arXiv subjects

Zvonimir Bujanović

Publications and source records attributed to Zvonimir Bujanović.

12 recordsLinked to original sources

arXiv2026

Stochastic trace estimation for positive trace-class operators

Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\mathcal{O}(\varepsilon^{-2})$ products to achieve accuracy $\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\mathcal{O}(\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\mathcal{O}(\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning & Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.

math.NA↗

arXiv2026

Stochastic trace estimation with tensor train random vectors

Stochastic trace estimation is a standard tool for approximating the trace of a large-scale matrix available only through matrix-vector products. However, in tensor-structured settings, unstructured Gaussian or Rademacher test vectors may be prohibitively expensive to store and compute with, while cheaper rank-one tensor-product vectors can require sample complexities that grow exponentially with the tensor order. This work studies Gaussian random tensor train vectors as a structured alternative for stochastic trace estimation. We show that, with a suitable choice of the tensor train rank, random tensor train vectors recover dimension-independent guarantees for the Girard--Hutchinson estimator. In particular, a median-of-means variant with tensor train rank $r \geq d-1$ achieves the same dependence on the accuracy $\varepsilon$ and failure probability $δ$ as the classical estimator based on unstructured Gaussian vectors. We further prove an oblivious subspace injection result for sketches formed from independent Gaussian random tensor train vectors: tensor train rank $r\geq d-1$ and $\mathcal{O}(\varepsilon^{-2}(k+\log(1/δ)))$ samples suffice for a $k$-dimensional target subspace. Finally, we investigate the use of such sketches within the Nyström++ framework. We show that the resulting estimator can achieve the desired $\mathcal{O}(\varepsilon^{-1})$ sample complexity under an additional spectral-tail condition. These results provide clarififcation on both the potential and the limitations of random tensor train vectors in stochastic trace estimation.

stat.ML↗

arXiv2025

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums

Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods.

math.NT↗

arXiv2024

Subspace embedding with random Khatri-Rao products and its application to eigensolvers

Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods, and preconditioned solvers such as the so called LOBPCG method. All of these solvers rely on random matrices to determine, e.g., starting vectors that have, with high probability, a non-negligible overlap with the eigenvectors of interest. For this purpose, a safe and common choice are unstructured Gaussian random matrices. In this work, we investigate the use of random Khatri-Rao products in eigenvalue solvers. On the one hand, we establish a novel subspace embedding property that provides theoretical justification for the use of such structured random matrices. On the other hand, we highlight the potential algorithmic benefits when solving eigenvalue problems with Kronecker product structure, as they arise frequently from the discretization of eigenvalue problems for differential operators on tensor product domains. In particular, we consider the use of random Khatri-Rao products within a contour integral method and LOBPCG. Numerical experiments indicate that the gains for the contour integral method strongly depend on the ability to efficiently and accurately solve (shifted) matrix equations with low-rank right-hand side. The flexibility of LOBPCG to directly employ preconditioners makes it easier to benefit from Khatri-Rao product structure, at the expense of having less theoretical justification.

math.NA↗

arXiv2024

Murmurations of Mestre-Nagao sums

This paper investigates the detection of the rank of elliptic curves with ranks 0 and 1, employing a heuristic known as the Mestre-Nagao sum \[ S(B) = \frac{1}{\log{B}} \sum_{\substack{p<B \\ \textrm{good reduction}}} \frac{a_p(E)\log{p}}{p}, \] where $a_p(E)$ is defined as $p + 1 - \#E(\mathbb{F}_p)$ for an elliptic curve $E/\mathbb{Q}$ with good reduction at prime $p$. This approach is inspired by the Birch and Swinnerton-Dyer conjecture. Our observations reveal an oscillatory behavior in the sums, closely associated with the recently discovered phenomena of murmurations of elliptic curves. Surprisingly, this suggests that in some cases, opting for a smaller value of $B$ yields a more accurate classification than choosing a larger one. For instance, when considering elliptic curves with conductors within the range of $[40\,000,45\,000]$, the rank classification based on $a_p$'s with $p < B = 3\,200$ produces better results compared to using $B = 50\,000$. This phenomenon finds partial explanation in the recent work of Zubrilina.

math.NT↗

arXiv2022

Iterative Refinement of Schur decompositions

The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of $A$ from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.

math.NA↗

arXiv2020

Norm and trace estimation with random rank-one vectors

A few matrix-vector multiplications with random vectors are often sufficient to obtain reasonably good estimates for the norm of a general matrix or the trace of a symmetric positive semi-definite matrix. Several such probabilistic estimators have been proposed and analyzed for standard Gaussian and Rademacher random vectors. In this work, we consider the use of rank-one random vectors, that is, Kronecker products of (smaller) Gaussian or Rademacher vectors. It is not only cheaper to sample such vectors but it can sometimes also be much cheaper to multiply a matrix with a rank-one vector instead of a general vector. In this work, theoretical and numerical evidence is given that the use of rank-one instead of unstructured random vectors still leads to good estimates. In particular, it is shown that our rank-one estimators multiplied with a modest constant constitute, with high probability, upper bounds of the quantity of interest. Partial results are provided for the case of lower bounds. The application of our techniques to condition number estimation for matrix functions is illustrated.

math.NA↗

arXiv2020

A numerical comparison of solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.

math.NA↗

arXiv2019

New robust ScaLAPACK routine for computing the QR factorization with column pivoting

In this note we describe two modifications of the ScaLAPACK subroutines PxGEQPF for computing the QR factorization with the Businger-Golub column pivoting. First, we resolve a subtle numerical instability in the same way as we have done it for the LAPACK subroutines xGEQPF, xGEQP3 in 2006. [LAPACK Working Note 176 (2006); ACM Trans. Math. Softw. 2008]. The problem originates in the first release of LINPACK in the 1970's: due to severe cancellations in the down-dating of partial column norms, the pivoting procedure may be in the dark completely about the true norms of the pivot column candidates. This may cause miss-pivoting, and as a result loss of the important rank revealing structure of the computed triangular factor, with severe consequences on other solvers that rely on the rank revealing pivoting. The instability is so subtle that, e.g., inserting a WRITE statement or changing the process topology can drastically change the result. Secondly, we also correct a programming error in the complex subroutines PCGEQPF, PZGEQPF, which also causes wrong pivoting because of erroneous use of PSCNRM2, PDZNRM2 for the explicit norm computation.

cs.MS↗

arXiv2018

A Householder-based algorithm for Hessenberg-triangular reduction

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - λB$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A non-vanishing fraction of the total flop count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state of the art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multi-threaded BLAS. The design and evaluation of a parallel formulation is future work.

math.NA↗

arXiv2017

Parallel solver for shifted systems in a hybrid CPU-GPU framework

This paper proposes a combination of a hybrid CPU--GPU and a pure GPU software implementation of a direct algorithm for solving shifted linear systems $(A - σI)X = B$ with large number of complex shifts $σ$ and multiple right-hand sides. Such problems often appear e.g. in control theory when evaluating the transfer function, or as a part of an algorithm performing interpolatory model reduction, as well as when computing pseudospectra and structured pseudospectra, or solving large linear systems of ordinary differential equations. The proposed algorithm first jointly reduces the general full $n\times n$ matrix $A$ and the $n\times m$ full right-hand side matrix $B$ to the controller Hessenberg canonical form that facilitates efficient solution: $A$ is transformed to a so-called $m$-Hessenberg form and $B$ is made upper-triangular. This is implemented as blocked highly parallel CPU--GPU hybrid algorithm; individual blocks are reduced by the CPU, and the necessary updates of the rest of the matrix are split among the cores of the CPU and the GPU. To enhance parallelization, the reduction and the updates are overlapped. In the next phase, the reduced $m$-Hessenberg--triangular systems are solved entirely on the GPU, with shifts divided into batches. The benefits of such load distribution are demonstrated by numerical experiments. In particular, we show that our proposed implementation provides an excellent basis for efficient implementations of computational methods in systems and control theory, from evaluation of transfer function to the interpolatory model reduction.

cs.MS↗

arXiv2016

RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations

This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature -- all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.

math.NA↗