| 1. |
- Anton, R., et al.
(författare)
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Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise
- 2016
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Ingår i: Siam Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 54:2, s. 1093-1119
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Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.
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| 2. |
- Anton, Rikard, et al.
(författare)
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Full discretization of semilinear stochastic wave equations driven by multiplicative noise
- 2016
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 54:2, s. 1093-1119
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Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.
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| 3. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Asymptotic error expansions for the finite element method for second order elliptic problems in R_N, N>=2, I: Local interior expansions
- 2010
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 48:5, s. 2000-2017
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Tidskriftsartikel (refereegranskat)abstract
- Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in RN for N ≥ 2, studies by Schatz in [Math. Comp., 67 (1998), pp. 877-899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269-1293].
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| 4. |
- Berggren, Martin, et al.
(författare)
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Weak material approximation of holes with traction-free boundaries
- 2012
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications Online. - 0036-1429 .- 1095-7170. ; 50:4, s. 1827-1848
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Tidskriftsartikel (refereegranskat)abstract
- Consider the solution of a boundary-value problem for steady linear elasticity in which the computational domain contains one or several holes with traction-free boundaries. The presence of holes in the material can be approximated using a weak material; that is, the relative density of material rho is set to 0 < epsilon = rho << 1 in the hole region. The weak material approach is a standard technique in the so-called material distribution approach to topology optimization, in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material. The use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value rho is an element of [epsilon, 1] of the design variable. A finite-element approximation of the boundary-value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem in which the solution is continuously and elastically extended into the holes. The error in this approximation can be bounded by two terms that depend on epsilon. One term scales linearly with epsilon with a constant that is independent of the mesh size parameter h but that depends on the surface traction required to fit elastic material in the deformed holes. The other term scales like epsilon(1/2) times the finite-element approximation error inside the hole. The condition number of the weak material stiffness matrix scales like epsilon(-1), but the use of a suitable left preconditioner yields a matrix with a condition number that is bounded independently of epsilon. Moreover, the preconditioned matrix admits the limit value epsilon -> 0, and the solution of corresponding system of equations yields in the limit a finite-element approximation of the continuously and elastically extended problem.
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| 5. |
- Bogfjellmo, Geir, et al.
(författare)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 56:4, s. 2623-2647
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Tidskriftsartikel (refereegranskat)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized Bézier curves. To construct $C^2$-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for $C^2$-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.
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| 6. |
- Bogfjellmo, Geir, 1987, et al.
(författare)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Tidskriftsartikel (refereegranskat)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
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| 7. |
- Brännström, Åke, 1975-, et al.
(författare)
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On the convergence of the Escalator Boxcar Train
- 2013
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 51:6, s. 3213-3231
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Tidskriftsartikel (refereegranskat)abstract
- The Escalator Boxcar Train (EBT) is a numerical method that is widely used in theoretical biology to investigate the dynamics of physiologically structured population models, i.e., models in which individuals differ by size or other physiological characteristics. The method was developed more than two decades ago, but has so far resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate. In rigorously establishing the convergence of the EBT method, our results pave the way for wider acceptance of the EBT method beyond theoretical biology and constitutes an important step towards integration with established numerical schemes.Read More: http://epubs.siam.org/doi/abs/10.1137/120893215
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| 8. |
- Burman, Erik, et al.
(författare)
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Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications. - 0036-1429 .- 1095-7170. ; 56:6, s. 3480-3509
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Tidskriftsartikel (refereegranskat)abstract
- consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Darde, Hannukainen, and Hyvonen in [SIAM T. Numer. Anal., 51 (2013), pp. 2123-2148]. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.
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| 9. |
- Burman, Erik, et al.
(författare)
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The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem
- 2017
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 55:6, s. 2523-2539
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Tidskriftsartikel (refereegranskat)abstract
- We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 ( 2013), pp. 1295-1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.
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| 10. |
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