| 1. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
-
The discrete ordinates method for the neutron transport equation in an infinite cylindrical domain
- 1992
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Ingår i: Mathematical Models and Methods in Applied Science. ; 2:3, s. 317-338
-
Tidskriftsartikel (refereegranskat)abstract
- We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L1, and is proved using Besov space techniques. This result is applied in the error analysis of the discrete ordinates method for the numerical solution of the neutron transport equation. We derive an error estimate in the L1-norm for the scalar flux, and as a consequence, we obtain an error bound for the critical eigenvalue.
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| 2. |
- Riise, Gerdt C., 1956, et al.
(författare)
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The intrabronchial microbial flora in chronic bronchitis patients: a target for N-acetylcysteine therapy?
- 1994
-
Ingår i: Eur Respir J. - 0903-1936. ; 7:1, s. 94-101
-
Tidskriftsartikel (refereegranskat)abstract
- Chronic bronchitis is common among smokers, often together with recurrent infectious exacerbations. Streptococcus pneumoniae and Haemophilus influenzae are the pathogens traditionally considered most important. N-acetylcysteine (NAC) treatment has been shown to reduce the number of infectious exacerbations in patients with chronic bronchitis. The mechanism behind this is unknown. We attempted to characterize the intrabronchial bacterial flora in patients with chronic bronchitis in an infection-free interval, and to determine whether pharmacological and immunological factors effected the bacterial occurrence. Twenty two smokers with non-obstructive chronic bronchitis, 19 smokers with chronic bronchitis and chronic obstructive pulmonary disease (COPD) and 14 healthy nonsmokers underwent bronchoscopy. To obtain uncontaminated intrabronchial samples, a protected specimen brush was used. Quantitative bacterial cultures and virus isolations were performed. Significantly positive bacterial cultures (> 1,000 colony-forming units (cfu).ml-1) were found only in the patients. S. pneumoniae and H. influenzae were found in five patients, and only in the patients without NAC treatment. The most common bacterium was alpha-haemolytic streptococcus. Negative cultures were more common in the healthy controls. Of the various factors examined, only NAC medication had an influence on bacterial numbers. Significantly fewer patients with NAC medication had positive cultures (3 out of 16) than in the group of patients without NAC therapy (15 out of 21). Our results confirm that chronic bronchitis in smokers leads to increased intrabronchial bacterial colonization. We could also confirm that 1,000 cfu.ml-1 is an adequate cut-off level for significant bacterial growth when using the protected specimen brush. NAC medication was associated with low bacterial numbers.
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| 3. |
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| 4. |
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| 5. |
- Adolfsson, Klas, 1972, et al.
(författare)
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Discretization of integro-differential equations modeling dynamic fractional order viscoelasticity
- 2006
-
Ingår i: Lecture Notes in Computer Science, Springer, ''Proceedings of Large-Scale Scientific Computations, 2005, Sozopol, Bulgaria'', I. Lirkov, S. Margenov, and J. Wasniewski (Eds.). - 3540319948 ; 3743, s. 76-83
-
Konferensbidrag (refereegranskat)abstract
- We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro-differential equation with a weakly singular convolution kernel. We discretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical experiment is included.
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| 6. |
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| 7. |
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| 8. |
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| 9. |
- Agapiou, Sergios, et al.
(författare)
-
Posterior consistency of the Bayesian approach to linear ill-posed inverse problems
- 2012
-
Ingår i: arXiv:1203.5753v2 [math.ST]. ; :1203.5753, s. 1-30
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying the posterior distribution using its precision operator is presented. Work- ing with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior con- sistency in a frequentist sense, and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. We show how these rates may be optimized by a choice of the scale parameter in the prior covariance operator. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator; more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known minimax rates of convergence in the case where the forward operator and the prior and noise covariances are all simultaneously diagonaliz- able, and confirm that the PDE method provides the same rates for a wide range of parameters. An elliptic PDE inverse problem is used to illustrate the power of the general theory.
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| 10. |
- Agapiou, Sergios, et al.
(författare)
-
Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems
- 2013
-
Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 123:10, s. 3828-3860
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.
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| 11. |
- Akrivis, Georgios, et al.
(författare)
-
Linearly implicit finite element methods for the time-dependent Joule heating problem
- 2005
-
Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 45:3, s. 429-442
-
Tidskriftsartikel (refereegranskat)abstract
- Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions.
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| 12. |
- Andersson, Adam, 1979, et al.
(författare)
-
Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE
- 2016
-
Ingår i: Stochastic Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 4:1, s. 113-149
-
Tidskriftsartikel (refereegranskat)abstract
- We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.
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| 13. |
- Andersson, Adam, 1979, et al.
(författare)
-
Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE
- 2013
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we introduce a new family of refined Watanabe- Sobolev spaces that capture in a fine way integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the It¯o formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.
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| 14. |
- Andersson, Adam, 1979, et al.
(författare)
-
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
- 2012
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We find the weak rate of convergence of approximate solutions of the nonlinear stochastic heat equation, when discretized in space by a standard finite element method. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation only has a solution in one space dimension. In order to get results for higher dimensions, colored noise is considered here, besides the white noise case where considerably weaker assumptions on the noise term is needed. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.
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| 15. |
- Andersson, Adam, 1979, et al.
(författare)
-
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
- 2016
-
Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 85, s. 1335-1358
-
Tidskriftsartikel (refereegranskat)abstract
- We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an earlier result of Debussche in which time discretization is considered for the stochastic heat equation perturbed by white noise. It is known that this equation has a solution only in one space dimension. In order to obtain results for higher dimensions, colored noise is considered here, besides white noise in one dimension. Integration by parts in the Malliavin sense is used in the proof. The rate of weak convergence is, as expected, essentially twice the rate of strong convergence.
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| 16. |
- Andersson, Adam, 1979, et al.
(författare)
-
Weak error analysis for semilinear stochastic Volterra equations with additive noise
- 2014
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our weak convergence result concerns not only the solution at a fixed time but also integrals of the entire path with respect to any finite Borel measure. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.
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| 17. |
- Andersson, Adam, 1979, et al.
(författare)
-
Weak error analysis for semilinear stochastic Volterra equations with additive noise
- 2016
-
Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 437:2, s. 1283-1304
-
Tidskriftsartikel (refereegranskat)abstract
- We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.
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| 18. |
- Anton, Rikard, et al.
(författare)
-
Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise
- 2015
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal approximation. This explicit time integrator allows for mean-square error bounds indepen- dent of the space discretisation and thus do not suffer from a step size restriction as in the often used Störmer-Verlet- leap-frog 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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| 19. |
- Anton, R., et al.
(författare)
-
Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise
- 2016
-
Ingår i: Siam Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 54:2, s. 1093-1119
-
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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| 20. |
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| 21. |
- Bågmark, Kasper, 1995, et al.
(författare)
-
An energy-based deep splitting method for the nonlinear filtering problem
- 2023
-
Ingår i: Partial Differential Equations and Applications. - : Springer Science and Business Media LLC. - 2662-2963 .- 2662-2971. ; 4:2
-
Tidskriftsartikel (refereegranskat)abstract
- The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution of (stochastic) partial differential equations. This is combined with an energy-based model for the approximation of functions by a deep neural network. This results in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received. The method is tested on four examples, two linear in one and twenty dimensions and two nonlinear in one dimension. The method shows promising performance when benchmarked against the Kalman filter and the bootstrap particle filter.
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| 22. |
- Cohen, David, et al.
(författare)
-
A trigonometric method for the linear stochastic wave equation
- 2012
-
Ingår i: arXiv.
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretisation and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretisation and thus do not have a step size restriction as in the often used Störmer-Verlet-leap-frog scheme. Moreover it enjoys a trace formula as does the exact solution of our problem. These favourable properties are demonstrated with numerical experiments.
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| 23. |
- Cohen, David, et al.
(författare)
-
A trigonometric method for the linear stochastic wave equation
- 2013
-
Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 51:1, s. 204-222
-
Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step-size restriction as in the often used Störmer--Verlet-leap-frog scheme. Moreover, it enjoys a trace formula as does the exact solution of our problem. These favorable properties are demonstrated with numerical experiments. Read More: http://epubs.siam.org/doi/abs/10.1137/12087030X
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| 24. |
- Crouzeix, Michel, et al.
(författare)
-
The stability of rational approximations of analytic semigroups
- 1993
-
Ingår i: BIT Numer. Math.. - 0006-3835 .- 1572-9125. ; 33:1, s. 74-84
-
Tidskriftsartikel (refereegranskat)abstract
- This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL p , 1
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| 25. |
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| 26. |
- Demlow, Alan, et al.
(författare)
-
Local pointwise a posteriori gradient error bounds for the Stokes equations
- 2013
-
Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 82:282, s. 625-649
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the standard Taylor-Hood finite element method for the stationary Stokes system on polyhedral domains. We prove local a posteriori error estimates for the maximum error in the gradient of the velocity field. Because the gradient of the velocity field blows up near reentrant corners and edges, such local error control is necessary when pointwise control of the gradient error is desirable. Computational examples confirm the utility of our estimates in adaptive codes.
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| 27. |
- Edelvik, Fredrik, 1972, et al.
(författare)
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An improved method for dipole modeling in EEG-based source localization
- 2009
-
Ingår i: International Federation for Medical and Biological Engineering Proceedings. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 1680-0737. - 9783642038884 ; 25:9, s. 146-149
-
Konferensbidrag (refereegranskat)abstract
- The inverse problem in EEG-based source localizationis to determine the location of the brain sources that areresponsible for the measured potentials at the scalp electrodes.The brain sources are usually modeled as current dipoles whichlead to a singularity in the right-hand side of the governing Poisson’sequation. Subtraction methods have been proposed as aremedy and in this paper an improved subtraction method formodeling the dipoles is presented. The accuracy is demonstratedfor radial and tangential sources in layered sphere models and isto the best of the authors’ knowledge superior to previous methodsfor superficial sources. An additional advantage is that itproduces a right hand side with few non-zeros which is beneficialfor efficient solution of the inverse problem.
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| 28. |
- Eisenmann, Monika, et al.
(författare)
-
Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
- 2022
-
Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62:3, s. 803-48
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525-589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.
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| 29. |
- Eisenmann, Monika, et al.
(författare)
-
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
- 2019
-
Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 19:6, s. 1387-1430
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.
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| 30. |
- Elliott, Charles M., et al.
(författare)
-
Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation
- 1992
-
Ingår i: Math. Comp.. - 0025-5718 .- 1088-6842. ; 58:198
-
Tidskriftsartikel (refereegranskat)abstract
- A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.
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| 31. |
- Enelund, Mikael, 1965, et al.
(författare)
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A computational mathematics education for students of mechanical engineering
- 2006
-
Ingår i: World Transactions on Engineering and Technology Education. - 1446-2257. ; 5:2, s. 329-332
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Tidskriftsartikel (refereegranskat)abstract
- In this article, the authors present a new model for computationally oriented mathematics education. This education combines traditional symbolic mathematics with computational mathematics and programming in the Matlab environment. Engineering applications are explored in computational exercises that are taught jointly with the courses in mechanics and thermodynamics at Chalmers University of Technology in Göteborg, Sweden.
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| 32. |
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| 33. |
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| 34. |
- Enelund, Mikael, 1965, et al.
(författare)
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Integration of Computational Mathematics Education in the Mechanical Engineering Curriculum
- 2011
-
Ingår i: Proceedings of 7th International CDIO Conference, Copenhagen, Denmark.
-
Konferensbidrag (refereegranskat)abstract
- The rapid development of computers and the internet has given new opportunities for engineering work as wells as for teaching and learning. The use of advanced modern mathematics is becoming increasingly more popular in the engineering community and and most problem solutions and developments incorporate high precision digital models, numerical analyses and simulations. However, this kind of mathematics has not been fully implemented into current engineering education programs. Students spend too much time on solving oversimplified problems that can be expressed analytically and with solutions that are already known in advance. Instead, we should be using computers to solve more general, real-world problems. Here we present the integration of a computationally oriented mathematics education into the CDIO-based MSc program in mechanical engineering at Chalmers. We found that the CDIO-approach was beneficial when designing a reformed mathematics education and integrating the mathematics in the curriculum. In the reform of the mathematics education, traditional symbolic mathematics is integrated with numerical calculations and the computer is used as a tool. Furthermore, the computer exercises and homework assignments are taken from applications of mechanical engineering and solutions are analyzed and discussed by means of simulations. The experience is very positive. The students’ interest for computation and simulation has increased. The students consider the the computer to be an important tool for learning and understanding of mathematics. Students spend more time training mathematics and solve more problems.
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| 35. |
- Enelund, Mikael, 1965, et al.
(författare)
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Space-time discretization of an integro-differential equation modeling quasi-static fractional-order viscoelasticity
- 2008
-
Ingår i: J. Vib. Control. - 1077-5463. ; 14:9-10, s. 1631-1649
-
Tidskriftsartikel (refereegranskat)abstract
- We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind, with a weakly singular kernel in the time variable, and which also involves partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used at each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, which can be used as the basis for an adaptive strategy.
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| 36. |
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| 37. |
- Forslund, Robert, et al.
(författare)
-
A greedy algorithm for optimal heating in powder-bed-based additive manufacturing
- 2021
-
Ingår i: Journal of Mathematics in Industry. - : Springer Science and Business Media LLC. - 2190-5983. ; 11:1
-
Tidskriftsartikel (refereegranskat)abstract
- Powder-bed-based additive manufacturing involves melting of a powder bed using a moving laser or electron beam as a heat source. In this paper, we formulate an optimization scheme that aims to control this type of melting. The goal consists of tracking maximum temperatures on lines that run along the beam path. Time-dependent beam parameters (more specifically, beam power, spot size, and speed) act as control functions. The scheme is greedy in the sense that it exploits local properties of the melt pool in order to divide a large optimization problem into several small ones. As illustrated by numerical examples, the scheme can resolve heat conduction issues such as concentrated heat accumulation at turning points and non-uniform melt depths.
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| 38. |
- Forslund, Robert, 1990, et al.
(författare)
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Analytical solution for heat conduction due to a moving Gaussian heat flux with piecewise constant parameters
- 2019
-
Ingår i: Applied Mathematical Modelling. - : Elsevier BV. - 0307-904X. ; 66, s. 227-240
-
Tidskriftsartikel (refereegranskat)abstract
- We provide an analytical solution of the heat equation in the half-space subject to a moving Gaussian heat flux with piecewise constant parameters. The solution is of interest in powder bed fusion applications where these parameters can be used to control the conduction of heat due to a scanning beam of concentrated energy. The analytical solution is written in a dimensionless form as a sum of integrals over (dimensionless) time. For the numerical computation of these integrals we suggest a quadrature scheme that utilizes pre-calculated look-up tables for the required quadrature orders. Such a scheme is efficient because the required quadrature orders are strongly dependent on the parameters in the heat flux. The possibilities of using the obtained computational technique for the control and optimization of powder bed fusion processes are discussed.
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| 39. |
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| 40. |
- Furihata, Daisuke, 1968, et al.
(författare)
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Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation
- 2016
-
Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method is space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.
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| 41. |
- Furihata, Daisuke, et al.
(författare)
-
Strong convergence of a fully discrete finite element approximation of the stochastic cahn–hilliard equation
- 2018
-
Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 56, s. 708-731
-
Tidskriftsartikel (refereegranskat)abstract
- © 2018 Society for Industrial and Applied Mathematics. We consider the stochastic Cahn–Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension d ≤ 3. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.
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| 42. |
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| 43. |
- Geissert, Matthias, et al.
(författare)
-
Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise
- 2009
-
Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 49:2, s. 343-356
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Tidskriftsartikel (refereegranskat)abstract
- The stochastic heat equation driven by additive noise is discretized in the spatial variables by a standard finite element method. The weak convergence of the approximate solution is investigated and the rate of weak convergence is found to be twice that of strong convergence.
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| 44. |
- Jareteg, Cornelia, 1986, et al.
(författare)
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Geometry Assurance Integrating Process Variation with Simulation of Spring-in for Composite Parts and Assemblies
- 2014
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Ingår i: Proc. of ASME 2014 International Mechanical Engineering Congress & Exposition. - 9780791846438 ; 2A
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Konferensbidrag (refereegranskat)abstract
- Geometrical variation and deviation in all manufacturing processes affect quality of the final product. Therefore geometry assurance is an important tool in the design phase of a new product. In the automotive and aviation industries where the use of composite parts is increasing drastically, new tools within variation simulations are needed. Composite parts tend to deviate more from nominal specification compared to metal parts. Methods to simulate the manufacturing process of composites have been developed before. In this paper we present how to combine the process variation simulation of composites with traditional variation simulations. The proposed method is demonstrated on a real complex subassembly, representing part of an aircraft wing-box. Since traditional variation simulation methods are not able to capture the spring-in and the special deviation behavior of composites,the proposed method adds a new feature and reliability to the geometry assurance process of composite assemblies.
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| 45. |
- Jareteg, Cornelia, 1986, et al.
(författare)
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Geometry Assurance Integrating Process Variation with Simulation of Spring-In for Composite Parts and Assemblies
- 2016
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Ingår i: Journal of Computing and Information Science in Engineering. - : ASME International. - 1530-9827 .- 1944-7078. ; 16:3
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Tidskriftsartikel (refereegranskat)abstract
- Copyright © 2016 by ASME.Geometrical variation and deviation in all the manufacturing processes affect the quality of the final product. Therefore, geometry assurance is an important tool in the design phase of a new product. In the automotive and aviation industries where the use of composite parts is increasing drastically, new tools within variation simulations are needed. Composite parts tend to deviate more from nominal specification compared to metal parts. Methods to simulate the manufacturing process of composites have been developed before. In this paper, we present how to combine the process variation simulation of composites with traditional variation simulations. The proposed method is demonstrated on a real complex subassembly, representing part of an aircraft wing-box. Since traditional variation simulation methods are not able to capture the spring-in and the special deviation behavior of composites, the proposed method adds a new feature and reliability to the geometry assurance process of composite assemblies.
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| 46. |
- Karlsson, Jesper, et al.
(författare)
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An a posteriori error estimate for symplectic Euler approximation of optimal control problems
- 2014
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns Symplectic Euler solutions of the Hamiltonian system connected with the optimal control problem. The error representation has a leading order term consisting of an error density that is computable from Symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading error density part in the error representation. With the error representation, it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. The performance is illustrated by numerical tests.
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| 47. |
- Karlsson, J., et al.
(författare)
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An Error Estimate for Symplectic Euler Approximation of Optimal Control Problems
- 2015
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 37:2
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Tidskriftsartikel (refereegranskat)abstract
- This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns symplectic Euler solutions of the Hamiltonian system connected with the optimal control problem. The error representation has a leading-order term consisting of an error density that is computable from symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading-error density part in the error representation. With the error representation, it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. The performance is illustrated by numerical tests.
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| 48. |
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| 49. |
- Kirchner, Kristin, 1987, et al.
(författare)
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Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise
- 2017
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 1090-2732 .- 0022-0396. ; 262:12, s. 5896-5927
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Tidskriftsartikel (refereegranskat)abstract
- The characterization of the covariance function of the solution process to a stochastic partial differential equation is considered in the parabolic case with multiplicative Lévy noise of affine type. For the second moment of the mild solution, a well-posed deterministic space–time variational problem posed on projective and injective tensor product spaces is derived, which subsequently leads to a deterministic equation for the covariance function.
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| 50. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Erratum: Finite element approximation of the Cahn-Hilliard-Cook equation
- 2014
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 52:5, s. 2594-2597
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We prove an additional result on the linearized Cahn-Hilliard-Cook equation to fill a gap in the main argument in our paper that was published in SIAM J. Numer. Anal., 49 (2011), pp. 2407-2429. The result is a pathwise error estimate, which is proved by an application of the factorization argument for stochastic convolutions.
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