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The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given interval. Our quantum algorithm is based on a method that approximates the eigenvalue counts by applying the numerical contour integral and the stochastic trace estimator applied to a matrix involving resolvent matrices. As components of our algorithm, the HHL solver is applied to an augmented linear system of the resolvent matrices, and the quantum Fourier transform (QFT) is adopted to represent the operation of the numerical contour integral. To reduce the size of the augmented system, we exploit a certain symmetry of the numerical integration. We also introduce a permutation formed by CNOT gates to make the augmented system solution consistent with the QFT input. The eigenvalue count in a given interval is derived as the probability of observing a bit pattern in a fraction of the qubits of the output state.

Unsourced random-access (U-RA) is a type of grant-free random access with a virtually unlimited number of users, of which only a certain number $K_a$ are active on the same time slot. Users employ exactly the same codebook, and the task of the receiver is to decode the list of transmitted messages. We present a concatenated coding construction for U-RA on the AWGN channel, in which a sparse regression code (SPARC) is used as an inner code to create an effective outer OR-channel. Then an outer code is used to resolve the multiple-access interference in the OR-MAC. We propose a modified version of the approximate message passing (AMP) algorithm as an inner decoder and give a precise asymptotic analysis of the error probabilities of the AMP decoder and of a hypothetical optimal inner MAP decoder. This analysis shows that the concatenated construction can achieve a vanishing per-user error probability in the limit of large blocklength and a large number of active users at sum-rates up to the symmetric Shannon capacity, i.e. as long as $K_aR < 0.5\log_2(1+K_a\SNR)$. This extends previous point-to-point optimality results about SPARCs to the unsourced multiuser scenario. Furthermore, we give an optimization algorithm to find the power allocation for the inner SPARC code that minimizes the required $\SNR$.

We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix-matrix products, respectively. For problems of the form $\exp(-iA)$, with $A$ a real and symmetric matrix, an improved version is presented that computes the sine and cosine of $A$ with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Pad\'e approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schr\"odinger equation.

The multi-user Holographic Multiple-Input and Multiple-Output Surface (MU-HMIMOS) paradigm, which is capable of realizing large continuous apertures with minimal power consumption, has been recently considered as an energyefficient solution for future wireless networks, offering the increased flexibility in impacting electromagnetic wave propagation according to the desired communication, localization, and sensing objectives. The tractable channel modeling of MU-HMIMOS systems is one of the most critical challenges, mainly due to the coupling effect induced by the excessively large number of closely spaced patch antennas. In this paper, we focus on this challenge for downlink multi-user communications and model the electromagnetic channel in the wavenumber domain using the Fourier plane wave representation. Based on the proposed channel model, we devise the maximum-ratio transmission and Zero-Forcing (ZF) precoding schemes capitalizing on the sampled channel variance that depends on the number and spacing of the patch antennas in MU-HMIMOS, and present their analytical spectral efficiency performance. Moreover, we propose a low computational ZF precoding scheme leveraging Neumann series expansion to replace the matrix inversion, since it is practically impossible to perform direct matrix inversion when the number of patch antennas is extremely large. Our extensive simulation results showcase the impact of the number of patch antennas and their spacing on the spectral efficiency of the considered systems. It is shown that the more patch antennas and larger spacing results in improved performance due to the decreased correlation among the patches.

Consider the real symbol $f_{n}(\thetat)\coloneqq\sum_{j=0}^{n-1}h^{jh}|\thteta|^{2-jh}$ with $h\coloneqq\frac{1}{n}$. In this note we obtain a real interval where the smallest eigenvalue of the Toeplitz matrix $A_{n}\coloneqq T_{n}(f_{n})$ belongs to, with $f_{n}$ being the standard generating function of $A_{n}$. The considered type of matrix stems in the context of the numerical approximation of distributed order fractional differential equations (FDEs). In fact the new presented bounds improve those already present in the literature and give a more accurate spectral information, which are in fact used in the design of fast numerical algorithms for the associated linear systems, approximating the given distributed order FDEs.

In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quadratization approach and the analogous linearized strategy used in the construction of the linear momentum-preserving scheme. The proposed scheme is linear, high-order and can preserve a discrete quadratic energy exactly. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Optimization of hyperparameters of Gaussian process regression (GPR) determines success or failure of the application of the method. Such optimization is difficult with sparse data, in particular in high-dimensional spaces where the data sparsity issue cannot be resolved by adding more data. We show that parameter optimization is facilitated by rectangularization of the defining equation of GPR. On the example of a 15-dimensional molecular potential energy surface we demonstrate that this approach allows effective hyperparameter tuning even with very sparse data.

In this paper, we present, evaluate and analyse the performance of parallel synchronous Jacobi algorithms by different partitioned procedures including band-row splitting, band-row sparsity pattern splitting and substructuring splitting, when solving sparse large linear systems. Numerical experiments performed on a set of academic 3D Laplace equation and on a real gravity matrices arising from the Chicxulub crater are exhibited, and show the impact of splitting on parallel synchronous iterations when solving sparse large linear systems. The numerical results clearly show the interest of substructuring methods compared to band-row splitting strategies.

The recurrence rebuild and retrieval method (R3M) is proposed in this paper to accelerate the electromagnetic (EM) validations of large-scale digital coding metasurfaces (DCMs). R3M aims to accelerate the EM validations of DCMs with varied codebooks, which involves the analysis of a group of similar but distinct coding patterns. The method transforms general DCMs to rigorously periodic arrays by replacing each coding unit with the macro unit, which comprises all possible coding states. The system matrix corresponding to the rigorously periodic array is globally shared for DCMs with arbitrary codebooks via implicit retrieval. The discrepancy of the interactions for edge and corner units are precluded by the basis extension of periodic boundaries. Moreover, the hierarchical pattern exploitation algorithm is leveraged to efficiently assemble the system matrix for further acceleration. Due to the fully utilization of the rigid periodicity, the computational complexity of R3M is theoretically lower than that of $\mathcal{H}$-matrix within the same paradigm. Numerical results for two types of DCMs indicate that R3M is accurate in comparison with commercial software. Besides, R3M is also compatible with the preconditioning for efficient iterative solutions. The efficiency of R3M for DCMs outperforms the conventional fast algorithms by a large margin in both the storage and CPU time cost.

We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed algorithms.