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The current paper introduces a new stabilization scheme for void and low‐density elements for geometrical nonlinear topology optimization. Frequently, certain localized regions in the geometrical nonlinear finite element analysis of the topology optimization have excessive artificial distortions due to the low stiffness of the void and low‐density elements. The present stabilization applies a hyperelastic constitutive material model for the numerical stabilization that is associated with the condition number of the deformation gradient and thereby, is associated with the numerical conditioning of the mapping between current configuration and reference configuration of the underlying continuum mechanics on a constitutive material model level. The stabilization method is independent upon the topology design variables during the optimization iterations. Numerical parametric studies show that the parameters for the constitutive hyperelasticity material of the new stabilization scheme are governed by the stiffness of the constitutive model of the initial physical system. The parametric studies also show that the stabilization scheme is independently upon the type of constitutive model of the physical system and the element types applied for the finite element modeling. The new stabilization scheme is numerical verified using both academic reference examples and industrial applications. The numerical examples show that the number of optimization iterations is significantly reduced compared to the stabilization approaches previously reported in the literature. [ABSTRACT FROM AUTHOR]

Slope stability analysis is a critical component of geotechnical engineering, with its implications reaching far and wide, from infrastructure development to environmental management. This study delves into the innovative Stress Deviator Increasing Method (SDIM) and its implications for slope analysis, focusing on the interaction of various components in this intricate process. This study acknowledges the evolution of slope stability analysis, transitioning from traditional methods such as the Limit Equilibrium Method (LEM) to modern approaches, and introduces SDIM as a promising alternative. Bouzid's SDIM combines the Finite Element Method (FEM) with Mohr's circles to offer a comprehensive understanding of slope behavior, particularly under complex stress conditions. The study meticulously examines SDIM's application through the S4DINA (soil stability study by Stress Deviator Increasing using Numerical Analysis) program, highlighting parameter sensitivity and the significance of considering specific conditions. The results underscore the sensitivity of SDIM to certain parameters, including the associated flow rule, finite element number, and embankment with or without foundations. Careful application of SDIM enhances the accuracy of slope stability assessments, allowing for more reliable results. This study represents a significant step in the field of geotechnical engineering, offering a dynamic and comprehensive approach that can address a wide range of scenarios and enhance the reliability of slope stability assessments. [ABSTRACT FROM AUTHOR]

The K-optimality criterion is proposed to avoid multicollinearity in regression analysis. By far the most, popular models for modeling the response of a mixture experiment are the Scheffé polynomial models. The Scheffé polynomial models have a small degree of multicollinearity. However, there have been no reports about constructing K-optimal designs for the Scheffé polynomial models. This article expands the K-optimality criterion to the second-order Scheffé polynomial model, and derives the K-optimal allocations for such model. We also investigate the construction method of K-optimal designs with the non linear constraints. In addition, the relative efficiencies of D-, A-, and K-optimal designs are compared. [ABSTRACT FROM AUTHOR]

Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend's spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements. [ABSTRACT FROM AUTHOR]

In this paper we present two approaches for estimating matrix-inverse quadratic forms x T A − 1 x , where A is a symmetric positive definite matrix of order n , and x ∈ R n. Using the first, analytic approach, we establish two families of estimates which are convenient for matrices with small condition number. Based on the second, heuristic approach, we derive two families of estimates which are suitable for matrices when vector x is close enough to an eigenvector. The low complexity and stability of the estimates is proved. Several numerical results illustrating the effectiveness of the methods are presented. [ABSTRACT FROM AUTHOR]

AbstractThis paper presents the design and optimization process of a Virtual Biomechanical Shoulder Robot Model (VBSRM) based on a 6-4 parallel mechanism. To address the challenges posed by parallel manipulators and the specific biomechanical constraints of the shoulder joint, a comprehensive design analysis is conducted. First, a kinematic model of the VBSRM is developed, followed by an investigation of its geometric model. To obtain an optimal VBSRM design, performance objectives such as condition number, norm of actuator force, and stiffness are identified and optimized. Initially, only condition number of the robot mechanism is optimized using Genetic Algorithm and performance objectives from the optimal design are analyzed. Later, the three objectives are grouped to form a single function and a single objective-based optimization is also conducted. However, further investigation revealed the conflicting nature of the objectives and hence these were simultaneously optimized using the Non-dominated Sorting Genetic Algorithm (NSGA II). The results obtained from various optimization routines are compared and it is found that the results from the NSGA II provide a better tradeoff between the performance objectives. The motion trajectories from the optimal design of the VBSRM are later analyzed vis-à-vis human shoulder motions for its intended use as a robotic model of the human shoulder joint in various applications. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the worst case optimal value of an interval LP problem when the nominal data are inflated to intervals according to given perturbation patterns. By definition, the derivative expresses how the optimal value can worsen when the data are subject to variations. In addition, it also gives a certain sensitivity measure or condition number of an LP problem. If the LP problem is nondegenerate, the derivatives are easy to calculate from the computed primal and dual optimal solutions. For degenerate problems, the computation is more difficult. We propose an upper bound and some kind of characterization, but there are many open problems remaining. We carried out numerical experiments with specific LP problems and with real LP data from Netlib repository. They show that the derivatives give a suitable sensitivity measure of LP problems. It remains an open problem how to efficiently and rigorously handle degenerate problems. [ABSTRACT FROM AUTHOR]

Summary: We present RCO (regularized Cholesky optimization): a numerical algorithm for finding a symmetric positive definite (PD) n×n$$ n\times n $$ matrix with a bounded condition number that minimizes an objective function. This task arises when estimating a covariance matrix from noisy data or due to model constraints, which can cause spurious small negative eigenvalues. A special case is the problem of finding the nearest well‐conditioned PD matrix to a given matrix. RCO explicitly optimizes the entries of the Cholesky factor. This requires solving a regularized non‐linear, non‐convex optimization problem, for which we apply Newton‐CG and exploit the Hessian's sparsity. The regularization parameter is determined via numerical continuation with an accuracy‐conditioning trade‐off criterion. We apply RCO to our motivating educational measurement application of estimating the covariance matrix of an empirical best linear prediction (EBLP) of school growth scores. We present numerical results for two empirical datasets, state and urban. RCO outperforms general‐purpose near‐PD algorithms, obtaining 10×$$ 10\times $$‐smaller matrix reconstruction bias and smaller EBLP estimator mean‐squared error. It is in fact the only algorithm that solves the right minimization problem, which strikes a balance between the objective function and the condition number. RCO can be similarly applied to the stable estimation of other posterior means. For the task of finding the nearest PD matrix, RCO yields similar condition numbers to near‐PD methods, but provides a better overall near‐null space. [ABSTRACT FROM AUTHOR]

We generalise our previous work [5] by giving a polynomial upper bound on the condition number of certain quasi-Vandermonde matrices to establish the equivalence between the RLWE and PLWE problems for the totally real subfield of the cyclotomic fields of conductor $ 2^r $, $ 2^rp $ and $ 2^rpq $ with $ r\geq 1 $ and $ p $, $ q $ arbitrary primes. Moreover, we give some cryptographic motivations for the study of these subfields. [ABSTRACT FROM AUTHOR]

In this paper, we focus on designing a well‐conditioned Galerkin spectral methods for solving a two‐sided fractional diffusion equations with drift in which the fractional operators are defined neither in Riemann–Liouville nor Caputo sense, and its physical meaning is clear. Based on the image spaces of Riemann–Liouville fractional integral operators on Lp([a, b]) space discussed in our previous work, after a step by step deduction, three kinds of Galerkin spectral formulations are proposed, the final obtained corresponding scheme of which shows to be well‐conditioned—the condition number of the stiff matrix can be reduced from O(N2α) to O(Nα), where N is the degree of the polynomials used in the approximation. Another point is that the obtained schemes can also be applied successfully to approximate fractional Laplacian with generalized homogeneous boundary conditions, whose fractional order α ∈ (0, 2), not only having to be limited to α ∈ (1, 2). Several numerical experiments demonstrate the effectiveness of the derived schemes. Besides, based on the numerical results, we can observe the behavior of mean first exit time, an interesting quantity that can provide us with a further understanding about the mechanism of abnormal diffusion. [ABSTRACT FROM AUTHOR]

The stability of an approximating sequence (A n) for an operator A usually requires, besides invertibility of A , the invertibility of further operators, say B , C , ... , that are well-associated to the sequence (A n). We study this set, { A , B , C , ... } , of so-called stability indicators of (A n) and connect it to the asymptotics of ‖ A n ‖ , ‖ A n − 1 ‖ and κ (A n) = ‖ A n ‖ ‖ A n − 1 ‖ as well as to spectral pollution by showing that lim sup Spec ε A n = Spec ε A ∪ Spec ε B ∪ Spec ε C ∪ .... We further specify, for each of ‖ A n ‖ , ‖ A n − 1 ‖ , κ (A n) and Spec ε A n , under which conditions even convergence applies. [ABSTRACT FROM AUTHOR]

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed "NP-completeness" as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm's potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts. [ABSTRACT FROM AUTHOR]

Slope stability analysis is a critical component of geotechnical engineering, with its implications reaching far and wide, from infrastructure development to environmental management. This study delves into the innovative Stress Deviator Increasing Method (SDIM) and its implications for slope analysis, focusing on the interaction of various components in this intricate process. This study acknowledges the evolution of slope stability analysis, transitioning from traditional methods such as the Limit Equilibrium Method (LEM) to modern approaches, and introduces SDIM as a promising alternative. Bouzid's SDIM combines the Finite Element Method (FEM) with Mohr's circles to offer a comprehensive understanding of slope behavior, particularly under complex stress conditions. The study meticulously examines SDIM's application through the S4DINA (soil stability study by Stress Deviator Increasing using Numerical Analysis) program, highlighting parameter sensitivity and the significance of considering specific conditions. The results underscore the sensitivity of SDIM to certain parameters, including the associated flow rule, finite element number, and embankment with or without foundations. Careful application of SDIM enhances the accuracy of slope stability assessments, allowing for more reliable results. This study represents a significant step in the field of geotechnical engineering, offering a dynamic and comprehensive approach that can address a wide range of scenarios and enhance the reliability of slope stability assessments. [ABSTRACT FROM AUTHOR]

Gaussian processes provide probabilistic surrogates for various applications including classification, uncertainty quantification, and optimization. Using a gradient‐enhanced covariance matrix can be beneficial since it provides a more accurate surrogate relative to its gradient‐free counterpart. An acute problem for Gaussian processes, particularly those that use gradients, is the ill‐conditioning of their covariance matrices. Several methods have been developed to address this problem for gradient‐enhanced Gaussian processes but they have various drawbacks such as limiting the data that can be used, imposing a minimum distance between evaluation points in the parameter space, or constraining the hyperparameters. In this paper a diagonal preconditioner is applied to the covariance matrix along with a modest nugget to ensure that the condition number of the covariance matrix is bounded, while avoiding the drawbacks listed above. The method can be applied with any twice‐differentiable kernel and when there are noisy function and gradient evaluations. Optimization results for a gradient‐enhanced Bayesian optimizer with the Gaussian kernel are compared with the use of the preconditioning method, a baseline method that constrains the hyperparameters, and a rescaling method that increases the distance between evaluation points. The Bayesian optimizer with the preconditioning method converges the optimality, that is, the ℓ2$$ {\ell}_2 $$ norm of the gradient, an additional 5 to 9 orders of magnitude relative to when the baseline method is used and it does so in fewer iterations than with the rescaling method. The preconditioning method is available in the open source Python library GpGradPy, which can be found at https://github.com/marchildon/gpgradpy/tree/paper_precon. [ABSTRACT FROM AUTHOR]

This paper is motivated by the need to stabilise the impact of deep learning (DL) training for medical image analysis on the conditioning of convolution filters in relation to model overfitting and robustness. We present a simple strategy to reduce square matrix condition numbers and investigate its effect on the spatial distributions of point clouds of well- and ill-conditioned matrices. For a square matrix, the SVD surgery strategy works by: (1) computing its singular value decomposition (SVD), (2) changing a few of the smaller singular values relative to the largest one, and (3) reconstructing the matrix by reverse SVD. Applying SVD surgery on CNN convolution filters during training acts as spectral regularisation of the DL model without requiring the learning of extra parameters. The fact that the further away a matrix is from the non-invertible matrices, the higher its condition number is suggests that the spatial distributions of square matrices and those of their inverses are correlated to their condition number distributions. We shall examine this assertion empirically by showing that applying various versions of SVD surgery on point clouds of matrices leads to bringing their persistent diagrams (PDs) closer to the matrices of the point clouds of their inverses. [ABSTRACT FROM AUTHOR]

This paper introduces a new version of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, characterized as a scaled memoryless, projection-based, and derivative-free method for finding approximate solutions of monotone nonlinear equations with convex constraints. The optimal value of the scaling parameter is achieved by minimizing the BFGS update matrix. The theoretical analysis is performed to demonstrate the global convergence of the approach. Numerical analysis and comparisons with prior results indicate that the proposed approach has superior performance for CPU time, iteration count, and function evaluations. The new algorithm is used to solve the motion control issue of a two-jointed coplanar robot manipulator.

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field Q (ζ n) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

I revisit the condition number of computing left and right singular subspaces from J.-G. Sun (1996) [19]. For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values corresponding to the selected singular subspace and those not selected. [ABSTRACT FROM AUTHOR]

The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann, and absorbing boundary conditions. This study focuses in particular on the spectral dependence on the polynomial degree p, mesh size h, regularity k, of the IGA discretization and on the time step size Δ t and parameter β of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing p and maximal regularity k = p - 1 . [ABSTRACT FROM AUTHOR]

This paper proposes a novel method of evaluating the Independent Double-Boost Interleaved Converter's performance, which can offer an overview of its behavior and can enhance the design stage of the converter. The concept of "condition number" is utilized and applied to the converter's model. Correlations between the condition number variation and the converter parameters lead to a qualitative assessment of the converter's behavior and offer the possibility to anticipate aspects like the stability and response of the converter. An in-depth analysis is conducted, starting from the state matrix and continuing with a series of presumptions regarding the converter and its operating mode, obtaining a series of expressions that define the condition number for the converter. [ABSTRACT FROM AUTHOR]

The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized by a Hessenberg form. We demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial. [ABSTRACT FROM AUTHOR]

In variational assimilation, the most probable state of a dynamical system under Gaussian assumptions for the prior and likelihood can be found by solving a least‐squares minimization problem. In recent years, we have seen the popularity of hybrid variational data assimilation methods for Numerical Weather Prediction. In these methods, the prior error covariance matrix is a weighted sum of a climatological part and a flow‐dependent ensemble part, the latter being rank deficient. The nonlinear least squares problem of variational data assimilation is solved using iterative numerical methods, and the condition number of the Hessian is a good proxy for the convergence behavior of such methods. In this article, we study the conditioning of the least squares problem in a hybrid four‐dimensional variational data assimilation (Hybrid 4D‐Var) scheme by establishing bounds on the condition number of the Hessian. In particular, we consider the effect of the ensemble component of the prior covariance on the conditioning of the system. Numerical experiments show that the bounds obtained can be useful in predicting the behavior of the true condition number and the convergence speed of an iterative algorithm [ABSTRACT FROM AUTHOR]

Some of the key features of the isogeometric analysis, IGA, are the capacity of exactly representing the problem geometry, the use of the same basis functions to describe the geometry and the solution field, and a straightforward and automatic discretization refining scheme. The higher order continuity of the isogeometric approximation, important to correctly represent the domain geometry, can be a problem to approximate the displacement field in the neighbourhood of a crack. The eXtended Isogeometric Analysis (XIGA) overcomes this obstacle, enlarging the approximate space of IGA. This is achieved by incorporating customized functions, using the enrichment strategy of the Generalized/eXtended Finite Element Method. When these functions are unknown, they can be computed from the solution of local boundary value problems embracing the crack, and a global–local iterative procedure is established. Here this procedure is firstly proposed to combine FEM and isogeometric approximations, denoted XIGA gl . The effectiveness of this approach is investigated in terms of convergence rates and numerical stability. The method is applied to two-dimensional fracture mechanics problems. The numerical experiments show the importance of using the isogeometric approximation to recover more accurate solutions and minimize the deterioration of the conditioning of the related stiffness matrix. [ABSTRACT FROM AUTHOR]

We state some widely satisfied hypotheses, depending only on two functions |$g$| and |$h$|⁠ , under which the composition of a forward stable algorithm for |$g$| and a forward stable algorithm for |$h$| is a forward stable algorithm for the composition |$g \circ h$|⁠. We show that the failure of these conditions can potentially lead to unstable algorithms. Finally, we list a number of examples to illustrate the new concepts and the usability of the results. [ABSTRACT FROM AUTHOR]

The water distribution system (WDS) hydraulic model is extensively used for design and management of WDS. The nodal water demand is the crucial parameter of the model that requires accurate estimating by the pressure measurements. Proper pressure sampling design is essential for estimating nodal water demand and improving model accuracy. Existing research has emphasized the need to enhance the observability of monitoring systems and mitigate the adverse effects of monitoring noise. However, methods that simultaneously consider both of these factors in sampling design have not been adequately studied. In this study, a novel two-objective sampling design method is developed to improve the system observability and mitigate the adverse effects of monitoring noise. The approach is applied to a realistic network and results demonstrate that the developed approach can effectively improve the observability and robustness of the system especially when considerable measurement noise is considered. [ABSTRACT FROM AUTHOR]

Phylogenetic comparative methods (PCMs) use phylogenetic tree and trait data to explore the evolutionary information among a group of related species. Given a phylogenetic tree of extant species, the evolutionary evidence among a group of species can be represented by a squared matrix C obtained by an isomorphic transformation using the shared branch lengths in the tree. The quantitative trait evolution for species along the branch can be described by stochastic processes. Currently, most statistical models for trait evolution are built under the assumption of Gaussian processes, where the trait vector Y follows a multivariate normal distribution with covariance matrix V, transformed from the C matrix and model parameters. This study investigates the effects on parameter estimation in phylogenetic comparative methods that are caused by ill-conditioned phylogenetic tree matrices (C matrix) and their associated model-adjusted variance–covariance matrices (V matrix). Several popular models—the Brownian motion model, the Ornstein–Uhlenbeck model, the early burst model, and the phylogenetic mixed model (Pagel's λ)—are evaluated and compared through an empirical dataset and extensive simulations. Suggestions are provided to users in the community for identifying and overcoming potential issues with their own data. [ABSTRACT FROM AUTHOR]

This paper explores the research topic of enhancing the reliability of unmanned mobile exploration using LiDAR SLAM. Specifically, it proposes a technique to analyze waypoints where 3D LiDAR SLAM can be smoothly performed in potential exploration areas and points where there is a risk of divergence in navigation estimation. The goal is to improve exploration performance by presenting a method that secures these candidate regions. The analysis employs a 3D geometric observability matrix and its condition number to discriminate waypoints. Subsequently, the discriminated values are applied to path planning, resulting in the derivation of a final destination path connecting waypoints with a satisfactory SLAM position and attitude estimation performance. To validate the proposed technique, performance analysis was initially conducted using the Gazebo simulator. Additionally, experiments were performed with an autonomous unmanned vehicle in a real-world environment. [ABSTRACT FROM AUTHOR]

This paper is motivated by the need to stabilise the impact of deep learning (DL) training for medical image analysis on the conditioning of convolution filters in relation to model overfitting and robustness. We present a simple strategy to reduce square matrix condition numbers and investigate its effect on the spatial distributions of point clouds of well- and ill-conditioned matrices. For a square matrix, the SVD surgery strategy works by: (1) computing its singular value decomposition (SVD), (2) changing a few of the smaller singular values relative to the largest one, and (3) reconstructing the matrix by reverse SVD. Applying SVD surgery on CNN convolution filters during training acts as spectral regularisation of the DL model without requiring the learning of extra parameters. The fact that the further away a matrix is from the non-invertible matrices, the higher its condition number is suggests that the spatial distributions of square matrices and those of their inverses are correlated to their condition number distributions. We shall examine this assertion empirically by showing that applying various versions of SVD surgery on point clouds of matrices leads to bringing their persistent diagrams (PDs) closer to the matrices of the point clouds of their inverses.

This paper proposes and implements an approach to evaluate human–robot cooperation aimed at achieving high performance. Both the human arm and the manipulator are modeled as a closed kinematic chain. The proposed task performance criterion is based on the condition number of this closed kinematic chain. The robot end-effector is guided by the human operator via an admittance controller to complete a straight-line segment motion, which is the desired task. The best location of the selected task is determined by maximizing the minimum of the condition number along the path. The performance of the proposed approach is evaluated using a criterion related to ergonomics. The experiments are executed with several subjects using a KUKA LWR robot to repeat the specified motion to evaluate the introduced approach. A comparison is presented between the current proposed approach and our previously implemented approach where the task performance criterion was based on the manipulability index of the closed kinematic chain. The results reveal that the condition number-based approach improves the human–robot cooperation in terms of the achieved accuracy, stability, and human comfort, but at the expense of task speed and completion time. On the other hand, the manipulability-index-based approach improves the human–robot cooperation in terms of task speed and human comfort, but at the cost of the achieved accuracy.

Summary: An important class of nonlinear weighted least‐squares problems arises from the assimilation of observations in atmospheric and ocean models. In variational data assimilation, inverse error covariance matrices define the weighting matrices of the least‐squares problem. For observation errors, a diagonal matrix (i.e., uncorrelated errors) is often assumed for simplicity even when observation errors are suspected to be correlated. While accounting for observation‐error correlations should improve the quality of the solution, it also affects the convergence rate of the minimization algorithms used to iterate to the solution. If the minimization process is stopped before reaching full convergence, which is usually the case in operational applications, the solution may be degraded even if the observation‐error correlations are correctly accounted for. In this article, we explore the influence of the observation‐error correlation matrix (R$$ \mathbf{R} $$) on the convergence rate of a preconditioned conjugate gradient (PCG) algorithm applied to a one‐dimensional variational data assimilation (1D‐Var) problem. We design the idealized 1D‐Var system to include two key features used in more complex systems: we use the background error covariance matrix (B$$ \mathbf{B} $$) as a preconditioner (B‐PCG); and we use a diffusion operator to model spatial correlations in B$$ \mathbf{B} $$ and R$$ \mathbf{R} $$. Analytical and numerical results with the 1D‐Var system show a strong sensitivity of the convergence rate of B‐PCG to the parameters of the diffusion‐based correlation models. Depending on the parameter choices, correlated observation errors can either speed up or slow down the convergence. In practice, a compromise may be required in the parameter specifications of B$$ \mathbf{B} $$ and R$$ \mathbf{R} $$ between staying close to the best available estimates on the one hand and ensuring an adequate convergence rate of the minimization algorithm on the other. [ABSTRACT FROM AUTHOR]

Instrumented indentation is a versatile method of extracting hyper-elastic material parameters, particularly useful for applications where stress-strain data are difficult to be in-situ measured. Because the analytical force-displacement relation is still unavailable for the indentation of hyper-elastic materials, identifying hyper-elastic parameters often requires an iterative optimization strategy that fits finite element simulations with experimental data. However, the optimization strategy is burdened by heavy computation and its prediction accuracy is greatly influenced by the choice of optimization algorithm. To address these challenges in this study, a bidirectional long short-term memory (BLSTM) neural network is presented that directly predicts hyper-elastic material parameters from indentation load-displacement data, focusing on Mooney-Rivlin hyper-elasticity as an example. To improve the predication accuracy, the condition numbers for the inverse identification of the hyper-elastic parameters are investigated. And, a normalization procedure is proposed to treat the input data, which can guarantee the BLSTM network is well-conditioned. During evaluation, the trained BLSTM network significantly outperforms the iterative optimization strategy using a genetic algorithm. Furthermore, the effect of the normalization procedure is demonstrated. [ABSTRACT FROM AUTHOR]

Uncertainty in modal characteristics due to output-only system identification methods has been a challenge in operational modal analysis. The present study aims to extract modal parameters of Karun IV Dam (the highest arch dam in Iran) using the balanced stochastic subspace identification (B-SSI) and investigate the influence of user-defined parameters (i.e., columns and block rows of Hankel Matrix) on the uncertainty of the results. The effects of noise caused by numerical instabilities were first filtered using the inverse process by the condition number. Subsequently, the modal properties were homogenized with spatial clustering of applications with noise (DBSCAN) to remove the outlier and spurious characteristics. Then, the physical modes were validated by inspecting the complexity of the mode shapes based on the mode complexity factor criterion. Finally, the coefficient of variation (CV) of the validated clusters was employed to conduct a sensitivity analysis performed concerning the dimensions of the Hankel matrix to find the optimal models (with the minimum error in estimating the modal characteristics). The results indicated that the proposed method prevented the emergence of computational and noisy modes by regulating the extracted models, such that the first model of the structure was extracted with an error of less than 10% compared to the numerical model. [ABSTRACT FROM AUTHOR]

The quality of the measured signature is influenced not only by the instrument's precision but also by the selected measurement configuration. In optical scatterometry, the purpose of measurement configuration optimization (MCO) is to select an optimal or suboptimal combination of measurement conditions, such as the angles of incidence, azimuth, polarization and wavelength, to achieve higher measurement accuracy. This analysis not only requires an effective optimization strategy but is also time-consuming. In this work, we propose a general MCO method that incorporates error propagation theory and condition-number-based error estimation technique, by which the MCO problem can be formulated as an optimization problem for the condition number of the coefficient matrix in the linear estimation of parameter deviations. The method is demonstrated on a multi-wavelength Mueller matrix scatterometry measuring a Si grating. With the help of the neural-network-based surrogate model, the feasibility of the method is verified by making a comparison with Latin hypercube sampling. Fitting results of the measured and calculated Mueller matrix spectra obtained at the selected optimal measurement configuration show a good agreement. The proposed method is promising to provide an alternate solution to globally evaluate the MCO problem in optical scatterometry and other measurement scenarios. [ABSTRACT FROM AUTHOR]

Canonical polyadic decomposition (CPD) is prevalent in chemometrics, signal processing, data mining, and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithms for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in signal processing, data analytics, and various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function differently from previous works. Computing these critical points in an alternating manner motivates an alternating optimization algorithm which corresponds to the ALS algorithm in the matrix case. However, for tensors with order greater than or equal to 3, it minimizes an objective function which is different from the commonly used least squares loss. Alternating optimization of this new objective leads to simple updates to the factor matrices with the same asymptotic computational cost as ALS. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD when the known rank is not larger than the mode lengths of the input tensor. We verify our theoretical arguments experimentally. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with the ground metric dependent on the other factors. This perspective allows us to generalize our approach to interpolate between updates corresponding to the ALS and the new algorithm to manage the tradeoff between stability and fitness of the decomposition. Our experimental results show that for approximating synthetic and real-world tensors, this algorithm and its variants converge to a better conditioned decomposition with comparable and sometimes better fitness as compared to the ALS algorithm. [ABSTRACT FROM AUTHOR]

A peak effect minimization problem in the free motion of linear systems is considered in the paper. The paper proposes an iterative procedure for the peak effect minimization using a combination of the recently proposed gramian-based approach and the theory of using the condition number of an eigenvectors matrix for the upper bound estimations of the system state processes. Minimization of control costs is based on the analysis of the singular value decomposition of a gramian of control costs, followed by the formation of major and minor estimations of the gramian. Minimization of peak effect in the trajectories of free movement of systems is carried out by minimizing the condition number of the eigenvectors matrix of the matrix of a stable closed-loop system, while the state matrix with the desired eigenvalues and eigenvectors is designed with the generalized modal control. The development of an iterative algorithm for the peak effect minimization in the trajectories of linear systems under any non-zero initial conditions with restricted control is based on an aggregated index. The index takes into account both the estimate of the gramian of control costs and the condition number of the eigenvectors matrix of the stable closed-loop system. Minimization of the aggregated index makes it possible to ensure minimal deviations in the trajectories of free movement of systems of the considered class. The procedure is applied to a system of two satellites with restricted control, where peak effects in satellites relative trajectories are minimized. Two cases of peak affect minimization are considered. In the first case, the peak effect minimization in the trajectories of free movement of satellites is carried out only by minimizing the gramian of control costs. In the second case, the peak effect minimization is realized using the developed algorithm. The results illustrate the efficiency of the procedure and indicate the decrease of the peak effect for the satellites relative trajectories.

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an 265 × 371 × 7 tensor of rank 3 from a food science application whose condition number was computed in 6.9 milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art. [ABSTRACT FROM AUTHOR]

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations. [ABSTRACT FROM AUTHOR]

The nonlinear programming makes use of quasi-Newton methods, a collection of optimization approaches when traditional Newton's method are challenging due to the calculation of the Jacobian matrix and its inverse. Since the Jacobian matrix is computationally difficult to compute and sometimes not available specifically when dealing with non-smooth monotone systems, quasi-Newton methods with superlinear convergence are preferred for solving nonlinear system of equations. This paper provides a new version of the derivative-free David–Fletcher–Powell (DFP) approach for dealing with nonlinear monotone system of equations with convex constraints. The optimal value of the scaling parameter is found by minimizing the condition number of the DFP matrix. Under certain assumptions, the proposed method has global convergence, required minimal storage and is derivative-free. When compared to standard methods, the proposed method requires less iteration, function evaluations, and CPU time. The image restoration test problems demonstrate the method's reliability and efficiency. [ABSTRACT FROM AUTHOR]

This study proposes a method for designing a class of rotationally symmetric Stewart platforms (RSSPs) with an insensitive condition number (ICN), which is used to minimize the condition number to achieve a high accuracy for a multi-degree-of-freedom (multi-DOF) shaker. Considering the rotational symmetry of RSSPs, an analytical relationship between the architecture parameters and transfer coefficients is first established. Then, the decoupling conditions of the RSSPs are derived, and the transfer coefficient formulas are simplified by the given decoupling conditions and iso-length assumption. Following further analyses and discussions, the ICN condition and analytical form of the condition number are provided. The area of the ICN (AICN) is, subsequently, derived to evaluate the insensitivity of the condition number. To validate the effectiveness of the method, a design example (ICN-RSSP), along with a numerical analysis, is implemented, and, finally, a multi-DOF shaker is developed. The results of the numerical analysis show a smaller condition number and a larger AICN than those of the RSSP, for comparison. And the experiment results of the multi-DOF shaker show a high accuracy of vibration waveform reproduction. The method can reduce the condition number of RSSPs, improve the insensitivity, and further improve the accuracy of the multi-DOF shaker. [ABSTRACT FROM AUTHOR]

Aiming at the problem of how to select aero-engine health parameters when the number of sensors is limited, a method of selecting aero-engine health parameters based on correlation and condition number is proposed. Firstly, the engine health parameters are preliminarily selected based on correlation. When the influence of two health parameters on engine output parameters is strongly correlated, only one of them needs to be selected. Then health parameters are further selected based on condition number of sensors parameters degradation matrixes. The larger the condition number is, the more ill conditioned the matrix is and the worse the estimation effect of the on-board model is. According to this, the combination of health parameters with the minimum condition number is selected. The proposed method can quickly select optimal health parameters and improve the accuracy of engine adaptive estimation. The results demonstrate that the stronger the correlation between the two health parameters, the greater the impact on the accuracy of the on-board model. Compared with the singular value decomposition-Kalman filter (SVD-KF) method and other combinations of health parameters, the on-board model accuracy of the optimal combination selected by this method is greatly improved, and has the best state parameter tracking effect. [ABSTRACT FROM AUTHOR]

A recent study on the condition numbers of the mixed least squares-total least squares (MTLS) problem is due to Zheng and Yang (Numer Linear Algebra Appl. 2019;26(4):e2239). However, the associated expressions are not compact and the Kronecker-product operations make the computation costly. In this paper, we first present new and alternative closed formula for the first order perturbation estimate and condition numbers of the MTLS solution. Then we reveal the relationship between the new formula and Zheng and Yang's result. Several new computable formulae and perturbation bounds for the normwise condition number of the MTLS solution are also provided. Finally, mixed and componentwise condition numbers, structured condition numbers are investigated. Through a number of tests, they are shown to be tighter than the normwise condition numbers for sparse and structured problems. [ABSTRACT FROM AUTHOR]

This paper considers the structured perturbation analysis of Sylve-ster matrix equation with low-rank structures. When the coefficient matrix and the right-hand side of Sylvester matrix equation are $ \{1;1\} $-quasiseparable matrices, we propose the structured condition numbers and obtain explicit expressions for these structured condition numbers using the general parameter representation and the tangent-based Givens-vector representation. By comparing different condition numbers of Sylvester matrix equation, we analyze their mathematical relationship. Numerical experiments demonstrate that the structured condition number is significantly smaller than the unstructured condition number when the elements in the general representation of $ \{1;1\} $-quasiseparable matrices have different scales. This suggests that structured algorithms for low-rank structured matrix equations can effectively reduce the loss of numerical solution accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes and implements an approach to evaluate human–robot cooperation aimed at achieving high performance. Both the human arm and the manipulator are modeled as a closed kinematic chain. The proposed task performance criterion is based on the condition number of this closed kinematic chain. The robot end-effector is guided by the human operator via an admittance controller to complete a straight-line segment motion, which is the desired task. The best location of the selected task is determined by maximizing the minimum of the condition number along the path. The performance of the proposed approach is evaluated using a criterion related to ergonomics. The experiments are executed with several subjects using a KUKA LWR robot to repeat the specified motion to evaluate the introduced approach. A comparison is presented between the current proposed approach and our previously implemented approach where the task performance criterion was based on the manipulability index of the closed kinematic chain. The results reveal that the condition number-based approach improves the human–robot cooperation in terms of the achieved accuracy, stability, and human comfort, but at the expense of task speed and completion time. On the other hand, the manipulability-index-based approach improves the human–robot cooperation in terms of task speed and human comfort, but at the cost of the achieved accuracy. [ABSTRACT FROM AUTHOR]

Theoretical results and numerical experiments show that the linear systems originating from the last iterations of interior point methods (IPM) are very ill-conditioned. For this reason, preconditioners are necessary to approach this problem. In addition to that, in large-scale problems, the use of iterative methods and implicit preconditioners is essential because we only compute matrix–vector multiplications. Preconditioners with a lower computational cost than the splitting preconditioner only have good performance in the initial iterations of the IPM, so this preconditioner has become very important in the last iterations. The study of improvements thereof is justified. This paper studies the variation of the diagonal matrix D entries that appear in the linear systems to be solved to try to reuse or recycle some linearly independent columns of the splitting preconditioner base previously computed in a given IPM iteration to build another basis in the next one. It is justified by the fact that a subset of linearly independent columns remains linearly independent, and from that available subset, one may complete the number of columns necessary to form the new base. The numerical results show that the column recycling proposal improves the speed and robustness of the original approach for a test set, especially for large-scale problems. [ABSTRACT FROM AUTHOR]