromise the numerical stability and accuracy of the resulting $\mathcal{H}_h$-matrix-vector product. We perform a range of numerical experiments to validate our theoretical findings. Our numerical results show that the proposed adaptive mixed precision $\mathcal{H}_h$-matrices achieve significant storage reductions (up to $11 \times$) compared with uniform double precision standard admissibility-based $\mathcal{H}$-matrices, without compromising accuracy.

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arXiv2024

A hybrid interpolation ACA accelerated method for parabolic boundary integral operators

We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.

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arXiv2023

New Algebraic Fast Algorithms for $N$-body Problems in Two and Three Dimensions

We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for $N$-body problems in $d$ dimensions, namely efficient $\mathcal{H}^2_{*}$ (fully nested algorithm, i.e., $\mathcal{H}^2$ matrix-like algorithm) and $(\mathcal{H}^2 + \mathcal{H})_{*}$ (semi-nested algorithm, i.e., cross of $\mathcal{H}^2$ and $\mathcal{H}$ matrix-like algorithms). The efficient $\mathcal{H}^2_{*}$ and $(\mathcal{H}^2 + \mathcal{H})_{*}$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions, where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA and NCA) and develop both algorithms in a black-box fashion. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms in $2$D and $3$D. The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. Notably, all the algorithms are developed in a similar fashion in $\texttt{C++}$ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\mathcal{H}^2$ matrix algorithm with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/.

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arXiv2023

HODLR$d$D: A new Black-box fast algorithm for $N$-body problems in $d$-dimensions with guaranteed error bounds

In this article, we prove new theorems bounding the rank of different sub-matrices arising from these kernel functions. Bounds like these are often useful for analyzing the complexity of various hierarchical matrix algorithms. We also plot the numerical rank growth of different sub-matrices arising out of various kernel functions in $1$D, $2$D, $3$D and $4$D, which, not surprisingly, agrees with the proposed theorems. Another significant contribution of this article is that, using the obtained rank bounds, we also propose a way to extend the notion of \textbf{\emph{weak-admissibility}} for hierarchical matrices in higher dimensions. Based on this proposed \textbf{\emph{weak-admissibility}} condition, we develop a black-box (kernel-independent) fast algorithm for $N$-body problems, hierarchically off-diagonal low-rank matrix in $d$ dimensions (HODLR$d$D), which can perform matrix-vector products with $\mathcal{O}(pN \log (N))$ complexity in any dimension $d$, where $p$ doesn't grow with any power of $N$. More precisely, our theorems guarantee that $p \in \mathcal{O} (\log (N) \log^d (\log (N)))$, which implies our HODLR$d$D algorithm scales almost linearly. The $\texttt{C++}$ implementation with \texttt{OpenMP} parallelization of the HODLR$d$D is available at \url{https://github.com/SAFRAN-LAB/HODLRdD}. We also discuss the scalability of the HODLR$d$D algorithm and showcase the applicability by solving an integral equation in $4$ dimensions and accelerating the training phase of the support vector machines (SVM) for the data sets with four and five features.

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