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The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor completion. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.

Whole dose distribution results from well-conceived treatment plans including patient-specific (location, size and shape of tumor, etc.) and facility-specific (clinical policy and goal, equipment, etc.) information. To evaluate the whole dose distribution efficiently and effectively, we propose a method to apply spherical projection and real spherical harmonics (SH) expansion, thus leading to the expanded coefficients as a rank-2 tensor, SH coefficient tensor, for every patient-specific dose distribution. To verify the feature of this tensor, we introduce Isomap from the manifold learning method and multi-dimensional scaling (MDS). Subsequently, we obtained the MDS distance representing similarity, η, and the SH score, ζ, which is a Frobenius norm of the SH coefficient tensor. These were then validated in the intensity-modulated radiation therapy (IMRT) data sets of: (i) 375 mixing treated regions, (ii) 135 head and neck (HN), and (iii) 132 prostate cases, respectively. The MDS map indicated that the SH coefficient tensor enabled a quantitative feature extraction of whole dose distributions. In particular, the SH score systematically detected irregular cases as the deviation higher than +1.5 standard deviations (SD) from the average case, which matched up with clinically irregular case that required very complicated dose distributions. In summary, the proposed SH coefficient tensor is a useful representation of the whole dose distribution. The SH score from the SH coefficient tensor is a convenient and simple criterion used to characterize the entire dose distributions, which is not dependent on the data set.

We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.

Mueller polarized bi-directional scattering distribution functions (pBSDFs) are 4x4 matrix-valued functions which depend on acquisition geometry. The most popular pBSDF is a weighted sum between a Fresnel matrix and an ideal depolarizer. This work's main contribution is relating the relative weight between an ideal depolarizer and Fresnel matrix to a single depolarization parameter. Rather than a 16-dimensional matrix norm, this parameter can form a one-dimensional merit function. Then, instead of a full Mueller matrix measurement, a scheme for pBSDF fitting to only two polarimetric measurements is introduced. Depolarization can be mathematically expressed as the incoherent addition of coherent states. This work shows that, for a Mueller matrix to be in the span of a Fresnel matrix and an ideal depolarizer, the weights in the incoherent addition are triply degenerate. This triple degeneracy is observed in five different colored opaque plastics treated with nine different surface textures and measured at varying acquisition geometries and wavebands., 32 pages, 11 figures, Appendix, pre-print version

We consider the three-dimensional (3-D) seismic data as a tensor data of size ${n_1} \times {n_2} \times {n_3}$ contaminated with the white Gaussian noise. A new version of the tensor robust principal component analysis (TRPCA) is employed for denoising the 3-D seismic data. In the new TRPCA, the singular values are extracted using the optimal shrinkage method. We recover the low-rank matrices from noisy data by shrinkage of the singular values, in which the singular value thresholding in the Fourier domain is exploited to extract the low-rank component of the tensor. The algorithm is as follows. First, by assuming the incoherency conditions the whole tensor is modeled as a combination of a low-rank component and a sparse component. Second, the Fourier transform of the tensor is computed along the third dimension of the tensor, then the singular value decomposition (SVD) is computed in the Fourier domain, then the low-rank component is extracted by shrinkage of the singular values. Finally, the steps mentioned above are repeated until the Frobenius norm of the error matrix reaches the desired value. We evaluate the performance of the proposed method, which is called tensor optimal shrinkage of SVD based on the qualitative and quantitative measurements, and compare it with state-of-the-art methods such as iterative tensor singular value thresholding and 4-D block matching using different types of synthetic and real seismic data.

In this short note, we study some perturbation properties of the core inverse. We present the closed form and perturbation bounds for the core inverse under some conditions, which extend the classical result on the perturbation of the nonsingular matrix. Our expressions for the perturbation of the core inverse are simple and perturbation bounds are sharp.

Let $$ X_{1},\ldots ,X_{n} $$ be i.i.d. sample in $$ \mathbb {R}^{p} $$ with zero mean and the covariance matrix $$ \varvec{\Sigma }$$ . The problem of recovering the projector onto an eigenspace of $$ \varvec{\Sigma }$$ from these observations naturally arises in many applications. Recent technique from Koltchinskii and Lounici (Ann Stat 45(1):121–157, 2017) helps to study the asymptotic distribution of the distance in the Frobenius norm $$ \left\| \mathbf {P}_{r} - \widehat{\mathbf {P}}_{r} \right\| _{2} $$ between the true projector $$ \mathbf {P}_{r} $$ on the subspace of the rth eigenvalue and its empirical counterpart $$ \widehat{\mathbf {P}}_{r} $$ in terms of the effective rank of $$ \varvec{\Sigma }$$ . This paper offers a bootstrap procedure for building sharp confidence sets for the true projector $$ \mathbf {P}_{r} $$ from the given data. This procedure does not rely on the asymptotic distribution of $$ \left\| \mathbf {P}_{r} - \widehat{\mathbf {P}}_{r} \right\| _{2} $$ and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for Gaussian samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

We investigate first the sensitivity of the continuous Lyapunov tensor equation X×1A+X×2A+⋯+X×mA=P(miseven), where A is positively stable. On one hand, this leads to a Frobenius norm bound on the s...

We are concerned with the stability of non-autonomous linear discrete-time systems. Two necessary and sufficient conditions are derived for exponential stability of time-varying linear discrete-time systems. They show that through a nonsingular state transformation an exponentially stable system can be reduced to a simple one with the coefficient matrix whose spectral norm being less than unity. Furthermore, an estimation of solution of time-varying linear discrete-time system is provided which extends the results in the literature.

In this work, we obtain the matrix form of the conjugate direction (MCD) method for solving the coupled Sylvester matrix equations (CSMEs) $$\left\{ \begin{array}{ll} \sum _{i=1}^s A_iXB_i=C, &{} \hbox {} \\ \sum _{j=1}^t D_jXE_j=F, &{} \hbox {} \end{array}\right. $$ which are defined in the domain of real numbers. We prove that the MCD method converges to the solution of the CSMEs for any initial guess within a finite number of iterations in the absence of round-off errors. Also we show that the MCD method can find the least Frobenius norm solution of the CSMEs with special initial guess. Finally three numerical examples show that the MCD method is efficient to solve some matrix equations.

The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm ψ n ( m ) of ( 0 , 1 ) -matrices of size n × n with exactly m ones. It is shown that: (1) if n ≥ 2 and n m ≤ 2 n , then ψ n ( m ) ≤ m + 2 ( m − 1 ) , with equality if and only if m is a prime; (2) if n ≥ 4 and 2 n m ≤ 3 n , then ψ n ( m ) ≤ m + 2 2 ⌊ m / 3 ⌋ , with equality if and only if m is a prime or a double of a prime; (3) if 3 n m ≤ 4 n , then ψ n ( m ) ≤ m + 2 m − 2 , with equality if and only if there is an integer k ≥ 1 such that m = 12 k ± 2 and 4 k ± 1 , 6 k ± 1 , 12 k ± 1 are primes.

This article is devoted to the generalization of results obtained in 2002 by Jabin, Otto and Perthame. In their article they proved that planar vector fields taking value into the unit sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here, we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of any norm. Under natural assumptions on the norm, namely smoothness and a qualitative convexity property, that is to be of power type p, we prove that planar vector fields taking value into the unit sphere of such a norm and satisfying a certain kinetic equation are locally 1/(p−1)-Holder continuous. Furthermore we completely describe the behaviour of such a vector field around singular points as a vortex associated to the norm. As our kinetic equation implies for the vector field to be curl-free, this can be seen as a selection principle for curl-free vector fields valued in spheres of general norms which rules out line-like singularities.

We introduce a saddle-point dynamics, evolving isospectrally on the product of two homogeneous space of symmetric matrices, each orthogonally equivalent to a positive diagonal matrix. This dynamics is a saddle-point version of the so-called double-bracket flow and corresponds to a minimax version of the Frobenius norm minimization problem considered in (Brockett, 1989, 1991). We study the set of equilibria of this dynamics and prove its asymptotic convergence properties under suitable conditions. We also demonstrate how, under appropriate conditions, this dynamics provides a learning strategy for a two-player zero-sum game, where the players strategy set is the set of permutations.

The anisotropic norm of the linear discrete time invariant system is the measure of the system output sensitivity to a random input with the mean anisotropy bounded by some nonnegative value a. The mean anisotropy characterizes the degree of predictability of the stochastic signal. The anisotropic norm of a system is the induced norm with the extreme cases of H2 norm and H∞ norm, respectively, under a → 0 and a → ∞. The lower bound of the anisotropic norm of the linear discrete time invariant system was established in the present paper using the methods of linear algebra.

In this paper, both well-known and new properties of the spectral abscissa and the logarithmic norm are described. In addition to well-known formulas for the norm of a matrix and for its logarithmic norm in cubic, octahedral, spherical norms, various estimates for these quantities in an arbitrary Ho¨ lder norm are proved.

In this paper, we consider the perturbation analysis for the periodic generalized coupled Sylvester (PGCS) equation. The normwise backward error for this equation is first obtained. Then, we present its normwise and componentwise perturbation bounds, from which the normwise and effective condition numbers are derived. Moreover, the mixed and componentwise condition numbers for the PGCS equation are also given. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the statistical condition estimation method are applied. The obtained results are illustrated by numerical examples., 15 pages, 1 figure

Multisecant quasi-Newton methods have been shown to be particularly suited to solve nonlinear fixed-point equations that arise from partitioned multiphysics simulations where the exact Jacobian is inaccessible. In all these methods, the underdetermined multisecant equation for the approximate (inverse) Jacobian is enhanced by a norm minimization condition. The standard choice is the minimization of the Frobenius norm of the approximate inverse Jacobian. In this setting, it is well known that transient fluid-structure simulations typically require the use of secant information also from previous time steps to achieve a small enough number of iterations per implicit time step. The number of these time steps highly depends on the application, the physical parameters, the used solvers, and the mesh resolution. Using too few leads to a relatively high number of iterations, while using too many leads not only to a computational overhead but also to an increase in the number of iterations as well. Determining th...

A shrinkage curve optimization is proposed for weighted nuclear norm minimization and is adapted to image denoising. The proposed optimization method employs a penalty function utilizing the difference between a latent matrix and its observation and uses odd polynomials to shrink the singular values of the observation matrix. As a result, the coefficients of polynomial characterize the shrinkage operator fully. Furthermore, the Frobenius norm of the penalty function is converted into the corresponding spectral norm, and thus the parameter optimization problem can be easily solved by using off-and-shelf plain least-squares. In the practical application, the proposed denoising method does not work on the whole image at once, but rather a series of matrix termed Rank-Ordered Similar Matrix (ROSM). Simulation results on 256 noisy images demonstrate the effectiveness of the proposed algorithms.

In the corotational kinematics (EIRC), the movement is decomposed in deformational and rigid body components using projection operators. Large rigid body translations and rotations, and infinitesimal deformation tensors are adopted. In this way, a non-symmetric stiffness matrix is obtained for the finite element. Literature points that this matrix may be symmetrized in the usual way and the original non-symmetric matrix tends to be in a symmetric form as equilibrium is approached. This paper performed symmetrization studies using Frobenius norm and Absolute Maximum Coefficient of the anti-symmetric part of stiffness matrix in several numerical analyses with high non-linearity solved with an incremental-iterative scheme. An element independent corotational (EICR) kinematics is used to analyze non-linear spatial frames with 3D Euler–Bernoulli beam elements. The results show that this symmetrization not always occurs and depends significantly on the magnitude of displacements and finite-element mesh refinement. The authors demonstrated that, in some cases, the symmetrization would only take place with very refined meshes when the discretized structure approaches the continuum. In this way, the current literature and general mathematical proof of that symmetrization have to be restated.

This paper introduces a novel, model-based chemical exchange saturation transfer (CEST) magnetic resonance imaging (MRI), in which asymmetric spectra of interest are directly estimated from complete or incomplete measurements by incorporating subspace-based spectral signal decomposition into the measurement model of CEST MRI for a robust z-spectrum analysis. Spectral signals are decomposed into symmetric and asymmetric components. The symmetric component, which varies smoothly, is delineated by the linear superposition of a finite set of vectors in a basis trained from the simulated (Lorentzian) signal vectors augmented with data-driven signal vectors, while the asymmetric component is to be inherently lower than or equal to zero due to saturation transfer phenomena. Spectral decomposition is performed directly on the measured spectral data by solving a constrained optimization problem that employs the linearized spectral decomposition model for the symmetric component and the weighted Frobenius norm regularization for the asymmetric component while utilizing additional spatial sparsity and low-rank priors. The simulations and in vivo experiments were performed to demonstrate the feasibility of the proposed method as a reliable molecular MRI.

We derive a dynamical bound on the propagation of correlations in local random quantum circuits - lattice spin systems where piecewise quantum operations - in space and time - occur with classical probabilities. Correlations are quantified by the Frobenius norm of the commutator of two positive operators acting on space-like separated local Hilbert spaces. For times $t=O(L)$ correlations spread to distances $\mathcal{D}=t$ growing, at best, diffusively for any distance within that radius with extensively suppressed distance dependent corrections whereas for $t=o(L^2)$ all parts of the system get almost equally correlated with exponentially suppressed distance dependent corrections and approach the maximum amount of correlations that may be established asymptotically., Comment: 7 pages, 5 figures. Updated abstract

We consider a problem of determining the component-wise distance (called the radius) of a linear system of equations or inequalities to a system that is either solvable or unsolvable. We propose explicit characterization of these radii and show relations between them. Then the radii are classified in the polynomial vs. NP-hard manner. We also present a generalization to an arbitrary linear system consisting from both equations and inequalities with both free and nonnegative variables. Eventually, we extend the concept of the component-wise distance to a non-uniform one.

In this paper, the development of the conjugate direction CD method is constructed to solve the generalized nonhomogeneous Yakubovich-transpose matrix equation AXB + CXTD + EYF = R. We prove that the constructed method can obtain the least Frobenius norm solution pair X,Y of the generalized nonhomogeneous Yakubovich-transpose matrix equation for any special initial matrix pair within a finite number of iterations in the absence of round-off errors. Finally, two numerical examples show that the constructed method is more efficient than other similar iterative methods in practical computation.

Penalizing the model complexity is necessary to avoid overfittingwhen the number of data samples is low with respect to the number of model parameters. In this paper, we introduce a penalization term that places an independent prior distribution for each parameter of the multivariate von Mises distribution.We also propose a circular distance that can be used to estimate the Kullback–Leibler divergence between any two circular distributions as goodness-of-fit measure. We compare the resulting regularized von Mises models on synthetic data and real neuroanatomical data to show that the distribution fitted using the penalized estimator generally achieves better results than nonpenalized multivariate von Mises estimator.

In this work, we present a new class of diagonal quasi-Newton methods for solving large-scale unconstrained optimization problems. The methods are derived by means of variational principle under the generalized Frobenius norm. We show global convergence of our methods under the standard line search with Armijo condition. Numerical results are carried out in standard test problems and clearly indicate vast superiority over some classical conjugate gradient methods.

This paper concerns the numerical solution of three-dimensional degenerate Kawarada equations. These partial differential equations possess highly nonlinear source terms, and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes. Arbitrary fixed nonuniform spatial grids, which are not necessarily symmetric, are considered throughout this study. The numerical solution is advanced through a semi-adaptive Local One-Dimensional (LOD) integrator. The temporal adaptation is achieved via a suitable arc-length monitoring mechanism. Criteria for preserving the positivity and monotonicity are investigated and acquired. The numerical stability of the splitting method is proven in the von Neumann sense under the spectral norm. Extended stability expectations are proposed and investigated., Comment: 24 pages, 2 figures, this article is already accepted (but is being uploaded to keep all articles on arXiv)

In this paper, we prove that the mixed norm for an exponential type weighted integral of an entire function is equivalent to the mixed norm of its radial derivative and the distortion function from the weight function in the n-dimensional complex space.

A novel immersed boundary (IB) method based on an implicit direct forcing (IDF) scheme is developed for incompressible viscous flows. The key idea for the present IDF method is to use a block LU decomposition technique in momentum equations with Taylor series expansion to construct the implicit IB forcing in a recurrence form, which imposes more accurate no-slip boundary conditions on the IB surface. To accelerate the IB forcing convergence during the iterative procedure, a pre-conditioner matrix is introduced in the recurrence formulation of the IB forcing. A Jacobi-type parameter is determined in the pre-conditioner matrix by minimizing the Frobenius norm of the matrix function representing the difference between the IB forcing solution matrix and the pre-conditioner matrix. In addition, the pre-conditioning parameter is restricted due to the numerical stability in the recurrence formulation. Consequently, the present pre-conditioned IDF (PIDF) enables accurate calculation of the IB forcing within a few iterations. We perform numerical simulations of two-dimensional flows around a circular cylinder and three-dimensional flows around a sphere for low and moderate Reynolds numbers. The result shows that PIDF yields a better imposition of no-slip boundary conditions on the IB surfaces for low Reynolds number with a fairly larger time step than IB methods with different direct forcing schemes due to the implicit treatment of the diffusion term for determining the IB forcing. Finally, we demonstrate the robustness of the present PIDF scheme by numerical simulations of flow around a circular array of cylinders, flows around a falling sphere, and two sedimenting spheres in gravity.

Let B be a multiplicative perturbation of A ∈ C m × n given by B = D 1 ⁎ A D 2 , where D 1 ∈ C m × m and D 2 ∈ C n × n are both nonsingular. New norm upper bounds for ‖ B † − A † ‖ F and ‖ B † − A † ‖ 2 are derived, where A † and B † are the Moore–Penrose inverses of A and B, respectively. The main results of Meng and Zheng (2015) [10] are improved in the cases of the Frobenius norm and the spectral norm.

We pose the problem of determining the exact value of the operator norm of the Cauchy transform. A lower bound of the norm is presented.

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.

Let Ai ∈ B(H), (i = 1, 2, ..., n), and T = [ 0 ⋯ 0 A 1 ⋮ ⋰ A 2 0 0 ⋰ ⋰ ⋮ A n 0 ⋯ 0 ] $ T = \left[ {\matrix{ 0 & \cdots & 0 & {A_1 } \cr \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {A_2 } & 0 \cr 0 & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \cr {A_n } & 0 & \cdots & 0 \cr } } \right] $ . In this paper, we present some upper bounds and lower bounds for w(T). At the end of this paper we drive a new bound for the zeros of polynomials.

Two accelerated optimization algorithms are presented for computing approximate Tucker tensor decompositions, formulated using orthonormal factor matrices, by minimizing error as measured by the Frobenius norm. The first is a nonlinearly preconditioned conjugate gradient (NPCG) algorithm, wherein a nonlinear preconditioner is used to generate a direction which replaces the gradient in the nonlinear conjugate gradient iteration. The second is a nonlinear GMRES (N-GMRES) algorithm, in which a linear combination of past iterates and a tentative new iterate, generated by a nonlinear preconditioner, is minimized to produce an improved search direction. The Euclidean versions of these methods are extended to the manifold setting, where optimization on Grassmann manifolds is used to handle orthonormality constraints and to allow isolated minimizers. Several modifications are required for use on manifolds: logarithmic maps are used to determine required tangent vectors, retraction mappings are used in the line se...

This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.

The anisotropic norm of a linear discrete time invariant system is a measure of system output sensitivity to stationary Gaussian input disturbances with mean anisotropy bounded by some nonnegative parameter. The mean anisotropy characterizes the predictability (or coloredness) degree of stochastic signal. The anisotropic norm of a system is an induced norm, limiting cases of which are H 2 - and H ∞ -norms as a → 0 and a → ∞, respectively. In Vladimirov et al. (1996) a method for numerical computation of the anisotropic norm was proposed. This method involves cross-coupled Riccati and Lyapunov equations and associated special type equation. Another method for anisotropic norm computation via LMI-based approach was proposed in Tchaikovsky & Kurdyukov (2006). This paper develops a new method for getting the lower bound for anisotropic norm by linear algebra standard methods.

In a recent paper, Rostami [SIAM J. Sci. Comput, 37 (2015), pp. S447--S471] has presented an interesting algorithm for the computation of the real pseudospectral abscissa and the real stability radius (aka the distance to instability) of a square matrix $A \in \mathbb R^{n,n}$ in the spectral norm. The method is particularly well suited for large sparse matrices. The algorithm to compute the stability radius is based on the alternation of an iterative method in the rank-$2$ manifold of real matrices and a Lyapunov inverse iteration. The algorithm shows quadratic convergence in all experiments, but this peculiar property is only conjectured. In this paper, we provide a rigorous proof and propose a modification which replaces the Lyapunov inverse iteration and speeds up the method. Moreover, we interpret the algorithm to compute the real $\varepsilon$-pseudospectral abscissa as a nonstandard discretization of the ODE-based method given in Guglielmi and Lubich [SIAM J. Matrix Anal. Appl., 34 (2013), pp. 40--...

Orthonormal matrices are a class of well-conditioned matrices with the least spectral condition number. Here, at first it is shown that a recently proposed choice for parameter of the Dai–Liao nonlinear conjugate gradient method makes the search direction matrix as close as possible to an orthonormal matrix in the Frobenius norm. Then, conducting a brief singular value analysis, it is shown that another recently proposed choice for the Dai–Liao parameter improves spectral condition number of the search direction matrix. Thus, theoretical justifications of the two choices for the Dai–Liao parameter are enhanced. Finally, some comparative numerical results are reported.

This study develops numerical algorithms to compute an ellipsoidal set of input parameters, called a design space, that ensures that the system outputs lie within a set of design specifications in quality-by-design. The algorithm is based on a proposed skewed spherical structured singular value ν s , for which this study derives upper bounds and proves the (scaled) main loop theorems and small-gain theorem for the Frobenius norm. Three examples are included to illustrate applications of the numerical algorithms.

We consider the problem of recovering the superposition of $R$ distinct complex exponential functions from compressed non-uniform time-domain samples. Total Variation (TV) minimization or atomic norm minimization was proposed in the literature to recover the $R$ frequencies or the missing data. However, in order for TV minimization and atomic norm minimization to recover the missing data or the frequencies, the underlying $R$ frequencies are required to be well-separated, even when the measurements are noiseless. This paper shows that the Hankel matrix recovery approach can super-resolve the $R$ complex exponentials and their frequencies from compressed nonuniform measurements, regardless of how close their frequencies are to each other. We propose a new concept of orthonormal atomic norm minimization (OANM), and demonstrate that the success of Hankel matrix recovery in separation-free super-resolution comes from the fact that the nuclear norm of a Hankel matrix is an orthonormal atomic norm. More specifically, we show that, in traditional atomic norm minimization, the underlying parameter values must be well separated to achieve successful signal recovery, if the atoms are changing continuously with respect to the continuously-valued parameter. In contrast, for the OANM, it is possible the OANM is successful even though the original atoms can be arbitrarily close.

In this paper a homogenization procedure for theestimation of the failure surface of a quasi-periodic masonry,based on a mean stresses approach through the analysis ofthe Statistically Equivalent Periodic Unit Cell (SEPUC), isshown. The mean stresses approach consists in the identificationof critical states for the homogenized continuum bymeans of an overall failure criterion, function of the meanstress state of each constituent. These macroscopic tensorsare evaluated in the elastic field. The SEPUC definition refersto a statistical criterion applied to a population of PeriodicUnit Cells (PUCs) generated taking into account the geometricalfeatures of the quasi-periodic texture; moreover itis validated on the basis of the homogenized elastic propertiesin terms of components and Frobenius norm of theelastic matrix. By a multi-objective optimization approach, the obtained results highlight that the proposed SEPUC can be used to estimate the failure surface.

Let $ \lam$ be a simple eigenvalue of an $n$-by-$n$ matrix $A.$ Let $y$ and $ x$ be left and right eigenvectors of $A$ corresponding to $\lam,$ respectively. Then, for the spectral norm, the condition number $\cond(\lam, A) := \|x\|_2\, \|y\|_2 /{|y^*x|}$ measures the sensitivity of $\lam$ to small perturbations in $A$ and plays an important role in the accuracy assessment of computed eigenvalues. R. A. Smith [Numer. Math., 10(1967), pp.232-240] proved that $ \cond(\lam, A) = \|x\|_2\|y\|_2/{|y^*x|} = \|\adj(\lam I -A)\|_2/{|p'(\lam)|}$, where $ \adj(A)$ is the ``adjugate" of $A$ and $p'(\lam)$ is the derivative of $ p(z) :=\det(z I- A)$ at $\lam.$ We extend Smith's condition number to any matrix norm $\|\cdot\|$ and show that $$\cond(\lam, A) = \frac{\|yx^*\|_*}{|y^*x|} = \frac{\|\adj(\lam I - A)^*\|_*}{|p'(\lam)|}$$ measures the sensitivity of $\lam$ to small perturbations in $A,$ where $\norm_*$ is the dual norm of $\|\cdot\|.$ The {\sc matlab} command {\tt roots} computes roots of a polynomial $p(x)$ by computing the eigenvalues of a companion matrix $C_p$ associated with $p.$ We analyze the sensitivity of $\lam$ as a root of $p(x)$ as well as the sensitivity of $\lam$ as an eigenvalue of $C_p$ and compare their condition numbers.

We consider a linear combination of a trigonometric polynomial and its conjugate. A sharp estimate for the uniform norm of the linear combination via the uniform norm of the polynomial is obtained.

In this paper, we consider a cone problem of matrix optimization induced by spectral norm (MOSN). By Schur complement, MOSN can be reformulated as a nonlinear semidefinite programming (NLSDP) problem. Then we discuss the constraint nondegeneracy conditions and strong second-order sufficient conditions of MOSN and its SDP reformulation, and obtain that the constraint nondegeneracy condition of MOSN is not always equivalent to that of NLSDP. However, the strong second-order sufficient conditions of these two problems are equivalent without any assumption. Finally, a sufficient condition is given to ensure the nonsingularity of the Clarke’s generalized Jacobian of the KKT system for MOSN.

Given an order-d tensor A ∈ Rn×n×...×n, we present a simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements of A, keeps all sufficiently large elements of A, and retains some of the remaining elements with probabilities proportional to the square of their magnitudes. We analyze the approximation accuracy of the proposed algorithm using a powerful inequality that we derive. This inequality bounds the spectral norm of a random tensor and is of independent interest. As a result, we obtain novel bounds for the tensor sparsification problem. As an added bonus, we obtain improved bounds for the matrix (d = 2) sparsification problem.

We study tensor products of the Schatten classes S p , and their duals. It is shown that if p 1 − 1 + ⋯ + p k − 1 + q − 1 = 1 , then the space of multilinear forms B ( S p 1 , … , S p k ; C ) contains a complemented subspace isometric to S q . We construct explicit embeddings of S r n into S p n ⊗ ˆ S q n for r − 1 = p − 1 + q − 1 whose range is complemented by a “natural” norm-one projection. As a byproduct we compute the nuclear norm of some multiplication operators: if r − 1 + 1 = p − 1 + q − 1 , with p , q ≥ 1 , then, given an n-by-n matrix ϕ, the nuclear norm of the multiplication operator h ∈ S p n ↦ h ⋅ ϕ ∈ S q n is n times the norm of ϕ in S r ˆ n , where r ˆ = max ⁡ ( 1 , r ) . A number of results on general noncommutative L p spaces are also included.

Anterior astigmatic corneal surface elements may have their continuity disturbed by environmental factors. Powers detected along principal meridians define matrices whose natures distinguish the continuity of surface elements. This work aims to adapt the matrix nature of data to fit closely, uniquely and holistically with the matrix nature in a sample of data for smooth astigmatic corneas that have powers along rectangular meridians. Matrices from continuous surfaces approximate matrices with outliers from irregular surfaces explicitly. Only approximations that minimize the Frobenius norm of the residual matrices are selected. Data from smooth astigmatic corneal surfaces, in an augmented sample, may now produce more reliable analyses.

A system of linear coupled reaction-diffusion equations is considered, where each equation is a two-point boundary value problem and all equations share the same small diffusion coefficient. A finite element method using piecewise quadratic splines that are globally C1 is introduced; its novelty lies in the norm associated with the method, which is stronger than the usual energy norm and is "balanced", i.e., each term in the norm is O(1) when the norm is applied to the true solution of the system. On a standard Shishkin mesh with N subintervals, it is shown that the method is O(N?1lnN)$O(N^{-1}\ln N)$ accurate in the balanced norm. Numerical results to illustrate this result are presented.

We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) for many f. In other cases, we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.

A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo-matrix-fractional function, the nuclear norm, and multitask learning. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support functional and its subdifferential are derived. In particular, the support functional is shown to be continuously differentiable on the interior of its domain, and a formula for the derivative is given when it exists.