Abstract We present a numerical investigation on the mixed precision Generalized Minimal Residual (GMRES) method, preconditioned by deflation. GMRES is a widely used iterative solver for sparse, non-symmetric linear systems. The mixed precision version of restarted GMRES, or mixed precision GMRES(m), is known to achieve double precision accuracy while delivering substantial speedups by using single precision (32-bit) arithmetic for most computations and double precision (64-bit) for key corrective steps only. In this study, the mixed precision GMRES(m) is enhanced with a deflation-based preconditioner to accelerate its convergence. All preconditioning operations are carried out in single precision. We demonstrate the effectiveness of the mixed precision GMRES(m) preconditioned by deflation in terms of both convergence (iterations required) and computational performance. This is validated through numerical experiments on a multi-core environment using a variety of matrices. The results show that the preconditioner remains highly effective in reducing iterations even under reduced precision. Additionally, we compare the speedups achieved by mixed precision over double precision in the preconditioned GMRES(m) with those in the non-preconditioned version. For the majority of tested matrices, the speedups in the preconditioned and non-preconditioned cases are comparable.

Abstract The present work deals with the numerical resolution of the telegrapher equation. We present various numerical explicit and implicit schemes for problems defined in one-dimensional domains. In this case, we consider, on one hand, Dirichlet boundary value conditions and in the other hand, the mixed ones. In both cases, we analyze the stability by using the classical Von Neumann method or the matricial one. Moreover, the truncation error of the presented schemes are computed. The cases of two- and three-dimensional cases, is also briefly considered with only the Dirichlet boundary value conditions and in simple geometries. Concerning the implicit schemes, we briefly survey the main classical numerical linear algebra algorithms and we show that they can be applied successfully to the numerical solution of the target problem. We briefly consider the cases where, the coefficients of the problem are not constant and, also, the case where the linear problem is perturbed by a singlevalued or a multivalued diagonal operator. This study is completed by numerical simulations in the case of one-dimensional problems.

Abstract We propose an efficient algorithm for reconstructing large-scale one-dimensional line spectra from their Fourier data in a bounded interval $$[-\Omega ,\Omega ]$$ [ - Ω , Ω ] . While traditional subspace methods such as MUSIC achieve super-resolution for closely separated line spectra, their computational cost is high, particularly for large-scale ones. To address this issue, we proposed a scalable algorithm, termed SCAN-MUSIC, that scans the spectral domain using a fixed Gaussian window and then reconstructs the line spectra falling into the window at each time. For line spectra with cluster structure, we further refine the proposed algorithm using the annihilating filter technique. Both algorithms can significantly reduce the computational complexity of the standard MUSIC algorithm with a moderate loss of resolution. The algorithms are supplemented with theoretical analyses of error estimates, sampling complexity, computational complexity, and computational limit.