Condition number

We explore gauge actions for lattice QCD, which are constructed such that the occurrence of small plaquette values is strongly suppressed. Such actions originate from the admissibility condition in order to conserve the topological charge. The suppression of small plaquette values is expected to be advantageous for numerical studies in the ε-regime and also for simulations with dynamical quarks. Performing simulations at a lattice spacing of about 0.1 fm, we present numerical results for the static potential, the physical scale r 0 , the stability of the topological charge history, the condition number of the kernel of the overlap operator and the acceptance rate against the step size in the local HMC algorithm.

We explore gauge actions for lattice QCD, which are constructed such that the occurrence of small plaquette values is strongly suppressed. By choosing strong bare gauge couplings we arrive at values for the physical lattice spacings of O(0.1 fm). Such gauge actions tend to confine the Monte Carlo history to a single topological sector. This topological stability facilitates the collection of a large set of configurations in a specific sector, which is profitable for numerical studies in the ǫ-regime. The suppression of small plaquette values is also expected to be favourable for simulations with dynamical quarks. We use a local Hybrid Monte Carlo algorithm to simulate such actions, and we present numerical results for the static potential, the physical scale, the topological stability and the kernel condition number of the overlap Dirac operator. In addition we discuss the question of reflection positivity for a class of such gauge actions.

Predicting prognosis is the key factor in selecting the proper treatment modality for patients with spinal metastases. Therefore, various assessment systems have been designed in order to provide a basis for deciding the course of treatment. Such systems have been proposed by Tokuhashi, Sioutos, Tomita, Van der Linden, and Bauer. The scores differ greatly in the kind of parameters assessed. The aim of this study was to evaluate the prognostic value of each score. Eight parameters were assessed for 69 patients (37 male, 32 female): location, general condition, number of extraspinal bone metastases, number of spinal metastases, visceral metastases, primary tumour, severity of spinal cord palsy, and pathological fracture. Scores according to Tokuhashi (original and revised), Sioutos, Tomita, Van der Linden, and Bauer were assessed as well as a modified Bauer score without scoring for pathologic fracture. Nineteen patients were still alive as of September 2006 with a minimum follow-up of 12 months. All other patients died after a mean period of 17 months after operation. The mean overall survival period was only 3 months for lung cancer, followed by prostate (7 months), kidney (23 months), breast (35 months), and multiple myeloma (51 months). At univariate survival analysis, primary tumour and visceral metastases were significant parameters, while Karnofsky score was only significant in the group including myeloma patients. In multivariate analysis of all seven parameters assessed, primary tumour and visceral metastases were the only significant parameters. Of all seven scoring systems, the original Bauer score and a Bauer score without scoring for pathologic fracture had the best association with survival (P \ 0.001). The data of the present study emphasize that the original Bauer score and a modified Bauer score without scoring for pathologic fracture seem to be practicable and highly predictive preoperative scoring systems for patients with spinal metastases. However, decision for or against surgery should never be based alone on a prognostic score but should take symptoms like pain or neurological compromise into account.

Predicting prognosis is the key factor in selecting the proper treatment modality for patients with spinal metastases. Therefore, various assessment systems have been designed in order to provide a basis for deciding the course of treatment. Such systems have been proposed by Tokuhashi, Sioutos, Tomita, Van der Linden, and Bauer. The scores differ greatly in the kind of parameters assessed. The aim of this study was to evaluate the prognostic value of each score. Eight parameters were assessed for 69 patients (37 male, 32 female): location, general condition, number of extraspinal bone metastases, number of spinal metastases, visceral metastases, primary tumour, severity of spinal cord palsy, and pathological fracture. Scores according to Tokuhashi (original and revised), Sioutos, Tomita, Van der Linden, and Bauer were assessed as well as a modified Bauer score without scoring for pathologic fracture. Nineteen patients were still alive as of September 2006 with a minimum follow-up of 12 months. All other patients died after a mean period of 17 months after operation. The mean overall survival period was only 3 months for lung cancer, followed by prostate (7 months), kidney (23 months), breast (35 months), and multiple myeloma (51 months). At univariate survival analysis, primary tumour and visceral metastases were significant parameters, while Karnofsky score was only significant in the group including myeloma patients. In multivariate analysis of all seven parameters assessed, primary tumour and visceral metastases were the only significant parameters. Of all seven scoring systems, the original Bauer score and a Bauer score without scoring for pathologic fracture had the best association with survival (P \ 0.001). The data of the present study emphasize that the original Bauer score and a modified Bauer score without scoring for pathologic fracture seem to be practicable and highly predictive preoperative scoring systems for patients with spinal metastases. However, decision for or against surgery should never be based alone on a prognostic score but should take symptoms like pain or neurological compromise into account.

Assessing the severity and prognosis of patients with craniocerebral damage is a major research area in medicine since it is a prevalent clinical disease. Acute craniocerebral injury, a common traumatic condition, is often caused by traffic accidents, collisions, and falls in daily life. Secondary craniocerebral injury refers to symptoms such as brain edema and intracranial hemorrhage after acute craniocerebral injury, which will aggravate the injury. Secondary craniocerebral injury can be avoided by effective and timely treatment, and real-time detection of brain edema and intracranial hemorrhage by non-invasive medical imaging is a solution. Therefore, non-invasive medical imaging technology has recently emerged as a new area of study. A new imaging technology, namely the brain injury detection technology based on electromagnetic induction, has been discovered after years of research on non-invasive detection of brain injury. Initially, electromagnetic induction technology was widely used in metal nondestructive testing. The human body, as a conductor, also has electromagnetic induction, allowing this technology to be used on the human body. This study reviews the technologies for detecting electromagnetic induction in cases of craniocerebral damage, including induced current electrical impedance tomography, magneto-acoustic tomography, and eddy current damping sensors for detection and imaging.

Se estudia la in uencia de la topología de redes eléctricas y alimentadores en el condicionamiento de las matrices asociadas a su modelado matemático. Se observa que el número de condición de estas matrices tiene una gran dependencia de la estructura de la red y que es afectado por el grado de enmallamiento de la red. Se incluyen resultados de simulaciones computacionales de distintos casos que permiten establecer una relación de consecuencia. Palabras clave: Matrices de redes, condicionamiento numérico, estabilidad numérica.

Se estudia la in uencia de la topología de redes eléctricas y alimentadores en el condicionamiento de las matrices asociadas a su modelado matemático. Se observa que el número de condición de estas matrices tiene una gran dependencia de la estructura de la red y que es afectado por el grado de enmallamiento de la red. Se incluyen resultados de simulaciones computacionales de distintos casos que permiten establecer una relación de consecuencia. Palabras clave: Matrices de redes, condicionamiento numérico, estabilidad numérica.

The object of this paper is to outline, describe and solve a parallel kinematics mechanism's manufacturing process. A detailed study of the different options, variants and possible solutions of the design will be carried out, explaining the convenience or not of each one of them and justifying the election of each option, considering the different outlined specifications. To complete this objective, a prototype of a parallel manipulator will be built, which must be able to validate and take out the summations of the whole process. The interest in building a prototype of a parallel manipulator borns in a previous works made by the authors about the kinematic and stiffness analyses of these manipulators.

Let A be an arbitrary matrix in which the number of rows, m, is considerably larger than the number of columns, n. Let the submatrix Ai,i=1,…,m, be composed of the first i rows of A. Let βi denote the smallest singular value of Ai, and let ki denote the condition number of Ai. In this paper, we examine the behavior of the sequences β1,…,βm, and k1,…,km. The behavior of the smallest singular values sequence is somewhat surprising. The first part of this sequence, β1,…,βn, is descending, while the second part, βn,…,βm, is ascending. This phenomenon is called “the smallest singular value anomaly”. The sequence of the condition numbers has a different character. The first part of this sequence, k1,…,kn, always ascends toward kn, which can be very large. The condition number anomaly occurs when the second part, kn,…,km, descends toward a value of km, which is considerably smaller than kn. It is shown that this is likely to happen whenever the rows of A satisfy two conditions: all the row...

For the simulation of magneto-quasi-static fields with finite integration implicit time domain (FI 2 TD) and finite integration frequency domain (FIFD) methods, a new technique is introduced which allows an improved geometric modeling of curved shape boundaries while maintaining orthogonal tensor product grids. The reduction of the geometrical error results in higher condition numbers of the linear algebraic systems. The resulting additional computational costs for the solution of these systems are shown to be small. Numerical results for test problems show the improvement of the solution accuracy with the new approach.

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to enforce uniqueness and select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is not smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to solve inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early-stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set, i.e. redundant information. More precisely we show that the linear systems can be used more than once along the iteration. Despite the simplicity of the idea, we show that this procedure brings surprising advantages in the numerical applications. We discuss various approaches to take advantage of redundant information, that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction through total variation. We confirm the efficiency of the proposed procedures, comparing the results with state-of-the-art methods.

The paper concerns numerical modelling of complicated GeoTechnical problems as mine stability, performance of underground radioactive waste repositories or microstructure modelling of geocomposites. The modelling uses finite elements and parallel computing based on overlapping domain decomposition. Important implementation issues for this approach are discussed: inexact solution of subproblems, creation of a global approximation of the problem by aggregation, stabilization of the outer CG type method and programming with the aid of MPI or OpenMP. The domain decomposition technique is also applied to the solution of saddle point systems arising from the use of mixed finite element method (FEM).

International audienceGiven a full column rank matrix A ∈ R m×n (m ≥ n), we consider a special class of linear systems of the form A ⊤ Ax = A ⊤ b + c with x, c ∈ R n and b ∈ R m. The occurrence of c in the right-hand side of the equation prevents the direct application of standard methods for least squares problems. Hence, we investigate alternative solution methods that, as in the case of normal equations, take advantage of the peculiar structure of the system to avoid unstable computations, such as forming A ⊤ A explicitly. We propose two iterative methods that are based on specific reformulations of the problem and we provide explicit closed formulas for the structured condition number related to each problem. These formula allow us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, that does not take into account the structure of the perturbations. The relevance of our estimates is shown on a set of synthetic test prob...

Abstract. Given a full column rank matrix A ∈ R (m ≥ n), we consider a special class of linear systems of the form AAx = Ab+ c with x, c ∈ R and b ∈ R. The occurrence of c in the right-hand side of the equation prevents the direct application of standard methods for least squares problems. Hence, we investigate alternative solution methods that, as in the case of normal equations, take advantage of the peculiar structure of the system to avoid unstable computations, such as forming AA explicitly. We propose two iterative methods that are based on specific reformulations of the problem and we provide explicit closed formulas for the structured condition number related to each problem. These formula allow us to compute a more accurate estimate of the forward error than the standard one used for generic linear systems, that does not take into account the structure of the perturbations. The relevance of our estimates is shown on a set of synthetic test problems. Numerical experiments hi...

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

Let A be a complex matrix with arbitrary Jordan structure and λ an eigenvalue of A whose largest Jordan block has size n. We review previous results due to Lidskii [U.S.S.R. Comput. Math. and Math. Phys., 1 (1965), pp. 73-85], showing that the splitting of λ under a small perturbation of A of order ε is, generically, of order ε 1/n . Explicit formulas for the leading coefficients are obtained, involving the perturbation matrix and the eigenvectors of A. We also present an alternative proof of Lidskii's main theorem, based on the use of the Newton diagram. This approach clarifies certain difficulties which arise in the nongeneric case and leads, in some situations, to the extension of Lidskii's results. These results suggest a new notion of Hölder condition number for multiple eigenvalues, depending only on the associated left and right eigenvectors, appropriately normalized, not on the Jordan vectors.

The following problem is addressed: given square matrices A and B, compute the smallest such that A + E and B + F have a common eigenvalue for some E, F with max( E 2 , F 2 ) ≤ . An algorithm to compute this quantity to any prescribed accuracy is presented, assuming that eigenvalues can be computed exactly.

The H∞ norm of a transfer matrix function for a control system is the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the stability region for the dynamical system (the left half-plane in the continuous-time case and the unit disk in the discrete-time case). After deriving some fundamental properties of spectral value sets, particularly the intricate relationship between the singular vectors of the transfer matrix and the eigenvectors of the corresponding perturbed system matrix, we extend an algorithm recently introduced by Guglielmi and Overton [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166-1192] for approximating the maximal real part or modulus of points in a matrix pseudospectrum to spectral value sets, characterizing its fixed points. We then introduce a Newton-bisection method to approximate the H∞ norm, for which each step requires optimization of the real part or the modulus over an ε-spectral value set. Although the algorithm is guaranteed only to find lower bounds on the H∞ norm, it typically finds good approximations in cases where we can test this. It is much faster than the standard Boyd-Balakrishnan-Bruinsma-Steinbuch algorithm to compute the H∞ norm when the system matrices are large and sparse and the number of inputs and outputs is small. The main work required by the algorithm is the computation of the spectral abscissa or radius of a sequence of matrices that are rank-one perturbations of a sparse matrix.

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii's analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra.

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

Principal Component Analysis (PCA) aims to reduce the dimensions of datasets by transforming them into uncorrelated Principal Components (PCs), retaining most of the data's variation with fewer components. However, standard PCA struggles in high-dimensional settings where there are more variables than observations due to limitations in covariance estimation. PCA relies on covariance matrices to measure variable relationships, using eigenvectors to determine data distribution directions and assessing eigenvalues' significance. This article examines the pros and cons of estimating high-dimensional covariance matrices and emphasizes the importance of well-conditioned covariance estimation for accurate finite sample PCA. Various methods are available for estimating population covariance, among which Ledoit-Wolf estimation is deemed optimal in scenarios where the number of observations is smaller than the number of variables. However, it tends to excessively shrink the sample covariates matrix, resulting in an underestimation of the true eigen spectrum and a dearth of sparsity. Therefore, there's a need for sparse and well-conditioned covariance matrix estimation to enhance PC estimation accuracy.

The concept of pseudospectrum was introduced by L. N. Trefethen to explain the behavior of nonnormal operators. Many phenomena (for example, hydrodynamic instability and convergence of iterative methods for linear systems) cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra. The straightforward way to compute pseudospectra involves many applications of the singular value decomposition. This paper presents several fast continuation methods to calculate pseudospectra of di erent types of matrices.

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an inÿnite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. Schwarz methods for nonoverlapping subdomains also exist but they have not been popular because of their slow convergence. These methods contain a free parameter in the Robin boundary condition of each subdomain problem. The slow convergence can be attributed to an improper choice of this parameter. In this paper, two models are proposed to give guidance to the choice of this parameter. For the Poisson equation on rectangular domains, these models suggest very simple expressions for the parameter in terms of the dimensions of the subdomain. Numerical experiments verify their e ectiveness. When used as a preconditioner, it is demonstrated numerically in some examples that the algorithm is quite e cient.

Consider a random sum η 1 v 1 + • • • + η n v n , where η 1 , . . . , η n are independently and identically distributed (i.i.d.) random signs and v 1 , . . . , v n are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(η In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v 1 , . . . , v n are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number.

Consider a random sum η 1 v 1 + • • • + η n v n , where η 1 , . . . , η n are independently and identically distributed (i.i.d.) random signs and v 1 , . . . , v n are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(η In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v 1 , . . . , v n are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number.

We present here the generic parallel computational framework in C++ called Feel++ for the mortar finite element method with the arbitrary number of subdomain partitions in 2D and 3D. An iterative method with block-diagonal preconditioners is used for solving the algebraic saddle-point problem arising from the finite element discretization. Finally we present a scalability study and the numerical results obtained using Feel++ library.

In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximation of the convex hull without loss of accuracy.

We consider a single outbreak susceptible-infected-recovered (SIR) model and corresponding estimation procedures for the effective reproductive number R(t). We discuss the estimation of the underlying SIR parameters with a generalized least squares (GLS) estimation technique. We do this in the context of appropriate statistical models for the measurement process. We use asymptotic statistical theories to derive the mean and variance of the limiting (Gaussian) sampling distribution and to perform post statistical analysis of the inverse problems. We illustrate the ideas and pitfalls (e.g., large condition numbers on the corresponding Fisher information matrix) with both synthetic and influenza incidence data sets.

Arithmetic systems such as those based on IEEE standards currently make no attempt to track the propagation of errors. A formal error analysis, however, can be complicated and is often confined to the realm of experts in numerical analysis. In recent years, there has been a resurgence of interest in automated methods for accurately monitoring the error propagation. In this article, a floatingpoint system based on significance arithmetic will be described. Details of the implementation in Mathematica will be given along with examples that illustrate the design goals and differences over conventional fixed-precision floating-point systems.

Distributed matrix computations over large clusters can suffer from the problem of slow or failed worker nodes (called stragglers) which can dominate the overall job execution time. Coded computation utilizes concepts from erasure coding to mitigate the effect of stragglers by running "coded" copies of tasks comprising a job; stragglers are typically treated as erasures. While this is useful, there are issues with applying, e.g., MDS codes in a straightforward manner. Several practical matrix computation scenarios involve sparse matrices. MDS codes typically require dense linear combinations of submatrices of the original matrices which destroy their inherent sparsity. This is problematic as it results in significantly higher worker computation times. Moreover, treating slow nodes as erasures ignores the potentially useful partial computations performed by them. Furthermore, some MDS techniques also suffer from significant numerical stability issues. In this work we present schemes that allow us to leverage partial computation by stragglers while imposing constraints on the level of coding that is required in generating the encoded submatrices. This significantly reduces the worker computation time as compared to previous approaches and results in improved numerical stability in the decoding process. Exhaustive numerical experiments on Amazon Web Services (AWS) clusters support our findings.

An upper bound for the infinity norm for the inverse of Dashnic-Zusmanovich type matrices is given. It is proved that the upper bound is sharper than the well-known Varah's bound for strictly diagonally dominant matrices. By introducing a new subclass of P -matrices: Dashnic-Zusmanovich type B-matrices (DZ-type-B-matrices), and using the proposed infinity norm bound, an error bound is given for the linear complementarity problems of DZ-type-B-matrices. We also give a new pseudospectra localization to measure the distance to instability.

The problem of determining matrix inertia is very important in many applications, for example, in stability analysis of dynamical systems. In the (point-wise) H-matrix case, it was proven that the diagonal entries solely reveal this information. This paper generalises these results to the block H-matrix cases for 1, 2, and ∞ matrix norms. The usefulness of the block approach is illustrated on 3 relevant numerical examples, arising in engineering.

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.

In this work we study the condition number of the least square matrix corresponding to scale free networks. We compute a theoretical lower bound of the condition number which proves that they are ill conditioned. Also, we analyze several matrices from networks generated with the linear preferential attachment model showing that it is very difficult to compute the power law exponent by the least square method due to the severe lost of accuracy expected from the corresponding condition numbers.

The paper deals with the debonding fiber-matrix process in composite materials. A couple of papers has been focused on this problem by the authors of this paper. As usual, the influence of separate pure normal and pure shear energy has been studied for obtaining the overall material properties. Such an approach is a simplification of the problem describing the mechanical behavior of the interface between fibres and the matrix in composite materials, since the standard procedure consisting of the superposition of both normal and shear influences is no longer admissible due to the strongly nonlinear behavior of the process. In this paper a more complex development of the debonding zones is shown, namely, responses of successively applied normal and then shear load and also first shear load and then normal load are observed. The physical behaviour of the interface is non-convex. This assertion immediately follows from the penalty formulation of the problem as published recently by the first author. The penetration of fiber into the matrix is not allowed in every case. In this contribution we start with a definition of the model describing the transfer of elastic stresses from the matrix to the fiber. Then, the contact problem is formulated in a manner leading to a very fast Uzawa's algorithm for its solution. In order to speed up the iterative solution influence matrices are created before the iteration. The approach turns to a similar one known as generalized transformation field analysis. Debonding processes in the interfacial zone are illustrated by examples.

Numerical methods seem to be the cheapest tool for assessing underground structures. However, there exists one obstacle in applications of any numerical method which is a lack of information concerning the input data, particularly the knowledge of material properties. If the theory of damage should be involved in the formulation of the problem to be solved, special treatment is required. The methods, which are extensively used, start with realization of the trial body by a continuum. We can name "Cohesive zone method", which deals with Barenblatt's theory, for example. In our problem such methods are difficult to apply and exhibit unreal behavior, according to a couple of test examples. This is why test experiments have been carried out to gain further information towards a reasonable approach for solving the problem. The free hexagon method seems to be very promising. The method will be described and basic formulas will be derived, and then applications to assessment of structural strength will be presented.

The pull-out problem has frequently been solved in cracking of composite structures. Previously, several numerical studies were carried out using the FEM, [4], and the BEM, , and the results were compared with available literature having mainly been focused on experiments. The comparison of both numerical and experimental results were in a good agreement. A remarkable feature of the mathematical solution occurred: a cracking was initiated not only at the face of the fiber-matrix system but also inside the trial body. As the previous formulations accepted the linear behavior of both the fiber and the matrix only, a natural question arises: what happens when the matrix will behave more realistic, i. e. the material of the matrix will obey the von Mises-Huber-Heiicky criterion of plasticity. The answer to this question is the objective of this paper. In this study of lag model the geometry of the problem is simplified in order to use high accuracy boundary element procedure together with Uzawa's algorithm. The fine mesh of the boundary elements is necessary for the stability of an iterative process which covers the Uzawa/s algorithm and plastic behavior of the matrix, using the TEA, [2], . Note that high accuracy approximation is always needed when solving crack problem (although this crack problem is solved as a contact problem, see e. g. references in ). The deboiiding can be caused due to several physical models. First, no tension tractions on the interface may occur. Also, some friction law (as Coulomb law, Mohr-Coulomb law, shear bond strength r&) are often iirtro-Typeset by

Numerical methods seem to be the cheapest tool for assessing different types of structures. If the theory of damage should be involved into the formulation of the problem to be solved, special treatment is required. The methods, which are extensively used, start with realization of the trial body by a continuum. For example, we can name "Cohesive zone method", which deals with Barenblatt's theory. In our problem such methods are on one side uneasily applicable and on the other side exhibit unreal behavior, according to a couple of test examples. This is why test experiments have been carried out to get knowledge about a reasonable approach for solving the problem. The free hexagon method seems to be very promising. First, the method will be described and basic formulas will be derived, and then some applications to rock bumps will be presented. In comparison with some previous works of the author of this paper, the method presented here involves a time-dependent problem with Newtonian forces, which are caused by contact forces of moving particles. It simplifies the body of the earth (soil) to a set of hexagons, which are, or are not in mutual contact. The material properties of the hexagons are determined from the state of stresses. In this paper we adjust the idea of PFC (particle flow code), which starts with balls instead of hexagons and dynamical equilibrium, while in our case the static equilibrium is considered. The hexagons represent a typical shape of the grains the earth consists of. The model proposed in this paper may, in contrast to modern numerical methods (FEM, BEM, etc.), enable one to disconnect the medium described by the balls, when needed (e.g. providing certain requirements on tensile strength is violated). The most natural contact conditions -Mohr-Coulomb hypotheses -may be simply introduced and, after imposing such contact conditions, the localized damage, or "cracking" can be discovered. The stability then depends on the "measure" of the touched zone.

In this paper, a time dependent (dynamical equilibrium) free hexagon DEM is formulated and solved. The main application is found in geomechanics, namely in bumps occurrence in deep mines. The time factor is included in a natural way in the model of discrete elements created by the boundary element method. One of the most important phenomena is the velocity of excavation. In the deep mines the method of depositing packs and its mechanical properties are also decisive. Their mutual coupling can principally influence the safety against bumps. For a correct understanding of the behavior of the rock aggregate (coal seam vs. overburden), the nucleation of cracks finally leading to bumps has to be treated as time dependent, while so far it was observed only from statical equilibrium. According to new experiments and results from accessible literature, a dynamical effect has to be included in thr formulation. Contact problems leading to bumps occurrence in deep mines have been solved in many of the papers of the present author for the static case. Either Lagrangian multipliers or penalty formulation were used. The new formulation has to be submitted in terms of a penalty, which if high enough (bond effect of adjacent elements) suppresses the influence of time. By including the interface properties with the lumped inertia mass of the elements, complex nucleation can be studied and the information on possible rock bursts is improved. From some examples it was shown in the static case that the behavior at the face of longwall mining is close to that near the crack tip, and the differences in material properties of coal and overburden are also not negligible. These factors are also expected to be important in the case of dynamic problems. Some examples show the application of the procedure proposed.

In this paper we obtain the formula for computing the condition number of a complex matrix, which is related to the outer generalized inverse of a given matrix. We use the Schur decomposition of a matrix. We characterize the spectral norm and the Frobenius norm of the relative condition number of the generalized inverse, and the level-2 condition number of the generalized inverse. The sensitivity for the generalized Drazin inverse solution of linear systems is presented. We also present the structured perturbation of the generalized inverse.

We prove mean comparison results from a different perspective, where we introduce the concept of partial convolutions. For a parabolic initial data problem on the whole domain of dimension n we consider data functions which live on a subspace of lower dimension k and coefficient functions which live on the whole space or some subspace of dimension l. The pair of natural numbers (k, l) is called a strong partial convolution pair if for some coordinate transformations the coefficients functions of the equation and the initial data functions live in complementary linear spaces of dimension k and l respectively. Here the qualification ’strong’ indicates that this transformation exists without any further restriction concerning the intersection of the original linear subspaces involved, and that refined concepts are possible in this respect. For purely second order parabolic equations we show that (1, n) for any n ≥ 2 is a strong partial convolution pair. As a consequence new criteria fo...

The results presented here constitute a brief summary of an on-going multi-year effort to investigate hierarchical/wavelet bases for solving PDE's and establish a rigorous foundation for these methods. A new, hierarchical, wavelet-Galerkin solution strategy based upon the Donovan-Geronimo-Hardin-Massopust (DGHM) compactly-supported multi-wavelet is presented for elliptic partial differential equations. This multi-scale wavelet-Galerkin method uses a wavelet transform to yield nearly mesh independent condition numbers for elliptic problems as opposed to the multi-scaling functions that yield condition numbers which increase as the square of the mesh size. In addition, the results of von Neumann analyses for the DGHM multi-wavelet element and the Reproducing Kernel Particle Method (RKPM) are presented for model hyperbolic partial differential equations. RKPM exhibits excellent dispersion characteristics using a consistent mass matrix with the proper choice of refinement parameter and mass matrix formulation. In comparison, the wavelet-Galerkin formulation using the DGHM element delivers a frequency response comparable to a Bubnov-Galerkin formulation with a quadratic element. and on the DGHM multi-wavelet element[2]. . This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recornmend;rtion, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Introduction Wavelet bases promise the capability to compute multi-scale solutions to partial differential equations with potentially higher convergence rates than conventional finite difference and finite element methods, and their built-in adaptive nature offers the possibility of "automatic" adaptivity. Similarly, the Reproducing Kernel Particle Method (RKPM) has many attractive properties that make it ideal for treating a broad class of physical problems. For example, RKPM may be implemented in a "mesh-full" or a "mesh-free" manner and provides the ability to selectively tune the method, via the selection of the refinement parameter and window function, in order to achieve the requisite numerical performance. RKPM also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi-scale problems. Despite the promise of wavelet bases and RKPM, the application of new numerical methods to physical proble...

The Reproducing Kernel Particle Method (RKPM) is a discretization technique for partial differential equations that uses the method of weighted residuals, classical reproducing kernel theory and modified kernels to produce either "mesh-bee" or '~mesh-full" methods. A1though RKPM has many appealing attributes, the method is new, and its numerical performance is just beginning to be quantified. In order to address the numerical performance of RKPM, von Neumann analysis is performed for semi-discretizations of three model one-dimensional PDEs. The von Neumann analyses results are used to examine the global and asymptotic behavior of the semi-discretizations. The model PDEs considered for this analysis include the parabolic and hyperbolic (first and-second-order wave) equations. Numerical diffusivity for the former and phase speed for the later are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero. Group speed is also presented for the hyperbolic problems. Excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter. In contrast, the row-sum lumped mass matrix formulation severely degraded performance. The asymptotic analysis indicates that very good rates of convergence are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter.

The adjoint method application in variational data assimilation provides a way of obtaining the exact gradient of the cost function J with respect to the control variables. Additional information may be obtained by using second order information. This paper presents a second order adjoint model (SOA) for a shallow-water equation model on a limited-area domain. One integration of such a model yields a value of the Hessian (the matrix of second partial derivatives, VzJ) multiplied by a vector or a column of the Hessian of the cost function with respect to the initial conditions. The SOA model was then used to conduct a sensitivity analysis of the cost function with respect to distributed observations and to study the evolution of the condition number (the ratio of the largest to smallest eigenvalues) of the Hessian during the course of the minimization. The condition number is strongly related to the convergence rate of the minimization. It is proved that the Hessian is positive definite during the process of the minimization, which in turn proves the uniqueness of the optimal solution for the test problem. Numerical results show that the sensitivity of the response increases with time and that the sensitivity to the geopotential field is larger by an order of magnitude than that to the u and v components of the velocity field. Experiments using data from an ECMWF analysis of the First Global Geophysical Experiment (FGGE) show that the cost function J is more sensitive to observations at points where meteorologically intensive events occur. Using the second order adjoint shows that most changes in the value of the condition number of the Hessian occur during the first few iterations of the minimization and are strongly correlated to major large-scale changes in the reconstructed initial conditions fields. * A brief description of the FOA model is provided in Appendix A.

This paper focuses on the spectral properties of the mass and stiffness matrices for acoustic wave problems discretized with Isogeometric analysis (IGA) collocation methods in space and Newmark methods in time. 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, varying the polynomial degree p, mesh size h, regularity k, of the IGA discretization and the time step ∆t and parameter β of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results are comparable with the available spectral estimates for IGA Galerkin matrices associated to the Poisson problem with Dirichlet boundary conditions, and in some cases the IGA collocation results are better than the corresponding IGA Galerkin estimates.

For the iterative solution of the Schur complement system associated with the discretization of an elliptic problem by means of a triangular spectral element method (TSEM), Neumann-Neumann (NN) type preconditioners are constructed and studied. The TSEM approximation, based on Fekete nodes, is a generalization to non-tensorial elements of the classical Gauss-Lobatto-Legendre quadrilateral spectral elements. Numerical experiments show that the TSEM Schur complement condition number grows linearly with the polynomial approximation degree, N, and quadratically with the inverse of the mesh size, h. NN preconditioners for the Schur complement allow to reduce the N-dependence of the condition number, by solving local Neumann problems on each spectral element, and to eliminate the h-dependence if an additional coarse solver is employed. Numerical results indicate that, in spite of the more severe ill-conditioning, the condition number of the TSEM preconditioned operator satisfies the same bound as that of the standard SEM, i.e., Ch -2 (1 + log N) 2 for one-level NN preconditioning and C(1 + log N) 2 for two-level Balancing Neumann-Neumann (BNN) preconditioning.