| 1. |
- Persson, Ingemar, et al.
(author)
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On the convergence of multigrid methods for flow problems
- 1999
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In: Electronic Transactions on Numerical Analysis. - 1068-9613. ; 8, s. 46-87
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Journal article (peer-reviewed)abstract
- We prove two theorems on the residual damping in multigrid methods when solving convection dominated diffusion equations and shock wave problems, discretized by the streamline diffusion finite element method. The first theorem shows that a V-cycle, including sufficiently many pre and post smoothing steps, damps the residual in LIloc for a constant coefficient convection problem with small diffusion in two space dimensions, without the assumption that the coarse grid is sufficiently fine. The proof is based on discrete Green's functions for the smoothing and correction operators on a uniform unbounded mesh aligned with the characteristic. The second theorem proves a similar result for a certain continuous version of a two grid method, with Isotropic artificial diffusion, applied to a two dimensional Burgers shock wave problem. We also present numerical experiments that verify the residual damping dependence on the equation, the choice of artificial diffusion and the number of smoothing steps. In particular numerical experiments show improved convergence of the multigrid method, with damped Jacobi smoothing steps, for the compressible Navier-Stokes equations in two space dimensions by using the theoretically suggested exponential increase of the number of smoothing steps on coarser meshes, as compared to the same amount of work with constant number of smoothing steps on each level.
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| 2. |
- Goodman, Jonathan, et al.
(author)
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A remark on the stability of viscous shock-waves
- 1994
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In: SIAM Journal on Mathematical Analysis. - PHILADELPHIA : SIAM PUBLICATIONS. - 0036-1410 .- 1095-7154. ; 25:6, s. 1463-1467
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Journal article (peer-reviewed)abstract
- Recently, Szepessy and Xin gave a new proof of stability of viscous shock waves. A curious aspect of their argument is a possible disturbance of zero mass, but log(t)t-1/2 amplitude in the vicinity of the shock wave. This would represent a previously unobserved phenomenon. However, only an upper bound is established in their proof. Here, we present an example of a system for which this phenomenon can be verified by explicit calculation. The disturbance near the shock is shown to be precisely of order t-1/2 in amplitude.
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| 3. |
- Hansbo, Peter F G, 1959, et al.
(author)
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A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations
- 1990
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In: Computer Methods in Applied Mechanics and Engineering. - LAUSANNE : Elsevier BV. ; 84:2, s. 175-192
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Journal article (peer-reviewed)abstract
- In this paper a streamline diffusion finite element method is introduced for the time-dependent incompressible Navier-Stokes equations in a bounded domnain in R^2 and R^3 in the case of high Reynolds number flow. An error estimate is proved and numerical results are given. The method is based on a mixed velocity-pressure formulation using the same finite element discretization of space-time for the velocity and the pressure spaces, which consists of piecewise linear functions, together with certain least-squares modifications of the Galerkin variational formulation giving added stability without sacrificing accuracy.
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| 4. |
- Hoel, Hakon, et al.
(author)
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Classical langevin dynamics derived from quantum mechanics
- 2020
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In: Discrete and continuous dynamical systems. Series B. - : AMER INST MATHEMATICAL SCIENCES-AIMS. - 1531-3492 .- 1553-524X. ; 25:10, s. 4001-4038
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Journal article (peer-reviewed)abstract
- The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.
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| 5. |
- Huang, Xin, et al.
(author)
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Canonical mean-field molecular dynamics derived from quantum mechanics
- 2022
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In: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - : EDP Sciences. - 2822-7840 .- 2804-7214. ; 56:6, s. 2197-2238
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Journal article (peer-reviewed)abstract
- Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.
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| 6. |
- Johnson, Claes, et al.
(author)
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Adaptive finite element methods for conservation laws based on a posteriori error estimates
- 1995
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In: Communications on Pure and Applied Mathematics. - NEW YORK : John Wiley & Sons. - 0010-3640 .- 1097-0312. ; 48:3, s. 199-234
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Journal article (peer-reviewed)abstract
- We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way.
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| 7. |
- Johnson, Claes, et al.
(author)
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On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws
- 1990
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In: Mathematics of Computation. - PROVIDENCE : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 54:189, s. 107-129
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Journal article (peer-reviewed)abstract
- We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size. With this term present, we prove a maximum norm bound for finite element solutionsof Burgers' equation an thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.
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| 8. |
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| 9. |
- Kammonen, Aku, 1984-, et al.
(author)
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Adaptive random fourier features with metropolis sampling
- 2019
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In: Foundations of Data Science. - : American Institute of Mathematical Sciences. - 2639-8001. ; 0:0, s. 0-0
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Journal article (peer-reviewed)abstract
- The supervised learning problem todetermine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$with one hidden layer is studied asa random Fourier features algorithm. The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$,are sampled using an adaptive Metropolis sampler.The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability$\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$,for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work.This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods.Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is testedboth on synthetic data and a real-world high-dimensional benchmark.
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| 10. |
- Kammonen, Aku, 1984-, et al.
(author)
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Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics
- 2018
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In: Annales Henri Poincaré. - : Springer Nature. - 1424-0637 .- 1424-0661. ; 19, s. 2727-2781
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Journal article (peer-reviewed)abstract
- It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.
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