Search

Gradient method is a simple optimization approach using minus gradient of the objective function as a search direction. Its efficiency highly relies on the choices of the stepsize. In this paper, the convergence behavior of a class of gradient methods, where the stepsize has an important property introduced in (Dai in Optimization 52:395–415, 2003), is analyzed. Our analysis is focused on minimization on strictly convex quadratic functions. We establish the R-linear convergence and derive an estimate for the R-factor. Specifically, if the stepsize can be expressed as a collection of Rayleigh quotient of the inverse Hessian matrix, we are able to show that these methods converge R-linearly and their R-factors are bounded above by $$1-\frac{1}{\varkappa }$$ , where $$\varkappa$$ is the associated condition number. Preliminary numerical results demonstrate the tightness of our estimate of the R-factor.

The doubly stochastic functional gradient descent algorithm (DSG) that is memory friendly and computationally efficient can effectively scale up kernel methods. However, in solving the highly ill-conditioned large-scale nonlinear machine learning problem, the convergence speed of DSG is quite slow. This is because the condition number of the Hessian matrix of this problem is quite large, which will make stochastic gradient methods converge very slowly. Fortunately, gradient preconditioning is a well-established technique in optimization aiming to reduce the condition number. Therefore, we propose a preconditioned doubly stochastic functional gradient descent algorithm (P-DSG) by combining DSG with gradient preconditioning. P-DSG first uses the gradient preconditioning to adaptively scale the individual components of the estimated functional gradient obtained by DSG, and then utilizes the preconditioned functional gradient as the descent direction in each iteration. Theoretically, an appropriate preconditioner is always the inverse of the Hessian matrix at the optimum, which is not easy to get due to its high computation cost. Therefore, we first choose an empirical covariance matrix of random Fourier features to approximate the Hessian matrix, and then perform a low-rank approximation to the empirical covariance matrix. P-DSG has a fast convergence rate $\mathcal {O}(1/t)$ and produces a smaller constant factor in the boundary than that of DSG while remains $\mathcal {O}(t)$ memory friendly and $\mathcal {O}(td)$ computationally efficient. Finally, we test the performance of P-DSG on the kernel ridge regression, kernel support vector machines, and kernel logistic regression, respectively. The experimental results show that P-DSG speeds up convergence and achieves better performance.

There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ norm and/or $$\ell _2$$ -norm squared. The common consensuses are (i) $$\ell _1$$ leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) $$\ell _2$$ (norm-squared) stabilizes the computation by improving the condition number of the problem resulting in strong out-of-sample performance; and (iii) there exist efficient numerical algorithms for those regularized portfolios with closed-form solutions each step. When combined such as in the well-known elastic net regularization, theoretical bounds are difficult to derive so as to limit extreme shorting of assets. In this paper, we propose a minimum variance portfolio with the regularization of $$\ell _1$$ and $$\ell _2$$ norm combined (namely $$\ell _{1, 2}$$ -norm). The new regularization enjoys the best of the two regularizations of $$\ell _1$$ norm and $$\ell _2$$ -norm squared. In particular, we derive a theoretical bound that limits short-sells and develop a closed-form formula for the proximal term of the $$\ell _{1,2}$$ norm. A fast proximal augmented Lagrange method is applied to solve the $$\ell _{1,2}$$ -norm regularized problem. Extensive numerical experiments confirm that the new model often results in high Sharpe ratio, low turnover and small amount of short sells when compared with several existing models on six datasets.

This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian matrix of the loss function, we establish an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, we establish its KL property of exponent 1/2 on the global optimal solution set under the noisy and full sample setting, and achieve this property at its certain class of critical points under the noisy and partial sample setting. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.

The multinode Shepard operator is a linear combination of local polynomial interpolants with inverse distance weighting basis functions. This operator can be rewritten as a blend of function values with cardinal basis functions, which are a combination of the inverse distance weighting basis functions with multivariate Lagrange fundamental polynomials. The key for simply computing the latter, on a unisolvent set of points, is to use a translation of the canonical polynomial basis and the P A = L U factorization of the associated Vandermonde matrix. In this paper, we propose a method to numerically solve a Poisson equation with Dirichlet conditions through multinode Shepard interpolants by collocation. This collocation method gives rise to a collocation matrix with many zero entrances and a smaller condition number with respect to the one of the well known Kansa method. Numerical experiments show the accuracy and the performance of the proposed collocation method.

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

Since in periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, an elemental basis set that is independent of physical properties of materials may lead to significant simulation inaccuracies. To avoid such a lack of material specificity within a given basis set, we present a material-specific Gaussian basis optimization scheme for solids, which simultaneously minimizes the total energy of the system and optimizes the band energies when compared to the reference plane wave calculation while taking care of the overlap matrix condition number. To assess this basis set optimization scheme, we compare the quality of the Gaussian basis sets generated for diamond, graphite, and silicon via our method against the existing basis sets. The optimization scheme of this work has also been tested on the existing Gaussian basis sets for periodic systems such as MoS2 and NiO, yielding improved results.

This paper presents a novel method to make denim-fabric-polishing robots perform their primary task flexibly and efficiently within a limited workspace. Link lengths are optimized based on an adaptive fireworks algorithm to improve the comprehensive dexterity index. A forward kinematics analysis of the denim-fabric-polishing robot is conducted via the D–H method; the workspace is analyzed according to the needs at hand to determine the range of motion of each joint. To solve the movement condition number of the Jacobian matrix, the concept of low-condition-number probability is established, and a comprehensive dexterity indicator is constructed. The influence of the robot's size on the condition number and comprehensive dexterity index is determined. Finally, the adaptive fireworks algorithm is used to establish the objective optimization function by integrating the dexterity index and other performance indicators. The optimization results show that when the comprehensive dexterity index is taken as the optimization objective, the dexterity comprehensive index and other performance indices of the robot are the lowest; that is, the robot is more flexible. Compared with the traditional genetic algorithm and particle swarm algorithm, the adaptive fireworks algorithm proposed in this paper has better solving speed and solving precision. The optimized workspace of the robot meets the requirements of the polishing task. The design also yields a sufficiently flexible, efficient, and effective robot.

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis.

The modeling of thermodynamic equilibria leads to complex nonlinear chemical systems which are often solved with the Newton-Raphson method. But this resolution can lead to a non convergence or an excessive number of iterations due to the very ill-conditioned nature of the problem. In this work, we combine a particular formulation of the equilibrium system called the Positive Continuous Fraction method with two iterative methods, Anderson Acceleration method and Vector extrapolation methods (namely the reduced rank extrapolation and the minimal polynomial extrapolation). The main advantage of this approach is to avoid forming the Jacobian matrix. In addition, a strategy is used to improve the robustness of the Anderson acceleration method which consists in reducing the condition number of matrix of the least squares problem in the implementation of the Anderson acceleration so that the numerical stability can be guaranteed. We compare our numerical results with those obtained with the Newton-Raphson method on the Acid Gallic test and the 1D MoMas benchmark test case and we show the high efficiency of our approach.

Author(s): Chen, Yongtai; Tang, William C; Chu, Jinkui; Zhang, Ran; Li, Song | Abstract: An accurate sky polarization field map is a prerequisite for polarization navigation applications. In this article, a detector for sky full-polarization imaging detection is described, the major error-influencing factors (MEIFS ) are obtained, and the error propagation is modeled and analyzed. We reveal the relationship between the error of the inversed Stokes vector and the condition number of the detector matrix, which shows that the error of the inversed Stokes vector is affected by the Stokes vector of the incident light itself and the MEIFS together, with the MEIFS playing a decisive role. With the MEIFS optimized, the impact of detector error on the inversed Stokes vector is attenuated. A control equation for system calibration is also deduced which can establish the connection between the detector matrix design and calibration process. The work in this article provides a reference for optimization and calibration of sky full-polarization imaging detectors.

In this paper, we propose a robust framework based on the scaled boundary finite element method to model implicit interfaces in two-dimensional differential equations in nonhomegeneous media. The salient features of the proposed work are: (a) interfaces can be implicitly defined and need not conform to the background mesh; (b) Dirichlet boundary conditions can be imposed directly along the interface; (c) does not require special numerical integration technique to compute the bilinear and the linear forms and (d) can work with an efficient local mesh refinement using hierarchical background meshes. Numerical examples involving straight interface, circular interface and moving interface problems are solved to validate the proposed technique. Further, the presented technique is compared with conforming finite element method in terms of accuracy and convergence. From the numerical studies, it is seen that the proposed framework yields solutions whose error is O ( h 2 ) in L 2 norm and O ( h ) in the H 1 semi-norm. Further the condition number increases with the mesh size similar to the FEM.

A novel hybrid perfusion manipulator (HPM) with five degrees of freedom (DOFs) is introduced by combining the 5PUS-PRPU (P, R, U, and S represent prismatic, revolute, universal, and spherical joint, respectively) parallel mechanism with the 5PRR reconfigurable base to enhance the perfusion efficiency of the large-scale spherical honeycomb thermal protection layer. This study mainly presents the dimensional synthesis of the proposed HPM. First, the inverse kinematics, including the analytic expression of the rotation angles of the U joint in the PUS limb, is obtained, and mobility analysis is conducted based on screw theory. The Jacobian matrix of 5PUS-PRPU is also determined with screw theory and used for the establishment of the objective function. Second, a global and comprehensive objective function (GCOF) is proposed to represent the Jacobian matrix’s condition number. With the genetic algorithm, dimensional synthesis is conducted by minimizing GCOF subject to the given variable constraints. The values of the designed variables corresponding to different configurations of the reconfigurable base are then obtained. Lastly, the optimal structure parameters of the proposed 5-DOF HPM are determined. Results show that the HPM with the optimized parameters has an enlarged orientation workspace, and the maximum angle of the reconfigurable base is decreased, which is conducive to improving the overall stiffness of HPM.

Conventional wisdom dictates that learning rate should be in the stable regime so that gradient-based algorithms don't blow up. This letter introduces a simple scenario where an unstably large learning rate scheme leads to a super fast convergence, with the convergence rate depending only logarithmically on the condition number of the problem. Our scheme uses a Cyclical Learning Rate (CLR) where we periodically take one large unstable step and several small stable steps to compensate for the instability. These findings also help explain the empirical observations of [Smith and Topin, 2019] where they show that CLR with a large maximum learning rate can dramatically accelerate learning and lead to so-called "super-convergence". We prove that our scheme excels in the problems where Hessian exhibits a bimodal spectrum and the eigenvalues can be grouped into two clusters (small and large). The unstably large step is the key to enabling fast convergence over the small eigen-spectrum.

In this paper, the upper bounds for two kinds of normiwse condition numbers are derived for $ \min\limits_{x}\|(A\otimes B)x-b\|_2 $ when the coefficient matrix is of rank deficient. In addition, the upper bounds on the mixed and componentwise condition numbers are also given. Numerical experiments are given to confirm our results.

One of the main advantages of second-order methods in a centralized setting is that they are insensitive to the condition number of the objective function's Hessian. For applications such as regression analysis, this means that less pre-processing of the data is required for the algorithm to work well, as the ill-conditioning caused by highly correlated variables will not be as problematic. Similar condition number independence has not yet been established for distributed methods. In this paper, we analyze the performance of a simple distributed second-order algorithm on quadratic problems and show that its convergence depends only logarithmically on the condition number. Our empirical results indicate that the use of second-order information can yield large efficiency improvements over first-order methods, both in terms of iterations and communications, when the condition number is of the same order of magnitude as the problem dimension.

We study the Low Rank Phase Retrieval (LRPR) problem defined as follows: recover an $n \times q$ matrix $X^*$ of rank $r$ from a different and independent set of $m$ phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover $X^*$ from $y_k := |A_k{}' x^*_k|, k=1,2,\dots, q$ when the measurement matrices $A_k$ are mutually independent. Here $y_k$ is an $m$ length vector, $A_k$ is an $n \times m$ matrix, and $'$ denotes matrix transpose. The question is when can we solve LRPR with $m \ll n$? A reliable solution can enable fast and low-cost phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens. In this work, we develop the first provably correct approach for solving this LRPR problem. Our proposed algorithm, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), is an AltMin based solution and hence is also provably fast (converges geometrically). Our guarantee shows that AltMinLowRaP solves LRPR to $\epsilon$ accuracy, with high probability, as long as $m q \ge C n r^4 \log(1/\epsilon)$, the matrices $A_k$ contain i.i.d. standard Gaussian entries, and the right singular vectors of $X^*$ satisfy the incoherence assumption from matrix completion literature. Here $C$ is a numerical constant that only depends on the condition number of $X^*$ and on its incoherence parameter. Its time complexity is only $ C mq nr \log^2(1/\epsilon)$. Since even the linear (with phase) version of the above problem is not fully solved, the above result is also the first complete solution and guarantee for the linear case. Finally, we also develop a simple extension of our results for the dynamic LRPR setting., Comment: A short version of this work is in ICML 2019, this longer version is published in IEEE Trans. Info. Th on March 2020. Fixing minor but important errors in Lemmas 3.10, 3.11, 3.12 statements and in proof of the Term1 bound. No change to Theorem statement

In this paper, we first consider the time-harmonic Maxwell equations with Dirichlet boundary conditions in three-dimensional anisotropic media, where the coefficients of the equations are general symmetric positive definite matrices. By using scaling transformations and coordinate transformations, we build the desired stability estimates between the original electric field and the transformed nonphysical field on the condition number of the anisotropic coefficient matrix. More importantly, we prove that the resulting approximate solutions generated by plane wave least squares (PWLS) methods have the nearly optimal L2 error estimates with respect to the condition number of the coefficient matrix. Finally, numerical results verify the validity of the theoretical results, and the comparisons between the proposed PWLS method and the existing PWDG method are also provided.

espanolEn este trabajo se presenta un manipulador paralelo reconfigurable asimetrico. Al modificar los angulos de cada una de las cadenas cinematicas pueden obtenerse configuraciones asimetricas. Gracias a la reconfiguracion propuesta pueden generarse mejoras significativas del desempeno del manipulador, con respecto a un robot paralelo tipo Delta. El calculo del mejor numero de condicion es obtenido, los resultados muestran que usando la redundancia todos los indices de desempeno que dependen de la matriz Jacobiana pueden ser mejorados tambien. EnglishIn this paper an asymmetric reconfigurable parallel manipulator is presented. Asymmetric configurations are obtained by modifying the angle between each of the kinematic chains. Thanks to the reconfiguration proposal significant improvement of the manipulator performance can be obtained with respect to a Delta-type parallel robot. The computation of the best condition number is obtained, the results show that by using the redundancy, all the performance indices that depend on the Jacobian matrix can be improved as well.

To guarantee heredity of positive definiteness under the popular Wolfe line search conditions, a modification is made on the symmetric rank–one updating formula, a simple quasi–Newton approximation for (inverse) Hessian of the objective function of an unconstrained optimization problem. Then, the scaling approach is employed on a memoryless version of the proposed formula, leading to an iterative method which is appropriate for solving large–scale problems. Based on an eigenvalue analysis, it is shown that the self–scaling parameter proposed by Oren and Spedicato is an optimal parameter for the proposed updating formula in the sense of minimizing the condition number. Also, a sufficient descent property is established for the method, together with a global convergence analysis for uniformly convex objective functions. Numerical experiments demonstrate computational efficiency of the proposed method with the self–scaling parameter proposed by Oren and Spedicato.

In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form $\sum_{t=1}^{p}A_{t}XB_{t}=C$ ∑ t = 1 p A t X B t = C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

In this paper, the fourth-order tensor Riccati equation with the Einstein product is investigated. Using the Einstein product, we introduce the M -tensor, the spectrum and the block tensor multipli...

The governing equation of the stochastic Galerkin method can be formulated as a generalized Sylvester equation. Therefore, developing solvers for it is attracting a lot of attention from the uncertainty quantification community. In this regard Krylov subspace based iterative solvers, that are used for standard linear systems are being used for the generalized Sylvester equations as well. This is achieved by converting the generalized Sylvester equation to a standard linear system using the Kronecker product. Accordingly the residual norm is used as a stopping criterion for the iterations, and the condition number of linear systems is used for the generalized Sylvester equations as well. For a linear system a small residual norm implies a small backward error, and hence using residual norm as a stopping criterion is justified. In this work we prove that this need not be the case for the generalized Sylvester equation. We introduce two definitions for the backward error, and then derive an upper bound on each of them. We also verify the predictions of the analysis using numerical experiments. For the special case of the stochastic Galerkin method we show that the upper bound on the backward error can be computed with minimal computational overhead, and hence it can be used as a stopping criterion in the iterative solvers. For the matrices stemming from the stochastic Galerkin method we numerically demonstrate that the actual backward error can be up to 3 orders of magnitude higher than the relative residual. Finally by taking into account the structure of the equation we derive an expression for the condition number, and discuss an algorithm for its computation in the special case of the stochastic Galerkin method.

This study proposes a random effects model based on inverse Gaussian process, where the mixture normal distribution is used to account for both unit-specific and subpopulation-specific heterogeneities. The proposed model can capture heterogeneities due to subpopulations in the same population or the units from different batches. A new Expectation-Maximization (EM) algorithm is developed for point estimation and the bias-corrected bootstrap is used for interval estimation. We show that the EM algorithm updates the parameters based on the gradient of the loglikelihood function via a projection matrix. In addition, the convergence rate depends on the condition number that can be obtained by the projection matrix and the Hessian matrix of the loglikelihood function. A simulation study is conducted to assess the proposed model and the inference methods, and two real degradation datasets are analyzed for illustration.

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$. From a lower bound perspective, relatively little is known. Ailon shows in 2013 an $\Omega(n\log n)$ bound for computing the normalized Fourier Transform assuming only unitary operations on two coordinates are allowed at each step, and no extra memory is allowed. In 2014, Ailon then improved the result to show that, in a $\kappa$-well conditioned computation, Fourier computation can be sped up by no more than $O(\kappa)$. The main conjecture is that Ailon's result can be exponentially improved, in the sense that $\kappa$-well condition cannot admit $\omega(\log \kappa)$ speedup. The main result here is that `algebraic' $\kappa$-well condition admits no more than $O(\sqrt \kappa)$ speedup. The definition of algebraic condition number is obtained by formally viewing multiplication by constants, as performed by the algorithm, as multiplication by indeterminates, giving rise to computation over polynomials. The algebraic condition number is related to the degree of these polynomials. Using the maximum modulus theorem from complex analysis, we show that algebraic condition number upper bounds standard condition number, and equals it in certain cases. Algebraic condition number is an interesting measure of numerical computation stability in its own right. Moreover, we believe that the approach of algebraic condition number has a good chance of establishing an algebraic version of the main conjecture., Comment: arXiv admin note: text overlap with arXiv:1404.1741

In the present paper we propose a new numerical methodology for the solution of 2D non-isothermal incompressible flows for natural and mixed convection in irregular geometries. The governing equations are the Incompressible Navier-Stokes Equations and the Energy Conservation Equation. Fluid velocity and temperature are coupled in the buoyancy term of the momentum equations according to the Oberbeck–Boussinesq approximation. The governing equations are discretized over unstructured triangular meshes satisfying the Delaunay property. Thanks to the Oberbeck–Boussinesq hypothesis, the flow and energy problems are solved in an uncoupled way, and two fractional time step procedures are sequentially applied to solve each problem. The prediction steps of both procedures are solved applying a Marching in Space and Time (MAST) numerical Eulerian scheme, which explicitly handles the non-linear terms in the momentum equations of the fluid problem, and allows numerical stability for Courant numbers greater than one. An analytical solution is applied for the prediction thermal problem. The correction steps of the two fractional time step procedures involve the solution of large linear systems, whose matrices are sparse and symmetric and have the “M” property if the mesh satisfies the Delaunay condition. This allows a well performing condition number. The matrix coefficients are constant in time, so that they are calculated and factorized only once, before the simulations loop starts, saving a lot of computational time. We present four numerical applications. For the first test, an analytical solution is available, and this makes it possible to analyze the spatial convergence order of the numerical solver. In the other applications, we investigate the capability of the proposed algorithm to handle irregular geometries, and we compare the computed results with experimental data and the outputs of literature models. A study of the required computational costs is also presented.

We obtain several quantitative bounds on the mixing properties of an “ideal” Hamiltonian Monte Carlo (HMC) Markov chain for a strongly log-concave target distribution π on Rd. Our main result says that the HMC Markov chain generates a sample with Wasserstein error ϵ in roughly O(κ2log(1ϵ)) steps, where the condition number κ=M2 m2 is the ratio of the maximum M2 and minimum m2 eigenvalues of the Hessian of −log(π). In particular, this mixing bound does not depend explicitly on the dimension d. These results significantly extend and improve previous quantitative bounds on the mixing of ideal HMC, and can be used to analyze more realistic HMC algorithms. The main ingredient of our argument is a proof that initially “parallel” Hamiltonian trajectories contract over much longer steps than would be predicted by previous heuristics based on the Jacobi manifold.

Robotic manipulators inevitably encounter singular configurations in the process of movement, which seriously affects their performance. Therefore, the identification of singular configurations is extremely important. However, serial manipulators that do not meet the Pieper criterion cannot obtain singular configurations through analytical methods. A joint angle parameterization method, used to obtain singular configurations, is here creatively proposed. First, an analytical method based on the Jacobian determinant and the proposed method were utilized to obtain their respective singular configurations of the Stanford manipulator. The singular configurations obtained through the two methods were consistent, which suggests that the proposed method can obtain singular configurations correctly. Then, the proposed method was applied to a seven-degree-of-freedom (7-DOF) serial manipulator and a planar 5R parallel manipulator. Finally, the correctness of the singular configurations of the 7-DOF serial manipulator was verified through the shape of the end-effector velocity ellipsoid, the value of the determinant, the value of the condition number, and the value of the manipulability measure. The correctness of singular configurations of the planar 5R parallel manipulator was verified through the value of the determinant, the value of the condition number, and the value of the manipulability measure.

Gust and turbulence events are of primary importance for the analysis of flight incidents, for the design of gust load alleviation systems and for the calculation of loads in the airframe. Gust and turbulence events cannot be measured directly but they can be obtained through direct or optimisation-based methods. In the direct method the discretisation of the Fredholm Integral equation is associated with an ill conditioned matrix. In this work the effects of regularisation methods including Tikhonov regularisation, Truncated Single Value Decomposition (TSVD), Damped Single Value Decomposition (DSVD) and a recently proposed method using cubic B-spline functions are evaluated for aeroelastic gust identification using in flight measured data. The gust identification methods are tested in the detailed aeroelastic model of FFAST and an equivalent low-fidelity aeroelastic model developed by the authors. In addition, the accuracy required in the model for a reliable identification is discussed. Finally, the identification method based on B-spline functions is tested by simultaneously using both low-fidelity and FFAST aeroelastic models so that the response from the FFAST model is used as measurement data and the equivalent low-fidelity model is used in the identification process.

Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centers use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data assimilation problem. In this article we examine the use of correlated OEC matrices in the preconditioned data assimilation problem for the first time. We consider the case where there are more state variables than observations, which is typical for applications with sparse measurements, for example, NWP and remote sensing. We find that similarly to the unpreconditioned problem, the minimum eigenvalue of the OEC matrix appears in new bounds on the condition number of the Hessian of the preconditioned objective function. Numerical experiments reveal that the condition number of the Hessian is minimized when the background and observation lengthscales are equal. This contrasts with the unpreconditioned case, where decreasing the observation error lengthscale always improves conditioning. Conjugate gradient experiments show that in this framework the condition number of the Hessian is a good proxy for convergence. Eigenvalue clustering explains cases where convergence is faster than expected.

Local Fourier analysis is a commonly used tool for the analysis of multigrid and other multilevel algorithms, providing both insight into observed convergence rates and predictive analysis of the performance of many algorithms. In this paper, for the first time, we adapt local Fourier analysis to examine variants of two- and three-level balancing domain decomposition by constraints (BDDC) algorithms, to better understand the eigenvalue distributions and condition number bounds on these preconditioned operators. This adaptation is based on a modified choice of basis for the space of Fourier harmonics that greatly simplifies the application of local Fourier analysis in this setting. The local Fourier analysis is validated by considering the two dimensional Laplacian and predicting the condition numbers of the preconditioned operators with different sizes of subdomains. Several variants are analyzed, showing the two- and three-level performance of the "lumped" variant can be greatly improved when used in multiplicative combination with a weighted diagonal scaling preconditioner, with weight optimized through the use of LFA., 25 pages, 6 figures

Let X = (Xt)t≥0 be a stochastic process issued from $x \in \mathbb{R}$ that admits a marginal stationary measure v, i.e. vPtf = vf for all t ≥ 0, where $\textbf{P}_t\,f(x)= \mathbb{E}_x[f(\textbf{X}_t)]$ . In this paper, we introduce the (resp. biorthogonal) spectral projections correlation functions which are expressed in terms of projections.” Also, update first published online date, if available. into the eigenspaces of Pt (resp. and of its adjoint in the weighted Hilbert space L2 (v)). We obtain closed-form expressions involving eigenvalues, the condition number and/or the angle between the projections in the following different situations: when X = X with X = (Xt)t ≥ 0 being a Markov process, X is the subordination of X in the sense of Bochner, and X is a non-Markovian process which is obtained by time-changing X with an inverse of a subordinator. It turns out that these spectral projections correlation functions have different expressions with respect to these classes of processes which enables to identify substantial and deep properties about their dynamics. This interesting fact can be used to design original statistical tests to make inferences, for example, about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-self-adjoint) and short-to-long-range dependence. To reveal the usefulness of our results, we apply them to a class of non-self-adjoint Markov semigroups studied in Patie and Savov (to appear, Mem. Amer. Math. Soc., 179p), and then time-change by subordinators and their inverses.

This article investigates the influence of low damping ratios on the performance of the multi-exciter stationary non-Gaussian random vibration control system. The basic theory of the multi-exciter stationary non-Gaussian random vibration method is reviewed first, and then the influences of low damping ratios on multi-output spectra and kurtoses are analyzed. The low damping ratios cause an ill-conditioned problem which will make the drive spectral matrix solution inaccurate; thus, some spectral lines located at resonance peaks in the response spectra cannot be modified within the preset tolerances by the control algorithms. The regularization method is used to alleviate the calculation error. The output kurtoses are dependent not only on the characteristics of the system but also on the input signals. It is found that the kurtosis control will be intractable if the damping ratios are very low. A two-input two-output cantilever beam simulation example is described to illustrate the analysis results.

In this article, the proper orthogonal decomposition (POD) is applied for parametric analysis in the gauged potential formulation of the Darwin model, considering both capacitive and inductive effects. Due to the large contrast in material parameters, the resulting system matrix is ill-conditioned. In addition, the condition number of the corresponding snapshot complex matrices is very huge. To improve the stability of the POD method, a structured-preserving strategy is considered and implemented for different unknown potentials, namely, the magnetic vector potential (MVP) $\mathbf {A}$ , the electric scalar potential (ESP) $\varphi $ , and the Lagrange multiplier $p$ . In addition, a greedy algorithm is proposed to select the snapshots adaptively. Two numerical examples, including a parallel plate capacitor and a modified $RLC$ device structure, are provided to illustrate the capability of the proposed POD in model-order reduction in the frequency-domain solvers.

The outcome of Jacobian singular values regularization was studied for supervised learning problems. In supervised learning settings for linear and nonlinear networks, Jacobian regularization allows for faster learning. It also was shown that Jacobian conditioning regularization can help to avoid the “mode-collapse” problem in Generative Adversarial Networks. In this paper, we try to answer the following question: Can information about policy network Jacobian conditioning help to shape a more stable and general policy of reinforcement learning agents? To answer this question, we conduct a study of Jacobian conditioning behavior during policy optimization. We analyze the behavior of the agent conditioning on different policies under the different sets of hyperparameters and study a correspondence between the conditioning and the ratio of achieved rewards. Based on these observations, we propose a conditioning regularization technique. We apply it to Trust Region Policy Optimization and Proximal Policy Optimization (PPO) algorithms and compare their performance on 8 continuous control tasks. Models with the proposed regularization outperformed other models on most of the tasks. Also, we showed that the regularization improves the agent's generalization by comparing the PPO performance on CoinRun environments. Also, we propose an algorithm that uses the condition number of the agent to form a robust policy, which we call Jacobian Policy Optimization (JPO). It directly estimates the condition number of an agent's Jacobian and changes the policy trend. We compare it with PPO on several continuous control tasks in PyBullet environments and the proposed algorithm provides a more stable and efficient reward growth on a range of agents.

In this paper, an extensive investigation is done on Stable Generalized Finite Element Method (SGFEM) performance through the analysis of 2-D fracture mechanics problems. Condition number, Stress Intensity Factors (SIFs), global and local measures of the energy norm are used to study SGFEM conditioning and accuracy. Computational time is also briefly discussed. The method is compared with the standard Generalized/eXtended Finite Element Method (G/XFEM). Numerical experiments corroborate and complement the knowledge available in the literature so far, and demonstrate SGFEM accuracy in 2-D cracked problems. Modified Heaviside Functions, combined with other enrichment functions, are also studied in the simulations. A simple and yet generic implementation for SGFEM is described, under the Object Oriented strategy, in a open source software. The implementation can be used in 2-D and 3-D problems, and it allows to generalize the implementation of any type of enrichment function under the SGFEM approach.

Coded illumination can enable quantitative phase microscopy of transparent samples with minimal hardware requirements. Intensity images are captured with different source patterns, then a nonlinear phase retrieval optimization reconstructs the image. The nonlinear nature of the processing makes optimizing the illumination pattern designs complicated. The traditional techniques for the experimental design (e.g., condition number optimization, and spectral analysis) consider only linear measurement formation models and linear reconstructions. Deep neural networks (DNNs) can efficiently represent the nonlinear process and can be optimized over via training in an end-to-end framework. However, DNNs typically require a large amount of training examples and parameters to properly learn the phase retrieval process, without making use of the known physical models. In this paper, we aim to use both our knowledge of the physics and the power of machine learning together. We propose a new data-driven approach for optimizing coded-illumination patterns for an LED array microscope for a given phase reconstruction algorithm. Our method incorporates both the physics of the measurement scheme and the nonlinearity of the reconstruction algorithm into the design problem. This enables efficient parameterization, which allows us to use only a small number of training examples to learn designs that generalize well in the experimental setting without retraining. We show experimental results for both a well-characterized phase target and mouse fibroblast cells, using coded-illumination patterns optimized for a sparsity-based phase reconstruction algorithm. Our learned design results using two measurements demonstrate similar accuracy to Fourier ptychography with 69 measurements.

A three-dof 2 P UR-2R P U redundantly-actuated parallel-kinematics machine, designed for the machining of complex curved surfaces that require high-speed and high-precision, is the object of study in this paper. The lower-mobility PKM, consisting of two pairs of symmetric, limited-dof limbs, has the advantages of high stiffness, simple kinematic chain, and reduced singularities. The mobility of the robot is investigated via Lie-groups, instead of the well-known Chebyshev–Grubler–Kutzbach formulas, which are not applicable to our case. Then, the inverse-displacement, direct-displacement and corresponding velocity relations are analyzed in detail. Next, by investigating the rank-deficiency of the corresponding Jacobian, three types of singularities, those associated with direct-kinematics, inverse-kinematics and combinations thereof, are analyzed in depth, while constraint singularities are investigated by resorting to constraint wrenches. Moreover, the workspace of both the reference point P and the tool head, when a tool is added to the moving platform, are derived. It is noteworthy that the local and global dexterity indices are evaluated by resorting to the characteristic length to homogenize the dimensionally inhomogeneous Jacobian matrix at hand, then the condition number is minimized over the independent posture parameters and the characteristic length via the first-order normality conditions.

In this paper we obtain approximated numerical solutions for the 2D Helmholtz equation using a radial basis function-generated finite difference scheme (RBF-FD), where weights are calculated by applying an oscillatory radial basis function given in terms of Bessel functions of the first kind. The problem of obtaining weights by local interpolation is ill-conditioned; we overcome this difficulty by means of regularization of the interpolation matrix by perturbing its diagonal. The condition number of this perturbed matrix is controlled according to a prescribed value of a regularization parameter. Different numerical tests are performed in order to study convergence and algorithmic complexity. As a result, we verify that dispersion and pollution effects are mitigated., 17 pages, 28 figures

The boundary knot method (BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number (ECN), which depends on the right-hand side vector of a linear system, is considered as an alternative criterion to the traditional condition number. In this paper, the effective condition number is used to help determine the position and distribution of the collocation points as well as the quasi-optimal collocation point numbers. During the solution process, we propose an NMN-search algorithm. Numerical examples show that the ECN is reliable to measure the feasibility of the BKM.

Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. We establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number of the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.

This note surveys recent strategies to estimate the condition number $$CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $$ of complex $$n\times n$$ matrices T with given spectrum. More precisely, we present a proof of the fact that if T acts on the Hilbert space $$\mathbb {C}^{n}$$ , then the supremum of CN(T) over all contractions T with smallest eigenvalues of modulus $$r>0$$ , is equal to $$1/r^{n}$$ , and is achieved by an analytic Toeplitz matrix. The same question is treated for n-dimensional Banach spaces. These strategies provide with explicit and constructive solutions to the so-called Halmos and Schaffer’s problems, and are also shown to be effective in a closely related situation, namely considering Kreiss matrices instead of contractions.

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.

Optimization is an important issue in machine learning because many machine learning models are reformulated as optimization problems. Different kinds of machine learning algorithms mainly focus on minimizing their empirical loss like deep learning, logistic regression, and support vector machine. Because data is explosively growing, it is challenging to deal with a large-scale optimization problem. Recently, stochastic second-order methods have emerged to attract much attention due to their efficiency in each iteration. These methods show good performance on training machine learning algorithms like logistic regression and support vector machine. However, the computational complexity of existing stochastic second-order methods heavily depends on the condition number of the Hessian. In this paper, we propose a new Newton-like method called Preconditioned Newton Conjugate Gradient with Sketched Hessian (PNCG). The runtime complexity of PNCG is at most logarithmic in the condition number of the Hessian. PNCG exhibits advantages over existing subsampled Newton methods especially when the Hessian matrix in question is ill-conditioned. We also show that our method has good performance on training machine learning algorithm empirically. The results show consistent improvements in computational efficiency.

To ensure the compatibility and interoperability with modernized GPS, Galileo satellites are capable of broadcasting navigation signals on carrier phase frequencies that overlap with GPS, i.e., GPS/Galileo L1-E1/L5-E5a. Moreover, the GPS/Galileo L2-E5b signals have different frequencies with wavelength differences smaller than 4.2 mm. Such overlapping and narrowly spaced signals between GPS and Galileo bring the opportunity to use the tightly combined double-differenced (DD) model for precise real-time kinematic (RTK) positioning, resulting in improved performance of ambiguity resolution and positioning with respect to the classical standard or loosely combined DD model. In this paper, we focus on the model and performance assessment of tightly combined GPS/Galileo L1-E1/L2-E5b/L5-E5a RTK for short and long baselines. We first investigate the tightly combined GPS/Galileo DD observational model for both short and long baselines with simultaneously considering the GPS/Galileo overlapping and non-overlapping frequencies. Particularly, we introduce a reparameterization approach to solve the rank deficiency that caused by the correlation between the DISB parameters and the DD ionospheric parameters for both overlapping and non-overlapping frequencies. Then we present performance assessment for the tightly combined GPS/Galileo RTK model with real-time estimation of the differential inter-system bias (DISB) parameters for short and long baselines in terms of ratio value, ambiguity dilution of precision (ADOP), ambiguity conditional number, decorrelation number, search count, empirical success rate, time-to-first-fix (TTFF), and positioning accuracy. Results from both static and kinematic experiments demonstrated that compared to the loosely combined model, the tightly combined model can deliver improved performance of ambiguity resolution and precise positioning with different satellite visibility. For the car-driven short baseline experiment with 10° elevation cut-off angle, the tightly combined model can not only significantly increase the ratio value by approximately 27.5% (from 16.0 to 20.4), but also reduce the ambiguity ADOP, the conditional number, and the search count in LAMBDA by approximately 22.2% (from 0.027 to 0.021 cycles), 14.9% (from 199.2 to 169.6), and 25.4% (from 150.1 to 112.0), respectively. Comparable decorrelation number, empirical success rate, and positioning accuracy are also obtained. For the car-driven long baseline experiment, it is also observed that the ambiguity resolution performance in terms of the ratio value, the decorrelation number, the condition number, and the search count are significantly improved by approximately 18.5% (from 2.7 to 3.2), 22.0% (from 0.186 to 0.227), 55.9% (from 937.6 to 413.7), and 10.3% (from 43.8 to 39.3), respectively. Moreover, comparable ADOP, empirical success rate, and positioning accuracy are obtained as well. Additionally, the TTFF can be reduced (from 54.1 to 51.8 epochs with 10° elevation cut-off angle) as well from the results of static experiments.

Purpose The purpose of this paper is to discuss about evaluating the integrals involving B-spline wavelet on the interval (BSWI), in wavelet finite element formulations, using Gauss Quadrature. Design/methodology/approach In the proposed scheme, background cells are placed over each BSWI element and Gauss quadrature rule is defined for each of these cells. The nodal discretization used for BSWI WFEM element is independent to the selection of number of background cells used for the integration process. During the analysis, background cells of various lengths are used for evaluating the integrals for various combination of order and resolution of BSWI scaling functions. Numerical examples based on one-dimensional (1D) and two-dimensional (2D) plane elasto-statics are solved. Problems on beams based on Euler Bernoulli and Timoshenko beam theory under different boundary conditions are also examined. The condition number and sparseness of the formulated stiffness matrices are analyzed. Findings It is found that to form a well-conditioned stiffness matrix, the support domain of every wavelet scaling function should possess sufficient number of integration points. The results are analyzed and validated against the existing analytical solutions. Numerical examples demonstrate that the accuracy of displacements and stresses is dependent on the size of the background cell and number of Gauss points considered per background cell during the analysis. Originality/value The current paper gives the details on implementation of Gauss Quadrature scheme, using a background cell-based approach, for evaluating the integrals involved in BSWI-based wavelet finite element method, which is missing in the existing literature.

A pointwise condition number associated to a representation of an interpolation operator is introduced. It is proved that the Lagrange formula is optimal with respect to this conditioning. For other representations of the interpolation operator, an upper bound for the conditioning is derived. A quantitative measure in terms of the Skeel condition number is used to compare the conditioning with the Lagrange representation. The conditioning of the Newton representation is considered for increasing nodes and for nodes in Leja order. For the polynomial Newton formula with n + 1 equidistant nodes in increasing order, it is proved that 3 n is the best uniform bound of its conditioning and it is attained at the last node. Numerical experiments are included.

In this study, the frequency and material independent reactions (FMIR) is combined with generalized single source tangential equivalence principle algorithm (GSST-EP A) to calculate the wideband RCS of finite period array structure. By Taylor series expansion of the exponent of Green function, a frequency independent matrix and a frequency dependent phase factor are generated. Therefore, the FMIR can accelerate the frequency sweep and improve the frequency sweep accuracy dynamically, while the memory requirement is large. To deal with this problem, the GSST-EPA can transfer the solution problem on the target to the equivalent surface, so it can improve the matrix condition number and decrease the number of unknowns. The characteristic basis function (CBF) is constructed on the equivalent surface to improve the efficiency of GSST-EPA. A numerical example is shown to prove the accuracy and efficiency of the method.

In this paper, we consider the condition number theory for a linear function of the solution to the constrained and weighted least squares problem. We first present two explicit expressions without Kronecker product of normwise condition number using the classical method for condition numbers. Then, we derive the explicit expression of mixed and componentwise condition numbers by the dual techniques. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator and the small-sample statistical condition estimation method and devise three algorithms. Numerical experiments are provided to illustrate the obtained results.

The time spectral method is a very popular reduced order frequency method for analyzing unsteady flow due to its advantage of being easily extended from an existing steady flow solver. Condition number of the inverse Fourier transform matrix used in the method can affect the solution convergence and stability of the time spectral equation system. This paper aims at evaluating the effect of the condition number of the inverse Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between the maximum allowable Courant number and the condition number will be derived. Different time instant groups which lead to the same condition number are also considered. Three numerical discretization schemes are provided for the stability analysis. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case, NASA rotor 67, and the other is a blade row interaction case, NASA stage 35. A series of numerical analyses will be performed for each case using different time instant groups corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis.