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  • 1. Björk, T.
    et al.
    Szepessy, A.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tempone, R.
    Zouraris, G. E.
    Monte Carlo Euler approximations of HJM term structure financial models2013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 2, p. 341-383Article in journal (Refereed)
    Abstract [en]

    We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

  • 2.
    Breiten, Tobias
    et al.
    Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Graz, 8010, Austria.
    Ringh, Emil
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Residual-based iterations for the generalized Lyapunov equation2019In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 59, no 4, p. 823-852Article in journal (Refereed)
    Abstract [en]

    This paper treats iterative solution methods for the generalized Lyapunov equation. Specifically, a residual-based generalized rational-Krylov-type subspace is proposed. Furthermore, the existing theoretical justification for the alternating linear scheme (ALS) is extended from the stable Lyapunov equation to the stable generalized Lyapunov equation. Further insights are gained by connecting the energy-norm minimization in ALS to the theory of H2-optimality of an associated bilinear control system. Moreover it is shown that the ALS-based iteration can be understood as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm.

  • 3.
    Carleson, Lennart
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Computations in pure mathematics2006In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 46, no SUPPL. 5, p. S19-S20Article in journal (Refereed)
  • 4. Collier, Nathan
    et al.
    Haji-Ali, Abdul-Lateef
    Nobile, Fabio
    von Schwerin, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tempone, Raul
    A continuation multilevel Monte Carlo algorithm2015In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 55, no 2, p. 399-432Article in journal (Refereed)
    Abstract [en]

    We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients.

  • 5.
    Engquist, Björn
    et al.
    Univ Texas Austin.
    Häggblad, Jon
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D2012In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 3, p. 615-637Article in journal (Refereed)
    Abstract [en]

    The Yee scheme is one of the most popular methods for electromagnetic wave propagation. A main advantage is the structured staggered grid, making it simple and efficient on modern computer architectures. A downside to this is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In this paper we present a method to improve the boundary treatment in two dimensions by, starting from a staircase approximation, modifying the coefficients of the update stencil so that we can obtain a consistent approximation while preserving the energy conservation, structure and the optimal CFL-condition of the original Yee scheme. We prove this in L_2 and verify it by numerical experiments.

  • 6.
    Hoffman, Johan
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Johnson, Claes
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Blow up of incompressible Euler solutions2008In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 48, no 2, p. 285-307Article in journal (Refereed)
    Abstract [en]

    We present analytical and computational evidence of blowup of initially smooth solutions of the incompressible Euler equations into non-smooth turbulent solutions. We detect blowup by observing increasing L-2-residuals of computed solutions under decreasing mesh size.

  • 7.
    Jarlebring, Elias
    et al.
    Katholieke Univ Leuven, B-3001 Heverlee, Belgium .
    Michiels, Wim
    Katholieke Univ Leuven, B-3001 Heverlee, Belgium .
    Analyzing the convergence factor of residual inverse iteration2011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 4, p. 937-957Article in journal (Refereed)
    Abstract [en]

    We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w (k) and we can use the formula for the convergence factor to analyze how it depends on the choice of w (k) . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.

  • 8.
    Kreiss, Heinz-Otto
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Ystrom, J.
    A note on viscous conservation laws with complex characteristics2006In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 46, p. S55-S59Article in journal (Refereed)
    Abstract [en]

    There are several physical set-ups involving multi-phase fluids that result in highly unstable behavior already at rather low flow rates. Mathematical models of these flow problems consist typically of conservation laws like conservation of mass and momentum for each phase together with coupling terms connecting the phases. For multi-phase flow the characteristics are often complex and without the dissipative terms the problem is ill-posed and not computable. We will discuss why the nonlinearity of the system can prevent blow-up.

  • 9.
    Mele, Giampaolo
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Jarlebring, Elias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    On restarting the tensor infinite Arnoldi method2018In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 58, no 1, p. 133-162Article in journal (Refereed)
    Abstract [en]

    An efficient and robust restart strategy is important for any Krylov-based method for eigenvalue problems. The tensor infinite Arnoldi method (TIAR) is a Krylov-based method for solving nonlinear eigenvalue problems (NEPs). This method can be interpreted as an Arnoldi method applied to a linear and infinite dimensional eigenvalue problem where the Krylov basis consists of polynomials. We propose new restart techniques for TIAR and analyze efficiency and robustness. More precisely, we consider an extension of TIAR which corresponds to generating the Krylov space using not only polynomials, but also structured functions, which are sums of exponentials and polynomials, while maintaining a memory efficient tensor representation. We propose two restarting strategies, both derived from the specific structure of the infinite dimensional Arnoldi factorization. One restarting strategy, which we call semi-explicit TIAR restart, provides the possibility to carry out locking in a compact way. The other strategy, which we call implicit TIAR restart, is based on the Krylov–Schur restart method for the linear eigenvalue problem and preserves its robustness. Both restarting strategies involve approximations of the tensor structured factorization in order to reduce the complexity and the required memory resources. We bound the error introduced by some of the approximations in the infinite dimensional Arnoldi factorization showing that those approximations do not substantially influence the robustness of the restart approach. We illustrate the effectiveness of the approaches by applying them to solve large scale NEPs that arise from a delay differential equation and a wave propagation problem. The advantages in comparison to other restart methods are also illustrated. 

  • 10.
    Moon, Kyoung-Sook
    et al.
    Department of Mathematics, University of Maryland.
    von Schwerin, Erik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tempone, R.
    School of Computational Science, Florida State University.
    Convergence rates for an adaptive dual weighted residual finite element algorithm2006In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 46, no 2, p. 367-407Article in journal (Refereed)
    Abstract [en]

    Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation, [GRAPHICS] applied to isoparametric d-linear quadrilateral finite element approximation of functionals of multi scale solutions to second order elliptic partial differential equations in bounded domains of R-d. In contrast to the usual aim to derive an a posteriori error estimate, this work derives, as the mesh size tends to zero, a uniformly convergent error expansion for the error density, with computable leading order term. It is shown that the optimal adaptive isotropic mesh uses a number of elements proportional to the d/2 power of the Ld/d+2 quasi-norm of the error density; the same error for approximation with a uniform mesh requires a number of elements proportional to the d/2 power of the larger L-1 norm of the same error density. A point is that this measure recognizes different convergence rates for multi scale problems, although the convergence order may be the same. The main result is a proof that the adaptive algorithm based on successive subdivisions of elements reduces the maximal error indicator with a factor or stops with the error asymptotically bounded by the tolerance using the optimal number of elements, up to a problem independent factor. An important step is to prove uniform convergence of the expansion for the error density, which is based on localized averages of second order difference quotients of the primal and dual finite element solutions. The averages are used since the difference quotients themselves do not converge pointwise for adapted meshes. The proof uses weak convergence techniques with a symmetrizer for the second order difference quotients and a splitting of the error into a dominating contribution, from elements with no hanging nodes or edges on the initial mesh, and a remaining asymptotically negligible part. Numerical experiments for an elasticity problem with a crack and different variants of the averages show that the algorithm is useful in practice also for relatively large tolerances, much larger than the small tolerances needed to theoretically guarantee that the algorithm works well.

  • 11. Olsson, K. Henrik A.
    et al.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Rational Krylov for eigenvalue computation and model order reduction2006In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 46, p. S99-S111Article in journal (Refereed)
    Abstract [en]

    A rational Krylov algorithm for eigenvalue computation and model order reduction is described. It is shown how to implement it as a modified shift-and-invert spectral transformation Arnoldi decomposition. It is shown how to do deflation, locking converged eigenvalues and purging irrelevant approximations. Computing reduced order models of linear dynamical systems by moment matching of the transfer function is considered. Results are reported from one illustrative toy example and one practical example, a linear descriptor system from a computational fluid dynamics application.

  • 12.
    Popovic, Jelena
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Analysis of a fast method for solving the high frequency Helmholtz equation in one dimension2011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 3, p. 721-755Article in journal (Refereed)
    Abstract [en]

    We propose and analyze a fast method for computing the solution of the high frequency Helmholtz equation in a bounded one-dimensional domain with a variable wave speed function. The method is based on wave splitting. The Helmholtz equation is split into one-way wave equations with source functions which are solved iteratively for a given tolerance. The source functions depend on the wave speed function and on the solutions of the one-way wave equations from the previous iteration. The solution of the Helmholtz equation is then approximated by the sum of the one-way solutions at every iteration. To improve the computational cost, the source functions are thresholded and in the domain where they are equal to zero, the one-way wave equations are solved with geometrical optics with a computational cost independent of the frequency. Elsewhere, the equations are fully resolved with a Runge-Kutta method. We have been able to show rigorously in one dimension that the algorithm is convergent and that for fixed accuracy, the computational cost is asymptotically just for a pth order Runge-Kutta method, where omega is the frequency. Numerical experiments indicate that the growth rate of the computational cost is much slower than a direct method and can be close to the asymptotic rate.

  • 13.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    BIT 502010In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 3, p. 451-453Article in journal (Refereed)
  • 14.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    BIT502011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 1, p. 1-5Article in journal (Other academic)
  • 15.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI).
    Editorial2012In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 4, p. 797-800Article in journal (Other academic)
  • 16.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Editorial2010In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 4, p. 689-692Article in journal (Other academic)
  • 17.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the contents of issue 47:32007In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 47, no 3, p. 485-486Article in journal (Refereed)
  • 18.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the contents of issue 49:22009In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 49, no 2, p. 247-248Article in journal (Other academic)
  • 19.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the Contents of Issue 49:32009In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 49, no 3, p. 475-476Article in journal (Other academic)
  • 20.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the contents of issue 50:12010In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 1, p. 1-2Article in journal (Other academic)
  • 21.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the contents of issue 50:22010In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 50, no 2, p. 221-222Article in journal (Other academic)
  • 22.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Introduction to the contents of issue 51:22011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 2, p. 251-252Article in journal (Other academic)
  • 23.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Introduction to the contents of issue 51:32011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 3, p. 481-482Article in journal (Refereed)
  • 24.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Introduction to the contents of issue 52:22012In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 2, p. 271-272Article in journal (Other academic)
  • 25.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Introduction to the contents of issue 52:32012In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 3, p. 537-538Article in journal (Other academic)
  • 26.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Preface and Introduction to the Contents of Issue 49:12009In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 49, no 1, p. 1-2Article in journal (Other academic)
  • 27.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Preface to 54-32014In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 54, no 3, p. 585-586Article in journal (Refereed)
  • 28.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Preface to BIT 52:12012In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 1, p. 1-2Article in journal (Other academic)
  • 29.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Preface to BIT 53:12013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 1, p. 1-2Article in journal (Other academic)
  • 30.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Preface to BIT 53:22013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 2, p. 283-284Article in journal (Refereed)
  • 31.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Preface to BIT 53:32013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 3, p. 565-566Article in journal (Other academic)
  • 32.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Preface to BIT 53:42013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, no 4, p. 821-825Article in journal (Refereed)
  • 33.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Preface to BIT 54:22014In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 54, no 2, p. 303-304Article in journal (Refereed)
  • 34.
    Ruhe, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Preface to BIT 54:4 PREFACE2014In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 54, no 4, p. 867-871Article in journal (Refereed)
  • 35.
    Ruhe, Axel
    KTH, School of Computer Science and Communication (CSC).
    Untitled2011In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 51, no 4, p. 777-780Article in journal (Refereed)
  • 36.
    Tornberg, Anna-Karin
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Multi-dimensional quadrature of singular and discontinuous functions2002In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 42, no 3, p. 644-669Article in journal (Refereed)
    Abstract [en]

    In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.

  • 37. Tornberg, Anna-Karin
    et al.
    Engquist, Björn
    Gustafsson, B.
    Wahlund, P.
    A new type of boundary treatment for wave propagation2006In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 46, p. S145-S170Article in journal (Refereed)
    Abstract [en]

    Numerical approximation of wave propagation can be done very efficiently on uniform grids. The Yee scheme is a good example. A serious problem with uniform grids is the approximation of boundary conditions at a boundary not aligned with the grid. In this paper, boundary conditions are introduced by modifying appropriate material coefficients at a few grid points close to the embedded boundary. This procedure is applied to the Yee scheme and the resulting method is proven to be L2-stable in one space dimension. Depending on the boundary approximation technique it is of first or second order accuracy even if the boundary is located at an arbitrary point relative to the grid. This boundary treatment is applied also to a higher order discretization resulting in a third order accurate method. All algorithms have the same staggered grid structure in the interior as well as across the boundaries for efficiency. A numerical example with the extension to two space dimensions is included.

  • 38. Wadbro, Eddie
    et al.
    Zahedi, Sara
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Kreiss, Gunilla
    Berggren, Martin
    A uniformly well-conditioned, unfitted Nitsche method for interface problems2013In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 53, p. 791-820Article in journal (Refereed)
1 - 38 of 38
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