Discrete Basis Research Articles

Emulating quantum computing with optical matrix multiplication

Optical computing harnesses the speed of light to perform vector-matrix operations efficiently. It leverages interference, a cornerstone of quantum computing algorithms, to enable parallel computations. In this work, we interweave quantum computing with classical structured light by formulating the process of photonic matrix multiplication using quantum mechanical principles such as state superposition and subsequently demonstrate a well-known algorithm, namely, Deutsch–Jozsa’s algorithm. This is accomplished by elucidating the inherent tensor product structure within the Cartesian transverse degrees of freedom of light, which is the main resource for optical vector-matrix multiplication. To this end, we establish a discrete basis using localized Gaussian modes arranged in a lattice formation and demonstrate the operation of a Hadamard gate. Leveraging the reprogrammable and digital capabilities of spatial light modulators, coupled with Fourier transforms by lenses, our approach proves adaptable to various algorithms. Therefore, our work advances the use of structured light for quantum information processing.

Analysis of Solution Existence, Uniqueness, and Stability of Discrete Basis Sinograms in Multispectral CT

Analysis of Solution Existence, Uniqueness, and Stability of Discrete Basis Sinograms in Multispectral CT

Robust methods for multiscale coarse approximations of diffusion models in perforated domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

A discrete basis for celestial holography

Celestial holography provides a reformulation of scattering amplitudes in four dimensional asymptotically flat spacetimes in terms of conformal correlators of operators on the two dimensional celestial sphere in a basis of boost eigenstates. A basis of massless particle states has been previously identified in terms of conformal primary wavefunctions labeled by a boost weight ∆ = 1+iλ with λ ∈ ℝ. Here we show that a discrete orthogonal and complete basis exists for ∆ ∈ ℤ. This new basis consists of a tower of discrete memory and Goldstone observables, which are conjugate to each other and allow to reconstruct gravitational signals belonging to the Schwartz space. We show how generalized dressed states involving the whole tower of Goldstone operators can be constructed and evaluate the higher spin Goldstone 2-point functions. Finally, we recast the tower of higher spin charges providing a representation of the w1+∞ loop algebra (in the same helicity sector) in terms of the new discrete basis.

Non-contact heart rate measurement from face video based on group sparse representation

Non-contact heart rate measurement based on face video is rapidly developed due to its comfort and wide application. However, it is difficult to extract the pulse signals for non-contact heart rate measurement due to the various interference factors, such as illumination variation, head motion and face expression. In this paper, we propose group sparse representation to reconstruct the pulse signals, then estimate the heart rate based on the fact that the real heart rate is consistent at the same time from different sub-regions. Specifically, we formulate the reconstruction of pulse signals as group sparse representation problem and require the raw signals of all the sub-regions to be similar sparse representation. Firstly, we use the face detection algorithm to obtain the region of interest (ROI) and divide it into sub-regions, followed by the distortion compensation of color signals from different sub-regions. Then we extract the sub-region chrominance signals and select the high-quality chrominance signals to construct the raw pulse signals matrix. After that, we construct a mixed pulse dictionary containing discrete cosine bases and wavelet bases considering the periodicity and pulsatility of the pulse signals. Finally, we conduct group sparse representation to reconstruct the pulse signals and estimate the heart rate via spectral analysis based on the reconstructed pulse signals. Experimental results on three public datasets show that this method outperforms most existing heart rate measurement methods.

An integer basis for celestial amplitudes

We present a discrete basis of solutions of the massless Klein-Gordon equation in 3 + 1 Minkowski space which transform as \U0001d530\U0001d529(2, ℂ) Lorentz/conformal primaries and descendants, and whose elements all have integer conformal dimension. We show that the basis is complete in the sense that the Wightman function can be expressed as a quadratic sum over the basis elements.

Quantum Dynamical Investigation of Dihydrogen-Hydride Exchange in a Transition-Metal Polyhydride Complex.

The ongoing shift toward clean, sustainable energy is a primary driving force behind hydrogen fuel research. Safe and effective storage of hydrogen is a major challenge (particularly for mobile applications) and requires a detailed understanding of the atomic level interactions of hydrogen with its host materials. The light mass of hydrogen, however, implies that quantum effects are important, so a quantum dynamical treatment is required to properly account for these effects in computational simulations. As one such example, we describe herein the hydrogen exchange dynamics between a hydride and a dihydrogen ligand in the [FeH(H2)(PH3)4]+ model complex. A global three-dimensional (3D) potential energy surface (PES) was constructed by fitting to and interpolating from a discrete set of grid points computed using density functional theory; exact quantum dynamical calculations were then carried out on the 3D PES using discrete variable representation basis sets. Energy levels and their quantum tunneling splittings were computed up to 3000 cm-1 above the ground state. Within that energy range, all three fundamentals have been identified using wave function plots, as well as the first three overtones of the exchange (reaction coordinate) motion and several of its combination bands. From the tunneling splittings, the Boltzmann-averaged tunneling rates were computed. The Arrhenius plot of the total exchange rate shows a clear transition around 150 K, below which the activation energy is essentially zero and above which it is less than half of the electronic structure barrier. This indicates that exchange rates are governed by quantum tunneling throughout the relevant temperature range with the low-temperature regime dominated by a single quantum (ground) state. This work is the first-ever fully quantum dynamical study to investigate the hydrogen exchange dynamics between hydride and dihydrogen ligands coordinated to a transition-metal complex.

Predicting Stock Market Volatility Using MODWT with HyFIS and FS.HGD Models

We enhance the precision of predicting daily stock market price volatility using the maximum overlapping discrete wavelet transform (MODWT) spectral model and two learning approaches: the heuristic gradient descent (FS.HGD) and hybrid neural fuzzy inference system (HyFIS). The FS.HGD approach iteratively updates the model’s parameters based on the error function gradient, while the HyFIS approach combines the advantages of neural networks and fuzzy logic systems to create a more robust and accurate learning model. The MODWT uses five mathematical functions to form a discrete wavelet basis. The dataset used includes the daily closing prices of the Tadawul stock market from August 2011 to December 2019. Inputs were selected based on multiple regression, tolerance, and variance inflation factor tests, and the oil price (Loil) and repo rate (Repo) were identified as input variables. The output variable is represented by the logarithm of the Tadawul stock market price (LSCS). MODWT-LA8 (ARIMA(1,1,0) with drift) outperforms other WT functions on the 80% dataset, with an ME of (0.00000532), MAE of (0.003214182), and MAPE of (0.06449683). The addition of WT functions to the FS.HGD and HyFIS models increases their forecasting ability. Based on the reduced RMSE (0.048), MAE (0.038), and MAPE (0.538), the MODWT-LA8-FS.HGD outperforms traditional models in predicting the remaining 20% of datasets.

Estimation of long-term series of total nutrient loads flowing into a large perialpine lake (Lake Como, Northern Italy) from incomplete discrete data by governmental monitoring

Continuous series of nutrient loads released into a water body are essential for nutrient budgeting, water quality modelling and watershed management activities. A quintessential example of pursued data are the series of total nutrient loads flowing from multiple tributaries into a lake. However, except for extraordinary cases in which high-frequency monitoring (HFM) stations are installed for both discharge and concentrations, measured nutrient loads are available on a discrete basis. Such observations are typically obtained by governmental agencies for environmental monitoring purposes, with an at best monthly resolution, yet commonly with gaps spanning years. Usually, monitoring activities are limited to major inflows, neglecting minor ones. However, the latters can play a more relevant role in nutrient load budgets than in hydrological ones, in response to different natural features and pollution among tributary watersheds. In this work, we present a methodology we developed to estimate long-term series of total nutrient loads flowing into a water body, employing as case study Lake Como, a large deep lake in the Italian Alps with manifold monitored and unmonitored tributary watersheds. The method uses observed long-term relationships between discharges and concentrations (Q – C) and available discharge measurements and hydrological estimations to estimate continuous load series for the monitored basins. For the unmonitored watersheds, Q – C relationships are estimated from those of the monitored basins, given the observed dependence of the power-law coefficients on basin hydromorphological parameters. This equals to extending the regionalisation approach applied in hydrology for rainfall and discharges to the ecohydrological field for nutrient load estimation. The application of the method to the case study led to overall annual load estimations congruent to traditional techniques and revealed interesting Q – C watershed dynamics at interannual time scales which could not be disclosed through previous approaches. This work represents an exploratory development and application of ecohydrological regionalisation techniques, whose future development is fostered.

The Position-Momentum Commutator as a Generalized Function: Resolution of the Apparent Discrepancy Between Continuous and Discrete Bases

The Position-Momentum Commutator as a Generalized Function: Resolution of the Apparent Discrepancy Between Continuous and Discrete Bases

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

Stress-based optimization of assembly surface mechanical properties oriented to stress-uniform assembly

Stress-uniform assembly is favorable for ensuring the precision and performance stability of precision electromechanical systems. Process compensation, assembly process optimization and assembly surface topography optimization are three typical ways to regulate the assembly stress distribution. Assembly surface mechanical properties optimization is a new and promising technique for achieving uniform stress distribution in assembled parts. This study takes that direction and investigates how variation in assembly surface material stiffness (i.e., Young's modulus) give rise to the variation in von Mises stress in assembled parts. To homogenize the von Mises stress distribution in assembled parts, the stress-uniform assembly-oriented material stiffness optimization model is constructed on the basis of finite element discretization. A stress-based material stiffness optimization method, which does not require sensitivity analysis, is developed by directly linking the amount of change in material stiffness to the value of von Mises stress. The feasibility and effectiveness of the stress-based assembly surface material stiffness optimization are demonstrated by a design case containing three elastic contact bodies, where both single-load and multi-loads conditions are considered. Some new phenomena during the assembly surface material stiffness optimization are also observed and explained.

A Robust Basis for Multi‐Bit Optical Communication with Vectorial Light

AbstractIncreasing the information capacity of communication channels is a pressing need, driven by growing data demands and the consequent impending data crunch with existing modulation schemes. In this regard, mode division multiplexing (MDM), where spatial modes of light form the encoding basis, has enormous potential but is impeded by noise due to imperfect channels. Here, this challenge is overcome by breaking the existing MDM paradigm of using the modes themselves as a discrete basis and instead exploiting the polarization inhomogeneity (vectorness) of vectorial light as the information carrier. It is shown that this vectorness communication basis is completely impervious to channel noise, which is verified by near perfect data fidelity maintained in multi‐bit information transfer through atmospheric turbulence, with negligible changes on the order of 1%. This allows for demonstration of a new state‐of‐the‐art of 50 vectorial modes in a communications channel with little cross‐talk. This approach replaces conventional amplitude modulation with a novel modal alternative for potentially orders of magnitude channel information enhancement, offering a new paradigm to exploiting the spatial mode basis for optical communication.

Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise

In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height h h driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant { h > 0 } \{h>0\} (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant measure and provide evidence for the fact that the finite difference discretization touches down almost surely while our discretization stays away from the ∂ { h > 0 } \partial \{h > 0\} .

Digital Self-Interference Cancellation for Full-Duplex UAV Communication System over Time-Varying Channels

Full-duplex unmanned aerial vehicle (UAV) communication systems are characterized by mobility, so the self-interference (SI) channel characteristics change over time constantly. In full-duplex UAV communication systems, the difficulty is to eliminate SI in time-varying channels. In this paper, we propose a pilot-aid digital self-interference cancellation (SIC) method. First, the pilot is inserted into the data sequence uniformly, and the time-varying SI is modeled as a linear non-causal function. Then, the time-varying SI channel is estimated by the discrete prolate spheroidal basis expansion model (BEM). The error of block edge channel estimation is reduced by cross-block interpolation. The result of channel estimation is convolved with the transmitted data to obtain the reconstructed SI, which is subtracted from the received signal to achieve SIC. The simulation results show that the SIC performance of the proposed method outperforms the dichotomous coordinate descent recursive least square (DCD-RLS) and normalized least mean square (NLMS) algorithms. When the interference to noise ratio (INR) is 25 dB, the performance index normalized least mean square (NMSE) is reduced by 5.5 dB and 4 dB compared with DCD-RLS and NLMS algorithms, which can eliminate SI to the noise floor, and the advantage becomes more obvious as the INR increases.

Optimally Band-Limited Noise Filtering for Single-Qubit Gates

We introduce a quantum control protocol that produces smooth, experimentally implementable control sequences optimized to combat temporally correlated noise for single-qubit systems. The control ansatz is specifically chosen to be a functional expansion of discrete prolate spheroidal sequences, a discrete time basis known to be optimally concentrated in time and frequency, and quite attractive when faced with experimental control hardware constraints. We leverage the filter function formalism to transform the control problem into a filter design problem, and show that the frequency response of a quantum system can be carefully tailored to avoid the most relevant dynamical contributions of noise processes. Using gradient ascent, we obtain optimized filter functions and exploit them to elucidate important details about the relationship between filter function design, control bandwidth, and noise characteristics. In particular, we identify regimes of optimal noise suppression and, in turn, optimal control bandwidth directly proportional to the size of the frequency bands where the noise power is large. In addition to providing guiding principles for filter design, our approach enables the development of controls that simultaneously yield robust noise filtering and high-fidelity single-qubit logic operations in a wide variety of complex noise environments.

A Two-Step Discrete Cosine Basis Oriented Motion Modeling Approach for Enhanced Motion Compensation.

Video coding algorithms attempt to minimize the significant commonality that exists within a video sequence. Each new video coding standard contains tools that can perform this task more efficiently compared to its predecessors. Modern video coding systems are block-based wherein commonality modeling is carried out only from the perspective of the block that need be coded next. In this work, we argue for a commonality modeling approach that can provide a seamless blending between global and local homogeneity information in terms of motion. For this purpose, at first a prediction of the current frame, the frame that need be coded, is generated by performing a two-step discrete cosine basis oriented (DCO) motion modeling. The DCO motion model is employed rather than traditional translational or affine motion model since it has the ability to efficiently model complex motion fields by providing a smooth and sparse representation. Moreover, the proposed two-step motion modeling approach can yield better motion compensation at a reduced computational complexity since an informed guess is designed for initializing the motion search procedure. After that the current frame is partitioned into rectangular regions and the conformance of these regions to the learned motion model is investigated. Depending on the non-conformance to the estimated global motion model, an additional DCO motion model is introduced to increase the local motion homogeneity. In this way, the proposed approach generates a motion compensated prediction of the current frame through the minimization of both global and local motion commonality. Experimental results show an improved rate-distortion performance of a reference high efficiency video coding (HEVC) encoder, specifically up to around 9% savings in bit rate, that employs the DCO prediction frame as a reference frame for encoding the current frame. When compared to the more recent video coding standard, the versatile video coding (VVC) encoder, a bit rate savings of 2.37% is reported.

The Discriminative Discrete Basis Problem: Definitions, Algorithms, Benchmarking, and Application to Brain's Functional Dynamics

In this paper, considering two Boolean matrices, we tackle the problem of finding a set of Boolean basis vectors that capture the latent components most discriminative of one matrix from another. We refer to this problem as the discriminative discrete basis problem (DDBP). Depending on the symmetry/asymmetry of the problem about the two input matrices, different classes of the DDBP are defined. We present an ensemble of new algorithms, called the discriminative-associative (DASSO) algorithms, to find solutions for these classes of the DDBP. Furthermore, we design a comprehensive, simulation-based benchmarking framework for evaluating the performance of such algorithmic solutions to the DDBP over a wide range of variables, including the effects of disturbances, noise, the geometric features of the latent components within each matrix, and the relationships among these components across the contrasted matrices. As an example application, we demonstrate the utility of the DASSO algorithms in studying the dynamics of the brain function by identifying the cortical activities that discriminate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">motor</i> from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">non-motor</i> tasks using recordings obtained via electroencephalography (EEG). The results highlight the location of such discriminating activities in the cortex and the time of their occurrence, which can lead to the design of efficient and accurate brain computer interfaces (BCIs). To the best of our knowledge, this is the first time the DDBP is recognized, defined, and addressed through the development of algorithmic solutions. The DASSO algorithms can be utilized in a variety of data mining and feature selection applications, in which finding discriminating structures across two datasets is of interest.

ОПТИМІЗАЦІЙНИЙ МЕТОД КОМРЕСІЇ БАГАТОСПЕКТРАЛЬНИХ ЦИФРОВИХ ЗОБРАЖЕНЬ ПРОЕКЦІЙНОЇ ПРИРОДИ

A method for compressing digital images of a projection nature obtained in an arbitrary amount of spectral radiation ranges - a carrier of specific information or under various positional conditions of formation, optimized according to the criteria for preserving the specified levels of signal image energy and their structural similarity, is proposed. The method uses the representation of a set of such images in the form of a single multidimensional geometric object, which is described by a two-dimensional data array ordered by raster and spectral intervals and includes preprocessing by means of singular decomposition of the specified array. Actual compression is carried out by means of decomposition the constituent images, represented by maximum singular numbers, according to discrete orthonormed functional bases, zeroing out part of the decomposition coefficients and subsequent reconstruction of the digital brightness codes of images. The determination of the thresholds for zeroing the decomposition coefficients of images digital levels is formulated in the form of a two-criteria optimization problem of obtaining the given ratios of signal energies of the compressed and initial images and indices of their structural similarity. The implementation of the proposed methods includes the following steps: pairwise orthogonalization of distributions of digital brightness codes of images in different spectral channels based on singular decompositions; preservation of the constituent distributions of brightness with maximum singular numbers; compression of orthogonalized representations according to the specified criteria; reconstructing the brightness codes of compressed images of spectral channels by a functional transformation reversed by the one used at the stage of decomposition of brightness distributions of directly recorded multispectral images. Comparison of various discretized functional bases as the basis for compression showed the greatest efficiency in the specified criteria of the discrete functional basis of Haar. Key words: multispectral image, orthogonalization, signal energy, structural similarity index, orthonormed basis.

Fast multi-source nanophotonic simulations using augmented partial factorization

Numerical solutions of Maxwell’s equations are indispensable for nanophotonics and electromagnetics but are constrained when it comes to large systems, especially multi-channel ones such as disordered media, aperiodic metasurfaces and densely packed photonic circuits where the many inputs require many large-scale simulations. Conventionally, before extracting the quantities of interest, Maxwell’s equations are first solved on every element of a discretization basis set that contains much more information than is typically needed. Furthermore, such simulations are often performed one input at a time, which can be slow and repetitive. Here we propose to bypass the full-basis solutions and directly compute the quantities of interest while also eliminating the repetition over inputs. We do so by augmenting the Maxwell operator with all the input source profiles and all the output projection profiles, followed by a single partial factorization that yields the entire generalized scattering matrix via the Schur complement, with no approximation beyond discretization. This method applies to any linear partial differential equation. Benchmarks show that this approach is 1,000–30,000,000 times faster than existing methods for two-dimensional systems with about 10,000,000 variables. As examples, we demonstrate simulations of entangled photon backscattering from disorder and high-numerical-aperture metalenses that are thousands of wavelengths wide.