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  • 1.
    Berg, Jan-Erik
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten E.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gradin, Per A.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    On the energy consumption for crack development in fibre wall in disc refining - A micromechanical approach2009In: Holzforschung, ISSN 0018-3830, E-ISSN 1437-434X, Vol. 63, no 2, p. 204-210Article in journal (Refereed)
    Abstract [en]

    An analytical model has been applied to calculate the acquired strain energy density in order to achieve a certain damage state in a softwood fibre by uniaxial tension or shear load. The energy density was found to be dependent on the microfibril angle in the middle secondary wall, the loading case, the thicknesses of the fibre cell wall layers, and conditions such as moisture content and temperature. At conditions, prevailing at the entrance of the gap between the plates in a refiner and at relative high damage states, more energy is needed to create cracks at higher microfibril angles. The energy density was lower for earlywood compared to latewood fibres. For low microfibril angles, the energy density was lower for loading in shear compared to tension for both earlywood and latewood fibres. Material parameters, such as initial damage state and specific fracture energy, were determined by fitting of input parameters to experimental data.

  • 2. Cheng, Aijie
    et al.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Finite difference methods for saturated-unsaturated flow in porous media2003Report (Other academic)
  • 3.
    Edström, Per
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Granberg, H
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Some Ideas on Models and Methods for Light Scattering in Paper2001Report (Other academic)
    Abstract [en]

    In this report a short overview is given on existing models for simulating light scattering in paper. The well known Kubelka-Munk model somewhat oversimplifies the problem. Newer discrete ordinate models take into account more aspects, e.g. angle resolved scattering, but have intrinsic ill-conditioned problems that have to be overcome. No existing models are designed to solve the inverse problem, i.e. finding model parameters given real light scattering measurements. Therefore we propose a Stable Multilayer Discrete Ordinate Radiative Transfer model, SM-DORT, to simulate the scattering of light in coated paper and similar structures, and to solve the corresponding inverse problem. We discuss improvements of existing models, for making SM-DORT stable to give reliable results in spite of intrinsic ill-conditioned problems, and for making it fast to efficiently solve the inverse problem. In this report we also cover some aspects of inverse problems in general, we give some ideas on further model improvement to take into account more aspects of light scattering in paper, and we discuss model validation, using experiments and existing models.

  • 4.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Gradin, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Isaksson, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    A Note on Wet Paper Web Adhesion Strength2012In: Journal of Testing and Evaluation, ISSN 0090-3973, E-ISSN 1945-7553, Vol. 40, no 4, p. 682-686Article in journal (Refereed)
    Abstract [en]

    This work is concerned with the determination of the adhesion strength between a paper web and an adhesive surface. Edvardsson et al. [Edvardsson, S., Gradin, P., and Isaksson, P., "On Dissipative Effects of Paper Web Adhesion Strength," Int. J. Solids Struct., Vol. 48(1), 2011, pp. 24-30] suggested recently a model that takes into account the energy dissipation caused by elastic plastic deformation in the bent structure of a paper specimen. This model is further developed and investigated in the present work. A linear relation in plastic dissipation is discovered facilitating a novel analysis of the peeling tension and a more convenient determination of the proper adhesion strength. Industrial relevant examples are made with wet newsprint and kraft stock. A straightforward experimental procedure for determining the consistent adhesion strength is suggested. It is found that the agreement between the model and the experimental observations is good.

  • 5.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Persson, Johan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    The dynamical functional particle method: an approach for boundary value problems2012In: Journal of applied mechanics, ISSN 0021-8936, E-ISSN 1528-9036, Vol. 79, no 2, p. art. no. 021012-Article in journal (Refereed)
    Abstract [en]

    The present work is concerned with new ideas of potential value for solving differential equations. First, a brief introduction to particle methods in mechanics is made by revisiting the vibrating string. The full case of nonlinear motion is studied and the corresponding nonlinear differential equations are derived. It is suggested that the particle origin of these equations is of more general interest than usually considered. A novel possibility to develop particle methods for solving differential equations in a direct way is investigated. The dynamical functional particle method (DFPM) is developed as a solution method for boundary value problems. DFPM is based on the concept of an interaction functional as a dynamical force field acting on quasi particles. The approach is not limited to linear equations. We exemplify by applying DFPM to several linear Schrödinger type of problems as well as a nonlinear case. It is seen that DFPM performs very well in comparison with some standard numerical libraries. In all cases, the convergence rates are exponential in time. © 2012 American Society of Mechanical Engineers.

  • 6.
    Edvardsson, Sverker
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Neuman, Magnus
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Edström, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Olin, Håkan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
    Role of the particle method DFPM for solving linear equationsManuscript (preprint) (Other academic)
  • 7. Eriksson, J
    et al.
    Gulliksson, Mårten
    Lindström, P
    Wedin, P-Å
    Regularization Tools for Training Feed-Forward Neural Networks1998In: Optimization Methods and Software, ISSN 1055-6788, Vol. 10, no 1, p. 49-69Article in journal (Refereed)
  • 8. Eriksson, J
    et al.
    Wedin, P.A.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Söderkvist, I
    Regularization methods for uniformly rank-deficient nonlinear least-squares problems2005In: Journal of Optimization Theory and Applications, ISSN 0022-3239, E-ISSN 1573-2878, Vol. 127, no 1, p. 1-26Article in journal (Refereed)
    Abstract [en]

    In solving the nonlinear least-squares problem of minimizing ||f(x)||22, difficulties arise with standard approaches, such as the Levenberg-Marquardt approach, when the Jacobian of f is rank-deficient or very ill-conditioned at the solution. To handle this difficulty, we study a special class of least-squares problems that are uniformly rank-deficient, i.e., the Jacobian of f has the same deficient rank in the neighborhood of a solution. For such problems, the solution is not locally unique. We present two solution tecniques: (i) finding a minimum-norm solution to the basic problem, (ii) using a Tikhonov regularization. Optimality conditions and algorithms are given for both of these strategies. Asymptotical convergence properties of the algorithms are derived and confirmed by numerical experiments. Extensions of the presented ideas make it possible to solve more general nonlinear least-squares problems in which the Jacobian of f at the solution is rank-deficient or ill-conditioned.

  • 9. Eriksson, Jerry
    et al.
    Gulliksson, Mårten E.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares2004In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 73, no 248, p. 1865-1883Article in journal (Refereed)
    Abstract [en]

    A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as $min_{x} 1/2Vert f_{2}(x) Vert _{2}^{2}$subject to the constraints $f_{1}(x) = 0$, the Jacobian $J_{1} = partial f_{1}/ partial x$ and/or the Jacobian $J = partial f/ partial x$, $f = [f_{1};f_{2}]$, may be ill conditioned at the solution. We analyze the important special case when $J_{1}$ and/or $J$ do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in $mathbb{R} ^{n}$. Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians. Finally we present computational tests on constructed problems verifying the local analysis.

  • 10.
    Feng, Tao
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Edström, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Levenberg-Marquardt Methods for Parameter Estimation Problems in the Radiative Transfer Equation2007In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 3, p. 879-891, article id 002Article in journal (Refereed)
    Abstract [en]

    A discrete ordinate method is developed for solving the radiative transfer equation, and the corresponding parameter estimation problem is given a least-squares formulation. Two Levenberg-Marquardt methods, a feasible-path approach and an SQP type method, are analyzed and compared. A sensitivity analysis is given, and it is shown how it can be used for designing measurements with minimal impact of measurement noise. Numerical experiments are performed to exemplify the usefulness of the theory.

  • 11.
    Feng, Tao
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Liu, W
    A Posteriori Error Estimators of Recovery Type for Parameter Estimation Problem in Linear Elastic Problem2007In: Free and Moving Boundaries: Analysis, Simulation and Control, Boca Raton: CRC Press, 2007, p. 395-410Chapter in book (Other academic)
  • 12.
    Feng, Tao
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Liu, W
    Institute of Mathematics and Statistics, University of Kent, Canterbury, CT2 7NF, United Kingdom.
    Adaptive finite element methods for the identification of elastic constants2006In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 26, no 2, p. 217-235Article in journal (Refereed)
    Abstract [en]

    In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.

  • 13.
    Feng, Tao
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Liu, W.B.
    Mesh Adaptation and Optimal Design of the Measurement Points for Parameter Estimation ProblemManuscript (Other academic)
  • 14.
    Feng, Tao
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Liu, Wenbin
    Adaptive finite element methods for parameter estimation problems in linear elasticity2009In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 6, no 1, p. 17-32Article in journal (Refereed)
    Abstract [en]

    In this paper, the Lame coefficients in the linear elasticity problem are estimated by using the measurements of displacement. Some a posteriori error estimators for the approximation error of the parameters are derived, and then adaptive finite element schemes are developed for the discretization of the parameter estimation problem, based on the error estimators. The Gauss-Newton method is employed to solve the discretized nonlinear least-squares problem. Some numerical results are presented.

  • 15.
    Gradin, Per
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Berg, Jan-Erik
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Nyström, Staffan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Measuring forces in a refiner plate gap2011In: Proceedings of International Mechanical Pulping Conference, 2011, p. 81-83Conference paper (Refereed)
  • 16.
    Gradin, Per
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Isaksson, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    A Note on the Co-linearity of forces and Displacements in an Elastic Structure2011In: Journal of Applied Fluid Mechanics, ISSN 1735-3572, E-ISSN 1735-3645, Vol. 78, no 6, p. Art. no. 4003912-Article in journal (Refereed)
    Abstract [en]

    Theconditions under which force vectors and corresponding displacement vectors becomeco-linear are investigated under the assumption of a linear elasticstructure and for an arbitrary number of loading points. Itis shown that there exist an infinite number of directionsalong which the load and displacement vectors in each loadingpoint coincide. Moreover, the problem of co-linearity is analogous tothe problem of finding the extreme values of the workperformed on an elastic structure under the constraint that eachforce has a given magnitude. The result for a finitenumber of loading points is extended to a continuous loaddistribution on the boundary of an elastic structure, i.e., itis possible to find an infinite number of load distributionssuch that the displacement in a point on the boundaryis co-linear with the boundary stress vector in that samepoint.

  • 17. Gulliksson, Mårten
    Algorithms for the partial procrustes problem1995Report (Other academic)
  • 18. Gulliksson, Mårten
    Backward error analysis for the constrained and weighted linear least squares problem when using the weighted $QR$ factorization1995In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 16, no 2, p. 675-687Article in journal (Refereed)
    Abstract [en]

    Backward errors are derived for the solution of the constrained and weighted linear least squares problem when using the weighted $QR$ factorization with column pivoting, [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 1298�1313]. On the way to achieving this goal we attain a detailed description of the errors produced when using our $M$-invariant reflections.

  • 19. Gulliksson, Mårten
    Iterative refinement for constrained and weighted linear least squares1994In: Bit: numerical mathematics, ISSN 0006-3835, Vol. 34, no 2, p. 239-253Article in journal (Refereed)
    Abstract [en]

    We present an algorithm for mixed precision iterative refinement on the constrained and weighted linear least squares problem, the CWLSQ problem. The approximate solution is obtained by solving the CWLSQ problem with the weightedQR factorization [6]. With backward errors for the weightedQR decomposition together with perturbation bounds for the CWLSQ problem we analyze the convergence behaviour of the iterative refinement procedure. In the unweighted case the initial convergence rate of the error of the iteratively refined solution is determined essentially by the condition number. For the CWLSQ problem the initial convergence behaviour is more complicated. The analysis shows that the initial convergence is dependent both on the condition of the problem related to the solution,x, and the vector lambda=Wr, whereW is the weight matrix andr is the residual. We test our algorithm on two examples where the solution is known and the condition number of the problem can be varied. The computational test confirms the theoretical results and verifies that mixed precision iterative refinement, using the system matrix and the weightedQR decomposition, is an effective way of improving an approximate solution to the CWLSQ problem.

  • 20. Gulliksson, Mårten
    KKT conditions for rank-deficient nonlinear least-square problems with rank-deficient nonlinear constraints1999In: Medietrender och papperskvalitet, ISSN 0022-3239, Vol. 100, no 1, p. 145-160Article in journal (Refereed)
    Abstract [en]

    In nonlinear least-square problems with nonlinear constraints, the function $$left. {(1/2)} right|left. {f_2 (x)} right|_2^2$$ , where f 2 is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f 1(x)=0. This problem is ill-posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill-posed if either the Jacobian of f 1 or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss�Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.

  • 21. Gulliksson, Mårten
    On the modified Gram-Schmidt algorithm for weighted and constrained linear least squares problems.1995In: Bit: numerical mathematics, ISSN 0006-3835, Vol. 35, no 4, p. 453-468Article in journal (Refereed)
    Abstract [en]

    A framework and an algorithm for using modified Gram-Schmidt for constrained and weighted linear least squares problems is presented. It is shown that a direct implementation of a weighted modified Gram-Schmidt algorithm is unstable for heavily weighted problems. It is shown that, in most cases it is possible to get a stable algorithm by a simple modification free from any extra computational costs. In particular, it is not necessary to perform reorthogonalization. Solving the weighted and constrained linear least squares problem with the presented weighted modified Gram-Schmidt algorithm is seen to be numerically equivalent to an algorithm based on a weighted Householder-likeQR factorization applied to a slightly larger problem. This equivalence is used to explain the instability of the weighted modified Gram-Schmidt algorithm. If orthogonality, with respect to a weighted inner product, of the columns inQ is important then reorthogonalization can be used. One way of performing such reorthogonalization is described. Computational tests are given to show the main features of the algorithm.

  • 22.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Carlsson, Nils
    Department of Mathematics, Luleå University, Sweden.
    Surface Fitting with NURBS - a Gauss Newton with Trust Region Approach: 13th WSEAS International Conference on APPLIED MATHEMATICS Puerto de la Cruz, SPAIN, DEC 15-17, 20082008In: APPLIED AND COMPUTATIONAL MATHEMATICS, 2ND EDITION, AG LOANNOU THEOLOGOU 17-23, 15773 ZOGRAPHOU, ATHENS, GREECE: World Scientific and Engineering Academy and Society, 2008, p. 169-174Conference paper (Refereed)
    Abstract [en]

    This paper shows a new approach for non-linear least squares fitting with NURBS as curves and surfaces to measured data by the Gauss-Newton method. A Trust Region algorithm is used to reach global convergence as well as variable substitution and simple bounds.

  • 23.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Edvardsson, Sverker
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    Lind, Andreas
    Mid Sweden University, Faculty of Science, Technology and Media, Department of applied science and design.
    The dynamical functional method2012Other (Other academic)
  • 24.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Jin, Xiao-Qing
    Wei, Yi-Min
    Perturbation bounds for constrained and weighted least squares problems2002In: Linear Algebra and Its Applications, ISSN 0024-3795, Vol. 349, p. 221-232Article in journal (Refereed)
    Abstract [en]

    We derive perturbation bounds for the constrained and weighted linear least squares (LS) problems. Both the full rank and rank-deficient cases are considered. The analysis generalizes some results of earlier works.

  • 25.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Söderkvist, I.
    Watson, G.A.
    Implicit surface fitting using directional constraints2001In: Bit: numerical mathematics, ISSN 0006-3835, Vol. 41, no 2, p. 308-321Article in journal (Refereed)
    Abstract [en]

    A commonly used technique for fitting curves and surfaces to measured data is that known as orthogonal distance regression, where the sum of squares of orthogonal distances from the data points to the surface is minimized. An alternative has recently been proposed for curves and surfaces which are parametrically defined, which minimizes the sum of squares in given directions which depend on the measuring process. In addition to taking account of that process, it is claimed that this technique has the advantage of complying with traditional fixed-regressor assumptions, enabling standard inference theory to apply. Here we consider extending this idea to curves and surfaces where the only assumption made is that there is an implicit formulation. Numerical results are given to illustrate the algorithmic performance.

  • 26. Gulliksson, Mårten
    et al.
    Söderkvist, Inge
    Wedin, Per-Åke
    Algorithms for constrained and weighted nonlinear least squares.1997In: SIAM Journal on Optimization, ISSN 1052-6234, Vol. 7, no 1, p. 208-224Article in journal (Refereed)
    Abstract [en]

    A hybrid algorithm consisting of a Gauss--Newton method and a second-order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed, and tested. One of the advantages of the algorithm is that arbitrarily large weights can be handled and that the weights in the merit function do not get unnecessarily large when the iterates diverge from a saddle point. The local convergence properties for the Gauss--Newton method are thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss--Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss--Newton method are not too ill conditioned, global convergence towards a first-order KKT point is proved.

  • 27. Gulliksson, Mårten
    et al.
    Wedin, Per-Åke
    Modifying the $QR$-decomposition to constrained and weighted linear least squares1992In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 13, no 4, p. 1298-1313Article in journal (Refereed)
    Abstract [en]

    A new way of looking at a class of methods for the weighted linear least squares problem $min _x | M^{ - ( 1/2 )} ( b - Ax ) |_2 $ where $M = {operatorname{diag}}(mu _i )$ is presented by introducing a modified QR-decomposition with $Q$$M$-invariant, i.e., $QMQ^T = M$. One of the main advantages with this approach is that linear constraints are easily incorporated by letting the corresponding diagonal elements in $M$ become zero. Householder reflections are generalized to $M$-invariant reflections, and an algorithm for solving the constrained and weighted linear least squares problem is described. The system equations (or the augmented system equations) are used to derive condition numbers, and the connection between these condition numbers and the rounding error in the solution is investigated.

  • 28.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Wedin, Per-Åke
    Perturbation theory for generalized and constrained linear least squares2000In: Numerical Linear Algebra with Applications, ISSN 1070-5325, Vol. 7, no 4, p. 181-195Article in journal (Refereed)
    Abstract [en]

    The perturbation analysis of weighted and constrained rank-deficient linear least squares is difficult without the use of the augmented system of equations. In this paper a general form of the augmented system is used to get simple perturbation identities and perturbation bounds for the general linear least squares problem both for the full-rank and rank-deficient problem. Perturbation identities for the rank-deficient weighted and constrained case are found as a special case. Interesting perturbation bounds and condition numbers are derived that may be useful when considering the stability of a solution of the rank-deficient general least squares problem. Copyright © 2000 John Wiley & Sons, Ltd.

  • 29. Gulliksson, Mårten
    et al.
    Wedin, Per-Åke
    Using the nonlinear L-curve and its dual1999In: Progress in industrial mathematics at ECMI 98: European conference on mathematics in industry, 1999, p. 162-170Conference paper (Other academic)
  • 30.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Wedin, P-Å
    The use and properties of Tikhonov filter matrices.2000In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, Vol. 22, no 1, p. 276-281Article in journal (Refereed)
    Abstract [en]

    We consider the concept of Tikhonov filter matrices in connection with discrete ill-posed and rank-deficient linear problems. Important properties of the Tikhonov filter matrices are given together with their filtering and regularization effects. We also present new perturbation identities for the Tikhonov regularized linear least squares problem using filter matrices generalizing well-known perturbation identities for the linear least squares problem and pseudoinverses.

  • 31.
    Gulliksson, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Wedin, P-Å
    Wei, Yimin
    Perturbation identities for regularized Tikhonov inverses and weighted pseudoinverses2000In: Bit: numerical mathematics, ISSN 0006-3835, Vol. 40, no 3, p. 513-523Article in journal (Refereed)
    Abstract [en]

    e consider the perturbation analysis of two important problems for solving ill-conditioned or rank-deficient linear least squares problems. The Tikhonov regularized problem is a linear least squares problem with a regularization term balancing the size of the residual against the size of the weighted solution. The weight matrix can be a non-square matrix (usually with fewer rows than columns). The minimum-norm problem is the minimization of the size of the weighted solutions given by the set of solutions to the, possibly rank-deficient, linear least squares problem. It is well known that the solution of the Tikhonov problem tends to the minimum-norm solution as the regularization parameter of the Tikhonov problem tends to zero. Using this fact and the generalized singular value decomposition enable us to make a perturbation analysis of the minimum-norm problem with perturbation results for the Tikhonov problem. From the analysis we attain perturbation identities for Tikhonov inverses and weighted pseudoinverses.

  • 32.
    Hellström, Lisbeth
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gradin, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Carlberg, Torbjörn
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    A laboratory wood chipper for chipping under realistic conditions2011In: Experimental mechanics, ISSN 0014-4851, E-ISSN 1741-2765, Vol. 51, no 8, p. 1309-1316Article in journal (Refereed)
    Abstract [en]

    In order to be able to study the damage mechanisms and in general the mechanisms active when a wood chip is created during the wood chipping process, it is crucial to have access to an experimental equipment in which chips can be produced under realistic conditions. In this paper is presented a laboratory chipper, which has been developed to admit chipping at rates that can be varied in a large interval i.e. at rates ranging from zero to 50 m/s. The knife used to cut the chips is mounted in a knife holder, which is instrumented in such a way that forces in three orthogonal directions can be measured. Since the actual force and the measured force differs due to inertia effects, a simple mathematical model is developed and used to evaluate the force acting on the knife. Some results are shown from the force measurements and it is concluded that the laboratory chipper is a versatile tool in the process of increasing the understanding of the chipping process.

  • 33. Hägglind, Richard
    et al.
    Gulliksson, Mårten
    Westerlind, Bo
    Nordstrand, Tomas
    Diffusion of water vapor in paper1998In: Forest products symposium (1998 : Miami Beach), 1998Conference paper (Other academic)
  • 34.
    Lund, Marie
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Ståhl, Fredrik
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Regularity Aspects in Inverse Musculoskeletal Biomechanics2008In: NUMERICAL ANALYSIS AND APPLIED MATHEMATICS / [ed] Simos, TE; Psihoyios, G; Tsitouras, C, American Institute of Physics (AIP), 2008, p. 368-371Conference paper (Refereed)
    Abstract [en]

    Inverse simulations of musculoskeletal models computes the internal forces such as muscle and joint reaction forces, which are hard to measure, using the more easily measured motion and external forces as input data. Because of the difficulties of measuring muscle forces and joint reactions, simulations are hard to validate. One way of reducing errors for the simulations is to ensure that the mathematical problem is well-posed. This paper presents a study of regularity aspects for an inverse simulation method, often called forward dynamics or dynamical optimization, that takes into account both measurement errors and muscle dynamics. Regularity is examined for a test problem around the optimum using the approximated quadratic problem. The results shows improved rank by including a regularization term in the objective that handles the mechanical over-determinancy. Using the 3-element Hill muscle model the chosen regularization term is the norm of the activation. To make the problem full-rank only the excitation bounds should be included in the constraints. However, this results in small negative values of the activation which indicates that muscles are pushing and not pulling, which is unrealistic but the error maybe small enough to be accepted for specific applications. These results are a start to ensure better results of inverse musculoskeletal simulations from a numerical point of view.

  • 35.
    Lund Ohlsson, Marie
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Least Squares Approach to Inverse Problems in Musculoskeletal Biomechanics2009Report (Other academic)
    Abstract [en]

    Inverse simulations of musculoskeletal models computes the internal forces such as muscle and joint reaction forces, which are hard to measure, using the more easily measured motion and external forces as input data. Because of the difficulties of measuring muscle forces and joint reactions, simulations are hard to validate. One way of reducing errors for the simulations is to ensure that the mathematical problem is well-posed. This paper presents a study of regularity aspects for an inverse simulation method, often called forward dynamics or dynamical optimization, that takes into account both measurement errors and muscle dynamics. The simulation method is explained in detail. Regularity is examined for a test problem around the optimum using the approximated quadratic problem. The results shows improved rank by including a regularization term in the objective that handles the mechanical over-determinancy. Using the 3-element Hill muscle model the chosen regularization term is the norm of the activation. To make the problem full-rank only the excitation bounds should be included in the constraints. However, this results in small negative values of the activation which indicates that muscles are pushing and not pulling. Despite this unrealistic behavior the error maybe small enough to be accepted for specific applications. These results is a starting point start for achieving better results of inverse musculoskeletal simulations from a numerical point of view.

  • 36. Mörtsell, Marlene
    et al.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    An overview of some non-linear techniques in chemometrics2001Report (Other academic)
  • 37.
    Noreus, Olof
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    Isaksson, Alf J.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Estimation of Missing Data for Model Validation2000In: Control Systems, Preprints, Conference, TAPPI Press, 2000, p. 89-95Conference paper (Other academic)
    Abstract [en]

    To be able to validate models based on signals logged from processes, it is necessary to have a continuous data set. Since missing data is a common problem when logging process variables, it can be almost impossible to find a continuous data set. In general it might be necessary to replace missing data also when estimating models for the signals and this paper compares different approaches for missing data estimation. The result shows that on the available data set there is no significant difference in the performance and for the model validation problem it is probably enough to use the simplest method. However, for other applications and on other data sets, there are probably important differences.

  • 38.
    Norgren, Sven
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gradin, P A
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Nyström, Staffan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gullikson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Investigation of the z-direction strength properties of paper by use of acoustic emission monitoring2008In: Journal of Acoustic Emission, ISSN 0730-0050, Vol. 26, p. 23-31Article in journal (Refereed)
  • 39.
    Norgren, Sven
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gradin, Per
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Nyström, Staffan
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Investigation of the Z-direction strength properties of paper by use of acoustic emission monitoring2008In: 28th European conferance on acoustic emission testing, 2008, p. 238-243Conference paper (Other academic)
  • 40.
    Scherzer, O
    et al.
    Mid Sweden University.
    Gulliksson, Mårten
    Adaptive strategy for the damping parameters in an iteratively regularized Gauss-Newton method1999In: Journal of Optimization Theory and Applications, ISSN 0022-3239, Vol. 100, no 1, p. 161-180Article in journal (Refereed)
    Abstract [en]

    In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss�Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.

  • 41.
    Sjöström, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Identifiability Analysis of Prandtl-Ishilinskii Hysteresis Model with Saturation2008In: Physica. B, Condensed matter, ISSN 0921-4526, E-ISSN 1873-2135, Vol. 403, no 2-3, p. 266-270Article in journal (Refereed)
    Abstract [en]

    A new class of Preisach operators based on play operators with an inverse in a closed form and allowing for saturation has recently been proposed. Its existence criteria and identification procedure were considered in earlier articles. The present paper analyses the identification procedure with respect to the sensitivity to underlying functions (i.e. intrinsic behaviour of the hysteretic system), to spline approximation, and to the least square error estimation procedure. The analysis shows that model errors are significantly influenced by large derivatives of the underlying functions. Spline approximations have generally little effect on model errors. In particular, an upper bound of for the relative parameter error due to measurement discrepancies has been derived for the least square error problem. The bound increases, the closer to saturation data are measured.

  • 42.
    Sjöström, Mårten
    et al.
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Information Technology and Media.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Identifiability Analysis of Prandtl-Ishilinskii Hysteresis Model with Saturation - Derivations2007Report (Other academic)
    Abstract [en]

    A new class of Preisach operators based on play operators with an inverse in a closed form and allowing for saturation has recently been proposed. Its existence criteria and identification procedure were considered in earlier articles. The present paper analyses the identification procedure with respect to the sensitivity to underlying functions (i.e. intrinsic behaviour of the hysteretic system), to spline approximation, and to the least square error estimation procedure. The analysis show that model errors are significantly influenced by large derivatives of the underlying functions. Spline approximations have generally little effect on model errors. In particular, an upper bound of for the relative parameter error due to measurement discrepancies has been derived for the least square error problem. The bound increases, the closer to saturation data are measured.

  • 43. Vexler, B
    et al.
    Feng, Tao
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
    Adaptive finite element methods for determining the elastic constants of paper from measured displacementsManuscript (preprint) (Other academic)
  • 44. Viklands, Thomas
    et al.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Optimization tools for solving nonlinear ill-posed problems2001In: Fast Solution of Discretized Optimization Problems: Workshop held at the Weierstrass Institute for Applied Analysis and Stochastics, Berlin, May 8-12, 2001, p. 255-264Conference paper (Other academic)
  • 45.
    Xie, Z Q
    et al.
    Hunan Normal Univ, Dept Math, Changsha 410081, Hunan Peoples R China.
    Gullikson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Hägglund, R
    SCA.
    Determining the elastic constants of paper with optimization methods2002In: Inverse Problems in Engineering, ISSN 1068-2767, E-ISSN 1029-0281, Vol. 10, no 5, p. 393-411Article in journal (Refereed)
    Abstract [en]

    In this article we describe a method for determining the elastic constants for paper from a measured displacement field. Our model is based on the finite element method using quadrilateral elements. A method for finding the first and second order derivatives with respect to the unknown parameters plays a crucial role in the optimization process used to solve the inverse problems in this article. To reduce the influence of the noise of the input data, an efficient continuation method is used for the regularization term in the cost function. Some numerical results for synthetic data and measurement will verify the efficiency of our approach.

  • 46.
    Xie, Ziqing
    et al.
    Dept of Mathematics, Hunan Normal Univ, China.
    Gulliksson, Mårten
    Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
    Hägglund, Richard
    SCA Packaging Research, Sundsvall, Sweden.
    Determining the elastic constants of paper from measured displacements2001Report (Other academic)
1 - 46 of 46
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