Diagonal Matrix Research Articles

A shrinkage based diagonal T 2 control chart for high dimensional data

Multivariate control charts are commonly used in manufacturing engineering to identify abnormal changes in the process. In a high-dimensional paradigm, where the number of variables (p) exceeds the number of Phase I subgroups (m), the sample covariance matrix is ill-conditioned and will become singular. This situation makes the classical T 2 control chart inefficient and even invalid to employ for monitoring a mean vector in statistical process control. This study proposes a new multivariate shrinkage-based diagonal T 2 control chart for high dimensional data, where p is very large compared to m with individual observation. A shrinkage estimator is used to estimate a diagonal covariance matrix to obtain an invertible, well-conditioned, and efficient estimate of the sample covariance matrix. The proposal's performance is also evaluated using the probability of detecting a shift. A simulation study reveals that the proposed control chart has an efficient sensitivity in detecting shifts for the high dimensional data. Results also show that m has a minor effect on the probability of shift detection for out-of-control data. In general, this probability improves for larger p, when shift is introduced in all p.

High-Dimensional Block Diagonal Covariance Structure Detection Using Singular Vectors

The assumption of independent subvectors arises in many aspects of multivariate analysis. In most real-world applications, however, we lack prior knowledge about the number of subvectors and the specific variables within each subvector. Yet, testing all these combinations is not feasible. For example, for a data matrix containing 15 variables, there are already 1 382 958 545 possible combinations. Given that zero correlation is a necessary condition for independence, independent subvectors exhibit a block diagonal covariance matrix. This paper focuses on the detection of such block diagonal covariance structures in high-dimensional data and therefore also identifies uncorrelated subvectors. Our approach exploits the fact that the structure of the covariance matrix is mirrored by the structure of its eigenvectors. However, the true block diagonal structure is masked by noise in the sample case. To address this problem, we propose to use sparse approximations of the sample eigenvectors to reveal the sparse structure of the population eigenvectors. Notably, the right singular vectors of a data matrix with an overall mean of zero are identical to the sample eigenvectors of its covariance matrix. Using sparse approximations of these singular vectors instead of the eigenvectors makes the estimation of the covariance matrix obsolete. We demonstrate the performance of our method through simulations and provide real data examples. Supplementary materials for this article are available online.

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Combinatorial reduction of set functions and matroid permutations through minor invertible product assignment

We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d indeterminates, whose exponents define a Zd-valued set function. By combining the Grassmann-Plücker relations for the two matrices, we derive a family of sparse polynomials, whose factorisation properties in a Laurent polynomial ring are studied and related to information-theoretic notions.Under a given genericity condition, we prove the equivalence between combinatorial reductions and determinantal expansions with invertible minor products; specifically, a deformation returns a determinantal expansion if and only if it is induced by a diagonal matrix of units in C(t) acting as a kernel in the original determinant expression. This characterisation supports the definition of a new method for checking and recovering combinatorial reductions for matroid permutations.

Nonthermal eigenstates and slow relaxation in quantum Fredkin spin chains

We study the dynamics and thermalization of the Fredkin spin chain, a system with local three-body interactions, particle conservation, and explicit kinetic constraints. We consider deformations away from its stochastic point in order to tune between regimes where kinetic energy dominates and those where potential energy does. By means of exact diagonalization, perturbation theory, and variational matrix product states, we show there is a sudden change of behavior in the dynamics that occurs, from one of fast thermalization to one of slow metastable (prethermal) dynamics near the stochastic point. This change in relaxation is connected to the emergence of additional kinetic constraints, which lead to the fragmentation of Hilbert space in the limit of a large potential energy. We also show that this change can lead to thermalization being evaded for special initial conditions because of nonthermal eigenstates (akin to quantum many-body scars). We provide clear evidence for the existence of these nonthermal states for large system sizes even when far from the large-potential-energy limit, and explain their connection to the emergent kinetic constraints. Published by the American Physical Society 2024

Fluorescence by incoherently pumped polar quantum dot placed in a low-frequency monochromatic field

Abstract We studied spectral properties of high-frequency fluorescent radiation from a two-level quantum system with broken inversion spatial symmetry, which can be implemented as a model of a one-electron two-level asymmetric polar semiconductor quantum dot whose electric dipole moment operator has permanent unequal diagonal matrix elements. The case of the excitation of this system by incoherent pumping of some sort supplemented with an additional driving monochromatic field, which frequency is much lower than the optical transition frequency of the quantum dot, was considered. An analytical expression for the fluorescence spectrum as a function of the amplitude and frequency of the monochromatic driving field, as well as of the pumping intensity, was derived. We also discussed feasible practical applications of the effects under study in the field of quantum optics and optoelectronics for the generation of high-frequency radiation, possessing stationary equidistant spectrum, and for the control of the radiative properties of such nanodevices as spasers (dipole nanolasers).

A continuous-discrete adaptive observer with a diagonal gain matrix and sigma-modification: applications in chaos synchronization and dynamical state estimation

In this paper, we propose an improved continuous-discrete observer with enhanced convergence properties and reduced sensitivity to persistent excitations. Our approach introduces several key innovations, including a customised diagonal gain matrix which improves the convergence speed, a sigma-modification technique that relaxes the persistent excitation conditions, and a weighting factor tailored to optimise the estimation of unknown parameters. Through comprehensive numerical experiments, we illustrate the effectiveness of our observer in two distinct scenarios. Firstly, we demonstrate its application in synchronising a 4-component chaotic dynamical system. Secondly, we showcase its utility in tracking a reduced 2-component chemical engineering process. Our simulation results reveal that the proposed observer outperforms existing methods by achieving quicker convergence and higher estimation accuracy. This study contributes to the advancement of adaptive observer design, offering valuable insights for its application across diverse fields.

Efficient algorithm for thermal nondestructive testing and evaluation by considering the heteroscedastic nature of noise sources in infrared thermography

Abstract Thermal Imaging is a promising Non Destructive Testing & Evaluation (NDT & E) approach to monitor the health
of composite materials. Among various post processing approaches adopted in thermal imaging for NDT & E, statistical
analysis schemes gained importance due to their reliability and data reduction capabilities. This paper provides an insight
to a factor analysis-based statistical approach to detect the hidden defects in the Glass Fiber Reinforced Polymer (GFRP)
sample. The proposed approach models the observed data covariance into combination of temporal signal covariance
and noise covariance matrices. The modeling of the diagonal covariance matrix (with different elements) is motivated by
the presence of heterogeneity in the experimental data obtained from GFRP sample. This novel method is based on the
coordinate descent technique, which estimates the covariance matrix of the noise variances iteratively by minimizing the
negative log likelihood function. The obtained results from the chosen GFRP samples compared with the widely used
statistical Principal Component Thermography (PCT) technique illustrate the improved performance in terms of defect
detection with the proposed technique.

Quantitatively Deciphering the Local Structure and Luminescence Spectroscopy of Pr3+-Doped Yttrium Lithium Fluoride.

Praseodymium (Pr3+)-doped LiYF4 nanophosphors have garnered significant interest for their potential applications in lasers, phosphors, and quantum memories. However, there remains a lack of comprehensive research on the local coordination environment and luminescence spectroscopy of Pr3+:LiYF4 nanocrystals. This study presents the first investigation of the ground-state structure of Pr3+:LiYF4 crystals by employing the crystal structure prediction method, and a [PrF8]5- ligand complex with S4 local symmetry is determined. The complete energy levels of the Pr3+ ions in LiYF4 nanocrystals are unveiled by using our newly developed well-established parametrization matrix diagonalization method. A novel set of free-ion and crystal-field parameters is derived through a good simulation with 45 experimental energy levels. Many of the emissions of Pr3+-doped LiYF4 are successfully reproduced based on Judd-Ofelt theory, and these transitions are comparable to the experimental ones. Moreover, two new prominent emission bands with their peaks at 675 and 849 nm originating from 1I6 → 3F4 and 1I6 → 1G4 transitions, respectively, are predicted by us for the first time. This study could provide a feasible method to search for practical laser transition channels of solid-state lasers based on Pr3+: LiYF4 nanophosphors.

Tuning diagonal scale matrices for HMC

Three approaches for adaptively tuning diagonal scale matrices for HMC are discussed and compared. The common practice of scaling according to estimated marginal standard deviations is taken as a benchmark. Scaling according to the mean log-target gradient (ISG), and a scaling method targeting that the frequency of when the underlying Hamiltonian dynamics crosses the respective medians should be uniform across dimensions, are taken as alternatives. Numerical studies suggest that the ISG method leads in many cases to more efficient sampling than the benchmark, in particular in cases with strong correlations or non-linear dependencies. The ISG method is also easy to implement, computationally cheap and would be relatively simple to include in automatically tuned codes as an alternative to the benchmark practice.

Spectrality and non-spectrality of a class of Moran measures

Let {Mn}n=1∞∈M2(R) be a sequence of real expanding matrices and {Dn}n=1∞ be a sequence of digit sets withDn={(00),(anbn),(cndn)}, where |andn−bncn|=ϕ(n)∈Z, and letμ{Mn},{Dn}=δM1−1D1⁎δ(M1M2)−1D2⁎δ(M1M2M3)−1D3⁎⋯ be the associated Moran measure. This paper establishes a necessary and sufficient condition for μMn,Dn to be a spectral measure when each Mn is a diagonal matrix, and characterizes the structure of the spectrum Λ. Furthermore, we demonstrate that for any sequence of matrices {Mn}n=1∞ ∈ M2(Z) with det⁡(Mn)∉3Z, the space L2(μ{Mn},{Dn}) contains at most 9 mutually orthogonal exponential functions. The precise maximum cardinality of such functions is also determined.

Distinguished varieties in a family of domains associated with spectral interpolation and operator theory

We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author and Shalit. We show that a distinguished variety $\Lambda$ in $\mathbb G_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which $\Gamma_n=\overline{\mathbb G_n}$ is a spectral set is called a $\Gamma_n$-contraction. To every $\Gamma_n$-contraction $(S_1, \dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, called the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with $\Gamma_n$ and by an application of that model we establish that the $\Gamma_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots , S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilations for a unique distinguished variety $\Lambda_{\Sigma}$ in $\mathbb G_n$, when $\Lambda_{\Sigma}$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots ,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of $P^*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$.

Observer-based decoupling control of air flow and pressure in fuel cell systems

This paper investigates the control of the air supply system for marine proton exchange membrane fuel cells. A mathematical model of the air supply system is established, and a sliding mode diagonal decoupling controller based on a state observer is proposed. The control effectiveness is validated through experiments and simulations. A decoupling control strategy for the air supply system of marine fuel cells is presented. First, the load input current of the fuel cell system is selected according to the load characteristics of the target vessel, ensuring that the load current meets the power level of the fuel cell system model. Second, the impact of the cathode pressure on the fuel cell system is analyzed. An observer-based measurement method is proposed to address the measurement issue of cathode pressure. On this basis, a decoupling control strategy combining a diagonal decoupling matrix and sliding mode variable structure control theory is proposed. Finally, the effectiveness of the decoupling controller is validated through Matlab/Simulink. The results show the proposed decoupling controller exhibits superior performance to existing controllers with environmental disturbances, continuous time-varying loads, and step loads. The maximum relative error is 2.95%, the average relative error is 0.022%, and the longest adjustment time is 0.5 s. These results indicate that the control performance of the proposed decoupling controller is superior to that of diagonal PI controllers and sliding mode controllers, thereby validating the superiority of the sliding mode decoupling controller. This article provides a useful method for the development and application of control technology for the air supply system of marine fuel cells.

Contribution analysis of the addition of laser ranging interferometry on GRACE-FO gravity field estimation under different accelerometer calibration schemes

To investigate the influence of K-Band microwave ranging (KBR) and Laser Ranging Interferometer (LRI) inter-satellite range-rates from the GRACE-FO mission on the recovery of gravity fields, this paper explores two distinct methods for accelerometer scale estimation, namely diagonal matrix parameterization and full matrix parameterization. Integrating both KBR and LRI data from GRACE-FO into gravity field recovery, we have derived six new time series of monthly gravity field solutions based on two different accelerometer calibration approaches, entitled KBRLRI_Com (incorporating both KBR and LRI data), KBR_Only (using pure KBR data), LRI_Only (solely utilizing LRI data) truncate to degree and order 60 over the period from January 2019 to June 2022. Analyses of KBRLRI_Com, KBR_Only, and LRI_Only reveal the following findings: (1) LRI demonstrates significant advantages over KBR in both diagonal and full scale matrix scenarios, across the time and frequency domains, which shows lower root mean square (RMS) values of the post-fit residuals in the time domain and substantially lower noise at higher frequencies; (2) the comparison between KBRLRI_Com, KBR_Only, and LRI_Only in terms of geoid degree error suggests that KBRLRI_Com agrees well with KBR_Only and LRI_Only at the low degrees (below degree 30), while effectively reducing the high-frequency noise, especially when the diagonal scale matrices are employed; the benefit of KBRLRI_Com solution is mainly attributed to the enhanced quantity of ranging observations. (3) by analyzing the signal and noise over the globe, ocean, Australian continent, and Amazon and Ganges river basins, we find that KBRLRI_Com is highly consistent with KBR_Only and LRI_Only in terms of spatial signals, while KBRLRI_Com exhibits less spatial noise; when employing diagonal and full matrices, the noise of gravity field solutions derived by incorporating LRI data is mitigated by 8.8 % and 3.9 % over the global ocean compared to KBR_Only, as well as 7.6 % and 3.2 % in the Australian continent; especially in the diagonal matrix case, the incorporation of LRI measurements for calculating monthly gravity fields results in a more pronounced decrease of spatial noise. (4) in the oceanic region, KBR_Only and LRI_Only have average RMS values of 11.3 cm and 11.6 cm in diagonal matrices, respectively, and 10.2 cm and 11.6 cm in the full matrix, indicating a generally comparable level of accuracy between the two models.

Simplified inverse distance weighting-immersed boundary method for simulation of fluid-structure interaction

When simulating fluid-structure interaction (FSI) problems involving moving objects, the implicit inverse distance weighting-immersed boundary method (IDW-IBM) developed by Du et al. [1] has to construct a large square correlation matrix and solve its inversion at each time step. In this work, a simplified inverse distance weighting-immersed boundary method (SIDW-IBM) is proposed to eliminate the intrinsic limitations in the original implicit IDW-IBM. Through error analysis using Taylor series expansion, a second order approximation can be derived, which allows us to approximate the large square correlation matrix into a diagonal matrix; thereby, we proposed the SIDW-IBM based on this second order approximation to explicitly evaluate the velocity corrections, where the needs to assemble the large correlation matrix and inverse it are circumvented. Owing to the fact that the inverse distance weighting interpolation removes the limitations in the Dirac delta function, the proposed SIDW-IBM has been successfully implemented on the non-uniform meshes to further improve the computational efficiency. The proposed SIDW-IBM is integrated with the reconstructed lattice Boltzmann flux solver (RLBFS) [2] to simulate some classic incompressible viscous flows, including flow past an in-line oscillating cylinder, flow past a heaving airfoil, and flow past a three-dimensional flexible plate. The good agreement between the present results and reference data demonstrates the capability and feasibility of the SIDW-IBM for simulating FSI problems with moving boundaries and large deformations.

Parallel cyclic reduction of padded bordered almost block diagonal matrices

The solution of linear systems is a crucial and indispensable technique in the field of numerical analysis. Among linear system solvers, the cyclic reduction algorithm stands out for its natural inclination to parallelization. So far, the cyclic reduction has been applied primarily to linear systems with almost block diagonal matrices. Some of its variants widen the usage to almost block diagonal matrices with a last block of rows introducing a set of non-zero elements in the first group of columns. In this work, we extend cyclic reduction to matrices with additional non-zero elements below and to the right without any limitations. These matrices, called padded bordered almost block diagonal, arise from the discretization of optimal control problems featuring arbitrary boundary conditions and free parameters. Nonetheless, they also appear in two-point boundary value problems with free parameters. The proposed algorithm is based on the LU factorizations, and it is designed to be executed in parallel on multi-thread architectures. The algorithm performance is assessed through numerical experiments with different matrix sizes and threads. The computation times and speedups obtained with the parallel implementation indicate that the suggested algorithm is a robust solution for solving padded bordered almost block diagonal linear systems. Furthermore, its structure makes it suitable for the use of different matrix factorization techniques, such as QR or SVD. This flexibility enables tailored customization of the algorithm on the basis of the specific application requirements.

On arbitrary matrix coefficient assignment for the characteristic matrix polynomial of block matrix linear control systems

For block matrix linear control systems, we study the property of arbitrary matrix coefficient assignability for the characteristic matrix polynomial. This property is a generalization of the property of eigenvalue spectrum assignability or arbitrary coefficient assignability for the characteristic polynomial from system with scalar $(s=1)$ block matrices to systems with block matrices of higher dimensions $(s>1)$. Compared to the scalar case $(s=1)$, new features appear in the block cases of higher dimensions $(s>1)$ that are absent in the scalar case. New properties of arbitrary (upper triangular, lower triangular, diagonal) matrix coefficient assignability for the characteristic matrix polynomial are introduced. In the scalar case, all the described properties are equivalent to each other, but in block matrix cases of higher dimensions this is not the case. Implications between these properties are established.

Matrix weighted $$\alpha $$-modulation spaces

AbstractGiven a matrix-weight W in the Muckenhoupt class $$\textbf{A}_p({{\mathbb {R}}}^n)$$ A p ( R n ) , $$1\le p<\infty $$ 1 ≤ p < ∞ , we introduce corresponding vector-valued continuous and discrete $$\alpha $$ α -modulation spaces $$M^{s,\alpha }_{p,q}(W)$$ M p , q s , α ( W ) and $$m^{s,\alpha }_{p,q}(W)$$ m p , q s , α ( W ) and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of Fourier multipliers on vector valued spaces is given.

Measuring, processing, and generating partially coherent light with self-configuring optics

Optical phenomena always display some degree of partial coherence between their respective degrees of freedom. Partial coherence is of particular interest in multimodal systems, where classical and quantum correlations between spatial, polarization, and spectral degrees of freedom can lead to fascinating phenomena (e.g., entanglement) and be leveraged for advanced imaging and sensing modalities (e.g., in hyperspectral, polarization, and ghost imaging). Here, we present a universal method to analyze, process, and generate spatially partially coherent light in multimode systems by using self-configuring optical networks. Our method relies on cascaded self-configuring layers whose average power outputs are sequentially optimized. Once optimized, the network separates the input light into its mutually incoherent components, which is formally equivalent to a diagonalization of the input density matrix. We illustrate our method with numerical simulations of Mach-Zehnder interferometer arrays and show how this method can be used to perform partially coherent environmental light sensing, generation of multimode partially coherent light with arbitrary coherency matrices, and unscrambling of quantum optical mixtures. We provide guidelines for the experimental realization of this method, including the influence of losses, paving the way for self-configuring photonic devices that can automatically learn optimal modal representations of partially coherent light fields.

Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024