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The radial basis function interpolation is an effective method for approximation on scattered data. However, this interpolation method suffers from the contradiction between the accuracy and the numerical stability. With considering the distribution of the interpolation points, a bounded random shape variation scheme for the radial basis function is developed to circumvent the problem of numerical stability. In this scheme, the shape variation is bounded by the value that is determined by the maximum distance and the minimal distance which are applied to describe the average density of the interpolation centers. Within this bound, the shape of the MQ is modified through a random scheme. With applying this bounded randomly variable shape scheme, the accuracy and the stability of the MQ interpolation are balanced. Comparisons on the accuracy and the condition number of the interpolation matrix between the constant shaped MQ interpolation and this bounded randomly variable shaped MQ interpolation have been made to verify the conclusion. Furthermore, this scheme is integrated in the dual reciprocity boundary element method in the analysis of three dimensional elastic problems. Results of the numerical examples demonstrated that the developed scheme improved the accuracy of the dual reciprocity boundary element method stably.

The method of auxiliary sources (MAS) is one of the numerical methods employed for the determination of the cutoff wavenumbers (eigenvalues) of cylindrical waveguides. According to the general scheme of MAS, the eigenvalues are obtained by requiring the square impedance matrix to be singular and therefore its determinant to be zero (or its condition number to be large). For large matrices, such computations are complicated and time-consuming. Additionally, these schemes are frequently contaminated with spurious eigenvalues. To overcome the difficulties, an efficient method based on MAS along with an excitation source has been proposed. The eigenvalues are obtained by measuring the physical response of the waveguide’s domain to this source; thus, an interior problem is always solved. This article is a comprehensive presentation of this method and its variants and aims to emphasize fundamental and often unfamiliar attributes. One of the main objectives is to show analytically and numerically that when an internal source is used, the eigenvalues are accurately computed without employing a regularization procedure. The method is applied to hollow simply and multiply connected waveguides with circular, elliptical, and rounded-triangular cross sections. The results are compared with those obtained from COMSOL Multiphysics. Moreover, the phenomenon of divergent and oscillating MAS currents—which may or may not occur depending on the relative position between the auxiliary curve and the excitation source—is discussed. It is emphasized that, with proper care, the phenomenon does not affect the accurate computation of the eigenvalues.

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.

A preconditioner for the mass matrix for high order finite element discretization on tetrahedra is presented and shown to give a condition number that is independent of both the mesh size and the p...

In this paper, three meshless methods are proposed to solve boundary value problems. The special purpose Trefftz function method, the method of fundamental solutions and the symmetry method of fundamental solutions are compared. We considered six numerical examples for harmonic and biharmonic problems as a flow through the cylindrical fibers arranged in a regular square array, deflection of a uniformly loaded square plate and torsion of a bar. The boundary value problems defined for a repeating element of the domain are solved by use of the collocation technique. The accuracy and stability of compared methods are investigated. Root mean square error on the boundary and condition number of the matrix are presented as functions of a number of collocation points.

In this paper, the technique of the Stable Generalized Finite Element Method (SGFEM) is applied to the numerically constructed functions of the Generalized Finite Element Method with Global-Local Enrichments (GFEMgl). The application of the resulting approach, named SGFEMgl, is expanded here to 2-D quasi-static crack propagation problems. Crack growth is performed by a two-scale strategy, using local problems generated at each propagation step – whose solutions enrich a single global problem defined on a coarse mesh. Stress Intensity Factors (SIFs) computed along crack growth, strain energy measures, performance in blending elements and the condition number are used to study the accuracy and conditioning of SGFEMgl. The method is compared with the standard GFEMgl. Numerical experiments demonstrate remarkable accuracy of SGFEMgl in linear elastic fracture mechanics problems, considering crack opening modes I and II. Convergence rates analyses also show the superiority of the method, especially with the use of geometrical enrichments.

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.

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

The existence of a figure of merit for measuring the degree of well-spread source points in noise parameter extraction has long been conjectured. This article proposes a measure based on noise waves that is physically motivated and is directly connected to linear algebra through the matrix condition number and/or determinant. The key to this figure of merit is the selection of the noise temperature equation and the removal of singularity due to the $1/(1-|\Gamma _{s}|^{2})$ factor. The result is a well-scaled source matrix with entries bounded within a unit circle. We demonstrate the effectiveness of this measure by extracting the noise parameters of an amplifier in the low-frequency Square Kilometre Array (SKA-Low) band of 50–350 MHz using seven tuner positions. The noise parameters in the 50–100-MHz band are successfully measured despite being below the 100-MHz tuner rating. This outcome is very well predicted by the condition number and the determinant of the source matrix in question.

In this paper we first consider a special class of nonhomogeneous and anisotropic Helmholtz equation with variable coefficients and the source term. Combined with local spectral element method, a generalized plane wave discontinuous Galerkin method for the discretization of such Helmholtz equation is designed. Then we define new generalized plane wave basis functions for two-dimensional anisotropic Helmholtz equation with the variable coefficient. Besides, the error estimates of the approximation solutions generated by the proposed discretization method are derived. Especially, the orders of the condition number ρ of the anisotropic matrix in the error estimates are optimal. Finally, numerical results verify the validity of the theoretical results, and indicate that the new method possesses high accuracy and is slightly affected by the pollution effect for the large wavenumbers.

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.

The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.

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 scalar wave equation is solved using higher order immersed finite elements. We demonstrate that higher order convergence can be obtained. Small cuts with the background mesh are stabilized by adding penalty terms to the weak formulation. This ensures that the condition numbers of the mass and stiffness matrix are independent of how the boundary cuts the mesh. The penalties consist of jumps in higher order derivatives integrated over the interior faces of the elements cut by the boundary. The dependence on the polynomial degree of the condition number of the stabilized mass matrix is estimated. We conclude that the condition number grows extremely fast when increasing the polynomial degree of the finite element space. The time step restriction of the resulting system is investigated numerically and is concluded not to be worse than for a standard (non-immersed) finite element method.

The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

Purpose The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable. Design/methodology/approach First of all the T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. Then, an approximate solutions of the problems are expressed in the form of a linear combination of the T-functions. The T-functions satisfy the equation modeling the problem under consideration. Therefore, approximating a solution of a problem with a linear combination of n T-functions one obtains a function that satisfies the equation. The unknown coefficients of the linear combination are obtained as a result of minimization of the functional that describes an inaccuracy of satisfying the initial and boundary conditions in a mean-square sense. Findings The sets of T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. An infinite set of these functions is a complete set of functions and stands for a base functions layout for the space of solutions for the equation used to generate them. Then, an approximate solutions of the initial-boundary problem have been found and compared to find out the effect of finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. Research limitations/implications The methods used in the literature to find an approximate solution of any bioheat transfer problems are more complicated than the one used in the presented paper. However, it should be pointed out that there is some limitation concerning the T-function method, namely, the greater number of T-function is used, the greater condition number becomes. This limitation usually can be overcome using symbolic calculations or conducting calculations with a large number of significant digits. Originality/value The T-functions for the Pennes’ equation and for the SPL equation with perfusion have been reported in this paper for the first time. In the literature, the T-functions are known for other linear partial differential equations (e.g. harmonic functions for Laplace equation), but for the first time they have been derived for the two aforementioned equations. The results are discussed with respect to practical applications.

In this paper, a new Boundary Integral Equation (BIE) is proposed for solution of the scalar Helmholtz equation. Applications include acoustic scattering problems, as occur in room acoustics and outdoor and underwater sound propagation. It draws together ideas from the study of time-harmonics and transient BIEs and spatial audio sensing and rendering, to produce an energy-inspired Galerkin BEM that is intended for use with oscillatory basis functions. Pivotal is the idea that waves at a boundary may be decomposed into incoming and outgoing components. When written in its admittance form, it can be thought of setting the Burton-Miller coupling parameter differently for each basis function based on its oscillation; this is a discrete form of the Dirichlet-to-Neumann map. It is also naturally expressed in a reflectance form, which can be solved by matrix inversion or by marching on in reflection order. Consideration of this leads to an orthogonality relation between the incoming and outgoing waves, which makes the scheme immune to interior cavity eigenmodes. Moreover, the scheme is seen to have two remarkable properties when solution is performed over an entire obstacle: i) it has a condition number of 1 for all positive-real wavenumber k on any closed geometry when a specific choice of cylindrical basis functions are used; ii) when modelling two domains separated by a barrier domain, the two problems are numerical uncoupled when plane wave basis functions are used - this is the case in reality but is not achieved by any other BIE representation that the authors are aware of. Normalisation and envelope functions, as would be required to build a Partition-of-Unity or Hybrid-Numerical-Asymptotic scheme, are introduced and the above properties are seen to become approximate. The modified scheme is applied successfully to a cylinder test case: accuracy of the solution is maintained and the BIE is still immune to interior cavity eigenmodes, gives similar conditioning to the Burton-Miller method and iterative solution is stable. It is seen that for this test case the majority of values in the interaction matrices are extremely small and may be set to zero without affecting conditioning or accuracy, thus the linear system become sparse - a property uncommon in BEM formulations.

In near-field scanning, the model parameters for the antenna under test can be determined from measured probe outputs by solving a linear system of equations in the least-squares sense. The model parameters are far-field pattern values, cylindrical or spherical-wave expansion coefficients, or equivalent surface-source values. The normal matrix of this linear system of equations is sometimes extremely ill-conditioned. This occurs when certain sets of model parameters lie outside the spatial bandwidth of the operator that computes the probe output. One remedy is to restrict the sets of model parameters allowed and perform upsampling if needed to achieve the desired accuracy. These ideas are illustrated through analysis and examples that involve both 2-D and 3-D scanning geometries.

We improve existing estimates for the condition number of matrices arising in radial basis function interpolation. To this end, we refine lower bounds on the smallest eigenvalue and upper bounds on the largest eigenvalue, where our upper bounds on the largest eigenvalue are independent of the matrix dimension (i.e., the number of interpolation points). We show that our theoretical results comply with recent numerical observations concerning the condition number of radial basis function interpolation matrices.

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.

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.

The energy spectrum of an X-ray tube plays an important role in computed tomography (CT), and is often estimated from physical measurement of dedicated phantoms. Usually, this estimation problem is reduced to solving a system of linear equations, which is generally ill-conditioned. In this paper, we optimize a phantom design to find the most effective combinations of thicknesses for different materials. First, we analyze the ill-posedness of the energy spectrum inversion when the number of unknown variables (N) and measurements (M) are equal, and show the condition number of the system matrix increases exponentially with N if the transmission thicknesses are linearly changed. Then, we present a genetic optimization algorithm to minimize the condition number of the system matrix in a general case (M < N) with respect to the selection of thicknesses and types of phantom materials. Finally, in the simulation with Poisson noise we study the accuracy of the spectrum estimation using the expectation-maximum algorithm. Our results indicate that the proposed method allows high-quality spectrum estimation, and the number of measurements is reduced over two thirds of that required by the widely-used method using a phantom with linearly-changed thicknesses.

The mixed formulation of the Local Discontinuous Galerkin (LDG) method is presented for a two boundary value problem that involves the Riesz operator with fractional order $$1< \alpha < 2$$ . Well posedness of the stabilized and non stabilized LDG method is proved. Using a penalty term of order $${{\mathcal {O}}}\left( h^{1-\alpha }\right) $$ a sharp error estimate in a mesh dependent energy semi-norm is developed for sufficiently smooth solutions. Error estimates in the $$L^2$$ -norm are obtained for two auxiliary variables which characterize the LDG formulation. Our analysis indicates that the non stabilized version of the method achieves higher order of convergence for all fractional orders. A numerical study suggests a less restrictive, $${{\mathcal {O}}}\left( h^{-\alpha }\right) $$ , spectral condition number of the stiffness matrix by using the proposed penalty term compared to the $${{\mathcal {O}}}\left( h^{-2}\right) $$ growth obtained when the traditional $${{\mathcal {O}}}\left( h^{-1}\right) $$ penalization term is chosen. The sharpness of our error estimates is numerically validated with a series of numerical experiments. The present work is the first attempt to elucidate the main differences between both versions of the method.

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa&rsquo, s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa&rsquo, s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

Heat transfer coefficient (HTC) is a useful concept to characterize the heat exchange performance of rock fracture. However, existing study has preliminarily found that some HTC formulas would lead to abnormal values, which claims a systematical evaluation of existing formulas. In this paper, 214 test records are employed to evaluate 8 existing formulas from literature. It is found that, for all these experimental data, Bai’s formula (Formula D) developed in 2017 shows fairly good numerical stability, while the rest show numerical oscillations and anomalies in different degrees. Moreover, the numerical oscillations and anomalies would be exacerbated with the increase of flow rates, which leads to poor applicability. Error propagation theory in numerical analysis is used to analyze the mechanism of the numerical oscillations and anomalies which can be effectively explained with the condition number and the relative error of the corresponding formula.

In this paper, we present a spectral method for solving the heat equation in cylindrical coordinates in a case where the data are axisymmetric and independent of the z-coordinate at the same time. The spectral method considered is of GalerkintypewithaGauss-Radaunumericalquadratureformula, itisbasedonaweightedweakvariationalformulationofthe continuous problem. The method considered is discret only in r-variable, the time variable remains continuous. Consequently, the discret problem leads to a system of ordinary differential equations, we solve the system and estimate the error, we also give some numerical examples.

In this paper, we study the Helmholtz equation by the method of fundamental solutions (MFS) using Bessel and Neumann functions. The bounds of errors are derived for bounded simply-connected domains, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Note that by using Bessel functions, the radius R of the source nodes is not necessarily to be larger than the maximal radius r max of the solution domain. This is against the well-known rule: r max R for the MFS. Numerical experiments are carried out, to support the analysis and conclusions made. This is the first novelty in this paper. The error analysis for the Helmholtz equation is more complicated than that for the modified Helmholtz equation in [35] , since the Bessel functions J n ( x ) have infinite zeros. We consider the curial and degenerate cases when J n ( k R ) ≈ 0 and J n ( k ρ ) ≈ 0 . There exist few reports for the analysis for such a degeneracy (e.g., Li [21] ). The error bounds are also explored for bounded simply-connected domains. The second novelty of this paper is for the analysis of the MFS in degeneracy. For the MFS using Neumann functions, the rule of the MFS, r max R , must obey. This paper is the first time to discover that the MFS using Bessel and Neumann functions suffer from the spurious eigenvalues. The spurious eigenvalues are not the true eigenvalues of the corresponding eigenvalue problems, but the correct solutions can not be obtained due to either algorithm singularity or divergence of numerical solutions. For the method of particular solutions (MPS) in [26] , however, the source nodes disappear. In this paper, we will briefly provide the analysis of the MFS using Neumann functions, and the polynomial convergence can be achieved for bounded simple-connected domains. The analysis of the MFS using Neumann functions and numerical comparisons for different methods are the third contribution in this paper.

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=\phi_t\ast f$ resulting from the convolution with a kernel $\phi_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space-time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $\widehat f$ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Tur\'an type inequalities. En route, we obtain a Remez-Tur\'an inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem., Comment: 29 pages

It is known that the convergence rate of the traditional iteration methods like the conjugate gradient method depends on the condition number of the stiffness matrix. Moreover the construction of fast solvers like multigrid and domain decomposition methods also need to estimate the condition number of the stiffness matrix. The main purpose of this paper is to give a rigorous condition number estimate of the stiffness matrix resulting from the linear and bilinear immersed finite element approximations of the high-contrast interface problem. It is shown that the condition number is $$C\rho h^{-2}$$, where $$\rho $$ is the jump of the discontinuous coefficients, h is the mesh size, and the constant C is independent of $$\rho $$ and the location of the interface on the triangulation. Numerical results are also given to verify our theoretical findings.

Fast and accurate resolution of electromagnetic problems via the boundary element method (BEM) is oftentimes challenged by conditioning issues occurring in three distinct regimes: (i) when the frequency decreases and the discretization density remains constant, (ii) when the frequency is kept constant while the discretization is refined and (iii) when the frequency increases along with the discretization density. While satisfactory remedies to the problems arising in regimes (i) and (ii), respectively based on Helmholtz decompositions and Calderon-like techniques have been presented, the last regime is still challenging. In fact, this last regime is plagued by both spurious resonances and ill-conditioning, the former can be tackled via combined field strategies and is not the topic of this work. In this contribution new symmetric scalar and vectorial electric type formulations that remain well-conditioned in all of the aforementioned regimes and that do not require barycentric discretization of the dense electromagnetic potential operators are presented along with a spherical harmonics analysis illustrating their key properties.

In this paper we analyze the nearly optimal block diagonal scalings of the rows of one factor and the columns of the other factor in the triangular form of the SR decomposition. The result is a block generalization of the result of the van der Sluis about the almost optimal diagonal scalings of the general rectangular matrices.

Condition numbers of pressure and velocity matrices obtained from an unstructured finite volume discretization of the three dimensional Navier–Stokes equations are studied for two distinct flow configurations. The exact condition number is calculated based on singular values. The continuous estimation of condition number of the velocity matrices with respect to the time varying velocity field is achieved through the Gershgorin theoretical bounds for the eigenvalues. The focus of the present study is on the analysis of flow dynamics with respect to the degree of ill-conditioning of the linear systems. For the two benchmark problems, condition numbers of pressure and velocity matrices are presented. The pressure matrix is shown to be more ill-conditioned than velocity and requires strong preconditioning. For a 3D lid-driven cavity, the appearance and disappearance of corner vortices at higher Reynolds number is clearly reflected in the condition number variation with time. In a second application of flow past a circular cylinder, condition number variation with time during the initial phase clearly reveals the onset of vortex shedding. The condition number based on Gershgorin bounds is also used to switch the SGS and ILU preconditioners for velocity matrices when they are well conditioned. Consequently, an overall reduction in the simulation time is demonstrated.

We present a new family of X-ray source scanning trajectories for large-angle cone-beam computed tomography. Traditional scanning trajectories are described by continuous paths through space, e.g., circles, saddles, or helices, with a large degree of redundant information in adjacent projection images. Here, we consider discrete trajectories as a set of points that uniformly sample the entire space of possible source positions, i.e., a space-filling trajectory (SFT). We numerically demonstrate the advantageous properties of the SFT when compared with circular and helical trajectories as follows: first, the most isotropic sampling of the data, second, optimal level of mutually independent data, and third, an improved condition number of the tomographic inverse problem. The practical implications of these properties in tomography are also illustrated by simulation. We show that the SFT provides greater data acquisition efficiency, and reduced reconstruction artifacts when compared with helical trajectory. It also possesses an effective preconditioner for fast iterative tomographic reconstruction.

We analyze two BPS preconditioners for a weak Galerkin (WG) finite element method for 2D diffusion equations with strongly discontinuous coefficients. The first preconditioner uses nonconforming linear elements on the coarse mesh, and the corresponding condition number is bounded from above by C ( 1 + ln ( H ∕ h ) ) 3 with C independent of the coefficients; the second one uses H 1 -conforming linear elements on the coarse mesh, and the corresponding condition number is bounded from above by C ( 1 + ln ( H ∕ h ) ) 2 with C depending on the coefficients. In addition, we construct and analyze a preconditioner for the sub-problems encountered in the procedure of applying the two preconditioners. The condition number of the preconditioned system is bounded from above by C ( 1 + ln ( H ∕ h ) ) 2 with C independent of the coefficients. Finally we carry out some numerical experiments to verify our theoretical results.

In the present paper, Multiquadric radial basis function (MQ RBF) and its integrated form are used to construct collocation methods for numerical solution of two-dimensional elliptic problems with curved or closed interface. The main purpose of this work is to perform a comparative analysis of both the methods via accuracy and condition number of the coefficient matrix for elliptic interface problems. In the classical RBF collocation method, the shape parameter is selected by using cross validation approach [1]. In the case of Integrated MQ RBF, a reasonable accuracy is obtained for a wide range of values of the shape parameter. Some of the benchmark problems such as linearized Poisson-Boltzmann problem [2], Poisson interface problem [3], Pennes Bioheat Equation [4] (with no exact solution, containing two phases), are considered to validate accuracy and efficiency of the RBFs collocation methods.

The flat-top partition of unity method can obtain a stiffness matrix with a small condition number and avoid the linear dependence problem. We aimed to extend the flat-top partition of unity method to rock dynamic analyses to simulate the influence of discontinuity and dynamic loading. Therefore, the flat-top partition of unity method was coupled with the discontinuous deformation analysis. In this paper, after the presentation of the mesh construction, the inertia force and contact algorithm for flat-top partition of unity-based discontinuous deformation analysis, several numerical examples are presented to verify the accuracy and efficiency of the method. Accepted version

In this paper, we consider the backward error and condition number of the indefinite linear least squares (ILS) problem. For the normwise backward error of ILS, we adopt the linearization method to derive the tight estimations for the exact normwise backward errors. We derive the explicit expressions of the normwise, mixed and componentwise condition numbers for the linear function of the solution for ILS. The tight upper bounds for the derived mixed and componentwise condition numbers are obtained, which can be estimated efficiently by means of the classical power method for estimating matrix 1-norm [N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, PA, 2002, Chapter 15] during using the QR-Cholesky method [S. Chandrasekaran, M. Gu, and A.H. Sayed, A stable and efficient algorithm for the indefinite linear least-squares problem, SIAM J. Matrix Anal. Appl. 20(2) (1999), pp. 354–362] for solving ILS. Moreover, we revisit the previous results on condition numb...

In this paper, we consider the mixed and componentwise condition numbers for a linear function Lx of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques under both unstructured and structured componentwise perturbations. The sharp upper bounds for condition numbers are obtained. An efficient condition estimation algorithm is proposed, which can be integrated into the iterative method for solving large scale TLS problems. Moreover, the new derived condition number expressions can recover the previous results on the condition analysis for the TLS problem when L = I n . Numerical experiments show the effectiveness of the introduced condition numbers and condition estimation algorithm.

In this paper, we present a new wavelet multigrid method for the numerical solution of elliptic type differential equations based on Daubechies (db4) high pass and low pass filter coefficients with modified intergrid operators. The proposed method is the robust technique for faster convergence with less computational cost which is justified through the error analysis and condition number of a system in comparison with integrated-RBF technique based on Galerkin formulation (Mai-Duy and Tran-Cong, 2009) and finite difference method. Some of the illustrative problems are presented to demonstrate the attractiveness of the proposed technique. Keywords: Wavelet multigrid, Daubechies filters, Elliptic differential equations, Condition number

Based on the classic definition of condition number, a structured condition number is proposed for a class of self-adjoint polynomial matrix equations. The explicit formula of the structured condition number is derived with applying newly defined linear operators. The structured condition number can be applied to some important polynomial matrix equations, including the continuous-time algebraic Riccati equation (CARE), the discrete-time algebraic Lyapunov equation (DALE), etc. Compared with the state-of-the-art condition numbers for CARE and DALE, the newly proposed structured condition number can measure the sensitivity of the solution better, which is validated by numerical examples.