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  • 1. Are, Sasanka
    et al.
    Katsoulakis, Markos A.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Coarse-Grained Langevin Approximations and Spatiotemporal Acceleration for Kinetic Monte Carlo Simulations of Diffusion of Interacting Particles2009In: Chinese Annals of Mathematics. Series B, ISSN 0252-9599, E-ISSN 1860-6261, Vol. 30, no 6, p. 653-682Article in journal (Refereed)
    Abstract [en]

    Kinetic Monte Carlo methods provide a powerful computational tool for the simulation of microscopic processes such as the diffusion of interacting paxticles oil a surface, at a detailed atomistic level. However such algorithms are typically computationally expensive and are restricted to fairly small spatiotemporal scales. One approach towards overcoming this problem was the development of coarse-gained Monte Carlo algorithms. In recent literature, these methods were shown to be capable of efficiently describing much larger length scales while still incorporating information on microscopic interactions and fluctuations. In this paper, a coarse-grained Langevin system of stochastic differential equations as approximations of diffusion of interacting particles is derived, based on these earlier coarse-grained models. The authors demonstrate the asymptotic equivalence of transient and long time behavior of the Langevin approximation and the underlying microscopic process, using asymptotics methods such as large deviations for interacting particles systems, and furthermore, present corresponding numerical simulations, comparing statistical quantities like mean paths, auto correlations and power spectra of the microscopic and the approximating Langevin processes. Finally, it is shown that the Langevin approximations presented here are much more computationally efficient than conventional Kinetic Monte Carlo methods, since in addition to the reduction in the number of spatial degrees of freedom in coarse-grained Monte Carlo methods, the Langevin system of stochastic differential equations allows for multiple particle moves in a single timestep.

  • 2. Bayer, C.
    et al.
    Hoel, Håkon
    Kadir, Ashraful
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Plechác, Petr
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces2015In: Applied Mathematics Research eXpress, ISSN 1687-1200, E-ISSN 1687-1197, no 2, p. 329-417Article in journal (Refereed)
    Abstract [en]

    The difference of the values of observables for the time-independent Schrödinger equation, with matrix-valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states, and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.

  • 3.
    Bayer, Christian
    et al.
    Weierstrass Institute for Applied Analysis and Stochastics.
    Hoel, Håkon
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Kadir, Ashraful
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Plechac, Petr
    Dept. of Mathematical Sciences, University of Delaware.
    Sandberg, Mattias
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Tempone, Raul
    Division of Mathematics, King Abdullah University of Science and Technology.
    How accurate is molecular dynamics?2012Report (Other academic)
    Abstract [en]

    Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on theBorn-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy.

  • 4.
    Bayer, Christian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tempone, Raúl
    Adaptive weak approximation of reflected and stopped diffusions2010In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 16, no 1, p. 1-67Article in journal (Refereed)
    Abstract [en]

    We study the weak approximation problem of diusions, which are reflected at a subset of the boundary of a domain and stopped at the remaining boundary. First, we derive an error representation for the projected Euler method of Costantini, Pacchiarotti and Sartoretto [Costantini et al., SIAM J. Appl. Math., 58(1):73–102, 1998], based on which we introduce two new algorithms. The first one uses a correction term from the representation in order to obtain a higher order of convergence, but the computation of the correction term is, in general, not feasible in dimensions d> 1. The second algorithm is adaptive in the sense of Moon, Szepessy, Tempone and Zouraris [Moon et al., Stoch. Anal. Appl., 23:511–558, 2005], using stochastic refinement of the time grid based on a computable error expansion derived from the representation. Regarding the stopped diusion, it is based in the adaptive algorithm for purely stopped di usions presented in Dzougoutov, Moon, von Schwerin, Szepessy and Tempone [Dzougoutov et al., Lect. Notes Comput. Sci. Eng., 44, 59–88, 2005]. We give numerical examples underlining the theoretical results.

  • 5. 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.

  • 6. Björk, T
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zourari, Georgios
    Monte Carlo euler approximation if HJM term structure financial models2001In: Stochastic Numerics 2001 at ETH, Zurich, Switzerland. February 19 - 21, 2001, 2001Conference paper (Other academic)
  • 7.
    Carlsson, Jesper
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Sandberg, Mattias
    Univ Oslo, CMA.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Symplectic Pontryagin Approximations for Optimal Design2009In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 43, no 1, p. 3-32Article in journal (Refereed)
    Abstract [en]

    The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L 2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

  • 8.
    Dzougoutov, Anna
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Moon, Kyoung-Sook
    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 Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tempone, Raul
    ICES, The University of Texas at Austin.
    Adaptive Monte Carlo Algorithms for Stopped Diffusion2005In: Multiscale Methods in Science and Engineering, Berlin: Springer-Verlag , 2005, 44, p. 59-88Chapter in book (Other academic)
    Abstract [en]

    We present adaptive algorithms for weak approximation of stopped diffusion using the Monte Carlo Euler method. The goal is to compute an expected value of a given function g depending on the solution X of an Itô stochastic differential equation and on the first exit time τ from a given domain.

    The main steps in the extension to stopped diffusion processes are to use a conditional probability to estimate the first exit time error and introduce difference quotients to approximate the initial data of the dual solutions.

  • 9. Goodman, Jonathan
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zumbrun, Kevin
    A remark on the stability of viscous shock-waves1994In: SIAM Journal on Mathematical Analysis, ISSN 0036-1410, E-ISSN 1095-7154, Vol. 25, no 6, p. 1463-1467Article in journal (Refereed)
    Abstract [en]

    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.

  • 10. Hall, E. J.
    et al.
    Hoel, H.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tempone, R.
    Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. A3773-A3807Article in journal (Refereed)
    Abstract [en]

    We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.

  • 11. Hansbo, P
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    A velocity pressure streamline diffusion finite element method for Navier-Stokes equations1990In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 84, no 2, p. 175-192Article in journal (Refereed)
    Abstract [en]

    A streamline diffusion finite-element method is introduced for the time-dependent incompressible Navier-Stokes equations in a bounded domain in R squared and R cubed in the case of a flow with a high Reynolds number. 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 consist of piecewise linear functions, together with certain least-squares modifications of the Galerkin variational formulation giving added stability without sacrificing accuracy.

  • 12. Hansbo, Peter
    et al.
    Szepessy, Anders
    A velocity pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations1990In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 84, no 2, p. 175-192Article in journal (Refereed)
  • 13. Hoel, H.
    et al.
    Von Schwerin, E.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tempone, R.
    Implementation and analysis of an adaptive multilevel Monte Carlo algorithm2014In: Monte Carlo Methods and Applications, ISSN 0929-9629, Vol. 20, no 1, p. 1-41Article in journal (Refereed)
    Abstract [en]

    We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm.

  • 14.
    Hoel, Håkon
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Von Schwerin, Erik
    Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Adaptive Multi Level Monte Carlo Simulation2012In: Numerical Analysis of Multiscale Computations: Proceedings of a Winter Workshop at the Banff International Research Station 2009, Springer, 2012, Vol. 82, p. 217-234Conference paper (Refereed)
    Abstract [en]

    This work generalizes a multilevel Forward Euler Monte Carlo methodintroduced in [5] for the approximation of expected values depending onthe solution to an Itˆo stochastic differential equation. The work [5] proposedand analyzed a Forward Euler Multilevel Monte Carlo method basedon a hierarchy of uniform time discretizations and control variates to reducethe computational effort required by a standard, single level, ForwardEuler Monte Carlo method. This work introduces an adaptive hierarchyof non uniform time discretizations, generated by adaptive algorithms introducedin [11, 10]. These adaptive algorithms apply either deterministictime steps or stochastic time steps and are based on a posteriori error expansionsfirst developed in [14]. Under sufficient regularity conditions, ournumerical results, which include one case with singular drift and one withstopped diffusion, exhibit savings in the computational cost to achieve anaccuracy of O(TOL), from O`TOL−3´to O“`TOL−1 log (TOL)´2”. Wealso include an analysis of a simplified version of the adaptive algorithmfor which we prove similar accuracy and computational cost results.

  • 15.
    Hoel, Håkon
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    von Schwerin, Erik
    King Abdullah University of Science and Technology.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Tempone, Raul
    King Abdullah University of Science and Technology.
    Implementation and Analysis of an Adaptive Multilevel Monte Carlo Algorithm2012Report (Other academic)
    Abstract [en]

    This work generalizes a multilevel Monte Carlo (MLMC) method in-troduced in [7] for the approximation of expected values of functions depending on the solution to an Ito stochastic differential equation. The work [7] proposed and analyzed a forward Euler MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O( TOL^(−3) ) to O( TOL^(−2) log( TOL^(−1))^2 ) for a meansquare error of size 2 . This work uses instead a hierarchy of adaptivelyrefined, non uniform, time discretizations, generated by an adaptive algo-rithm introduced in [20, 19, 5]. Given a prescribed accuracy TOL in theweak error, this adaptive algorithm generates time discretizations basedon a posteriori expansions of the weak error first developed in [24]. Atheoretical analysis gives results on the stopping, the accuracy, and thecomplexity of the resulting adaptive MLMC algorithm. In particular, it isshown that: the adaptive refinements stop after a finite number of steps;the probability of the error being smaller than TOL is under certain as-sumptions controlled by a given confidence parameter, asymptotically asTOL → 0; the complexity is essentially the expected for MLMC methods,but with better control of the constant factors. We also show that themultilevel estimator is asymptotically normal using the Lindeberg-FellerCentral Limit Theorem. These theoretical results are based on previouslydeveloped single level estimates, and results on Monte Carlo stoppingfrom [3]. Our numerical tests include cases, one with singular drift andone with stopped diffusion, where the complexity of uniform single levelmethod is O TOL−4 . In both these cases the results confirm the theoryby exhibiting savings in the computational cost to achieve an accuracy of O(TOL), from O( TOL^(−3) )for the adaptive single level algorithm toessentially O( TOL^(−2) log(TOL−1)^2 ) for the adaptive MLMC.

  • 16. Jaffre, Jerome
    et al.
    Johnson, Claes
    Szepessy, Anders
    Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws1995In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 5, no 3, p. 367-386Article in journal (Refereed)
    Abstract [en]

    We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q greater than or equal to 0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

  • 17. Johnson, Claes
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Adaptive finite element methods for conservation laws based on a posteriori error estimates1995In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 48, no 3, p. 199-234Article in journal (Refereed)
    Abstract [en]

    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.

  • 18. Johnson, Claes
    et al.
    Szepessy, Anders
    Hansbo, Peter
    On the convergence of shock-capturing  streamline diffusion finite element methods for hyperbolic conservation laws1990In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 54, no 189, p. 107-129Article in journal (Refereed)
  • 19.
    Kadir, Ashraful
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An Adaptive Mass Algorithm for Car-Parrinello and Ehrenfest ab initio molecular dynamicsManuscript (preprint) (Other academic)
    Abstract [en]

    Ehrenfest and Car-Parrinello molecular dynamics are computational alternatives to approximate Born-Oppenheimer molecular dynamics without solving the electron eigenvalue problem at each time-step. A non-trivial issue is to choose the artificial electron mass parameter appearing in the Car-Parrinello method to achieve  both good accuracy and high computational efficiency. In this paper, we propose an algorithm, motivated by the Landau-Zener probability, to systematically choose an artificial mass dynamically, which makes the Car-Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data. Numerical experiments for simple model problems show that the time-dependent adaptive artificial mass parameter improves the efficiency of the Car-Parrinello and Ehrenfest molecular dynamics.

  • 20.
    Kadir, Ashraful
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    von Schwerin, Erik
    An Adaptive Mass Algorithm for Ehrenfest and Car-Parrinello ab initio Molecular Dynamics: dynamics with several electronic statesManuscript (preprint) (Other academic)
    Abstract [en]

    This paper extends previous numerical studies on ficticious adaptive mass algorithms for Car-Parrinello and Ehrenfest dynamics to problems with more than two electron states. The main conclusion in this work is that it is necessary to resolve the near avoided conicial intersections between all electron eigenvalue gaps, also beween occupied states.

  • 21.
    Kammonen, Aku
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Plecháč, P.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Canonical Quantum Observables for Molecular Systems Approximated by Ab Initio Molecular Dynamics2018In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, Vol. 19, no 9, p. 2727-2781Article in journal (Refereed)
    Abstract [en]

    It is known that ab initio molecular dynamics based on the electron ground-state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the 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 semiclassical Weyl law to show that canonical 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 result covers observables that depend on time correlations. A combination of the Hilbert–Schmidt inner product for quantum operators and Weyl’s law shows 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 position and the observables are in the diagonal form with respect to the electron eigenstates.

  • 22. Karlsson, Jesper
    et al.
    Larsson, Stig
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tempone, Raul
    An error estimate for symplectic euler approximation of optimal control problems2015In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 37, no 2, p. A946-A969Article in journal (Refereed)
    Abstract [en]

    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.

  • 23. Katsoulakis, Markos A.
    et al.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Stochastic hydrodynamical limits of particle systems2006In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 4, no 3, p. 513-549Article in journal (Refereed)
    Abstract [en]

    Even small noise can have substantial influence on the dynamics of differential equations, e.g. for nucleation/coarsening and interface dynamics in phase transformations. The aim of this work is to establish accurate models for the noise in macroscopic differential equations, related to phase transformations/reactions, derived from more fundamental microscopic master equations. For this purpose the mathematical paradigm of the dynamic Ising model is considered in the relatively tractable case of stochastic spin flip dynamics and long range spin/spin interactions. More specifically, this paper shows that localized spatial averages, with width epsilon, of solutions to such Ising systems with long range interaction of range O(1), are approximated with error O(epsilon + (gamma/epsilon)(2d)) in distribution by a solution of an Ito stochastic differential equation, with drift as in the corresponding mean field model and a small diffusion coefficient of order (gamma/epsilon)(d/2), generating noise with spatial correlation length epsilon, where gamma is the distance between neighboring spin sites on a uniform periodic lattice in R-d. To determine the correct noise is subtle in the sense that there are expected values, i.e. observables, that require different noise: the expected values that can be accurately approximated by the Einstein-diffusion and the expected values that need an alternative diffusion related to large deviation theory are identified; for instance dendrite dynamics up to a bounded time needs Einstein diffusion while transition rates need a different diffusion model related to invariant measures. The elementary proofs use O((gamma/epsilon)(2d)) consistency of the Kolmogorov-backward equations for the averaged spin and the stochastic differential equation and show that the long range interaction yields smoothing, which contributes with the O(epsilon) error. A new aspect of the derivation is that the error, based on residuals and weights, is computable and suitable for adaptive refinements and modeling.

  • 24.
    Lindholm, Love
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. None.
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A mean field game model of an electricity market with consumers minimizing energy cost through dynamic battery usageManuscript (preprint) (Other academic)
    Abstract [en]

    This work contains a model of an electricity market consisting of consumers who own batteries that they charge and discharge in an optimal way. The goal of each individual customer is to minimize their total electricity cost, not by changing how much they consume, but by utilizing an optimal strategy for their battery usage. For each consumer we therefore have a value function. Since all consumers are assumed to be equal, their associated value functions are also equal. The optimization problem to determine the optimal battery usage depends on the electricity price, which in turn depends on the total electricity consumption. The consumption is given as a solution to a Kolmogorov forward equation, which involves the battery usage. Hence the Hamilton-Jacobi-Bellman and Kolmogorov equations need to be solved together as a coupled system of PDEs. We devise a numerical scheme for this system and show some simulations. We also prove a result on the existence and uniqueness of solutions to the system of PDEs.

  • 25.
    Lindholm, Love
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. None.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. None.
    Local volatility calibration with optimal control in a Hamiltonian frameworkManuscript (preprint) (Other academic)
    Abstract [en]

    We formulate the calibration of a local volatility function that makes a solution to Dupie's equation match market data as an optimal control problem for which optimality conditions are given by a Hamiltonian system. Regularization is added by mollifying the Hamiltonian functional in this system. We have direct access to the Jacobian matrix of the Hamiltonian system, and can therefore employ a Newton based method in the solving phase, whereas other studies tend to use gradient based methods or quasi Newton algorithms. We illustrate our method by calibrating a volatility function to market data on the Euro Stoxx 50 index and find that our algorithm is both accurate and robust.

  • 26. Moon, K. S.
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, G. E.
    Convergence rates for adaptive approximation of ordinary differential equations2003In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 96, no 1, p. 99-129Article in journal (Refereed)
    Abstract [en]

    This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by O(N). The alternative version with constant tolerance stops with O(N log N) total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the L 1/p+1 quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L-1-norm of the error density.

  • 27. Moon, K. S.
    et al.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone, Raúl
    Zouraris, G. E.
    A variational principle for adaptive approximation of ordinary differential equations2003In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 96, no 1, p. 131-152Article in journal (Refereed)
    Abstract [en]

    A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error = Sigma local error . weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms.

  • 28. Moon, K.-S.
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tempone, Raúl
    Zouraris, G
    Hyperbolic differential equations and adaptive numerics2001In: / [ed] J. F. Blowey, J. P. Coleman and A. Craig, 2001Conference paper (Refereed)
  • 29. Moon, Kyoung-Sook
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Convergence rates for adaptive weak approximation of stochastic differential equations2005In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 23, no 3, p. 511-558Article in journal (Refereed)
    Abstract [en]

    Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil optimal stochastic time steps or optimal deterministic time steps are studied. The analysis of their computational complexity combines the error expansions with a posteriori leading order term introduced in Szepessy et al. [Szepessy, A.. R. Tempone, and G. Zouraris. 2001. Comm. Pare Appl. Math. 54:1169-1214] and ail extension of the convergence results for adaptive algorithms approximating deterministic ordinary differential equations, derived in Moon et al. [Moon, K.-S., A. Szepessy, R. Tempone, and G. Zouraris. 2003. Numer. Malh. 93:99-129]. The main step in the extension is the proof of the almost sure convergence of the error density. Both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps.

  • 30.
    Moon, Kyoung-Sook
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Zouraris, Georgios
    Div of Applied Math - Statistics, Univ of Crete.
    Stochastic Dierential Equations: Model and Numerics2008Other (Refereed)
  • 31.
    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.

  • 32.
    Moon, Kyoung-Sook
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    von Schwerin, Erik
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    An Adaptive Algorithm for Ordinary, Stochastic and Partial Differential Equations2005In: Recent Advances in Adaptive Computation, Providence: American Mathematical Society , 2005, p. 325-343Chapter in book (Other academic)
    Abstract [en]

    The theory of a posteriori error estimates suitable for adaptive refinement is well established. This work focuses on the fundamental, but less studied, issue of convergence rates of adaptive algorithms. In particular, this work describes a simple and general adaptive algorithm applied to ordinary, stochastic and partial differential equations with proven convergence rates. The presentation has three parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. It is explained how to measure optimal convergence rates for approximation of functionals of the solution, and why convergence of the error density is always useful and subtle in the case of stochastic and partial differential equations. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor.

  • 33.
    Mordecki, Ernesto
    et al.
    Universidad de la República, Iguá 4225, Montevideo, Uruguay.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tempone, Raúl
    Zouraris, Georgios
    Div of Applied Math - Statistics, Univ of Crete.
    Adaptive weak approximation of diffusions with jumps2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 4, p. 1732-1768Article in journal (Refereed)
    Abstract [en]

    This work develops Monte Carlo Euler adaptive time stepping methods for the weak approximation problem of jump diffusion driven stochastic differential equations. The main result is the derivation of a new expansion for the omputational error, with computable leading order term in a posteriori form, based on stochastic flows and discrete dual backward problems which extends the results in [STZ]. These expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or quasi-deterministic time steps are described. Numerical examples show the performance of the proposed error approximation and of the described adaptive time-stepping methods.

  • 34. Moussa, Ben
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Scalar conservation laws with boundary conditions and rough data measure solutions2002In: Methods and Applications of Analysis, ISSN 1073-2772, E-ISSN 1945-0001, Vol. 9, no 4, p. 579-598Article in journal (Refereed)
    Abstract [en]

    Uniqueness and existence of $L^$#x221E;$ solutions to initial boundary value problems for scalar conservation laws, with continuous flux functions, is derived by $L^1$ contraction of Young measure solutions. The classical Kruzkov entropies, extended in Bardos, LeRoux and Nedelec’s sense to boundary value problems, are sufficient for the contraction. The uniqueness proof uses the essence of Kruzkov’s idea with his symmetric entropy and entropy flux functions, but the usual doubling of variables technique is replaced by the simpler fact that mollified measure solutions are in fact smooth solutions. The mollified measures turn out to have not only weak but also strong boundary entropy flux traces. Another advantage with the Young measure analysis is that the usual assumption of Lipschitz continuous flux functions can be relaxed to continuous fluxes, with little additional work

  • 35.
    Persson, Ingemar
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Samuelsson, Klas
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    On the convergence of multigrid methods  for flow problems1999In: Electronic Transactions on Numerical Analysis, ISSN 1068-9613, E-ISSN 1068-9613, Vol. 8, p. 46-87Article in journal (Refereed)
    Abstract [en]

    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.

  • 36.
    Sandberg, Mattias
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Convergence rates of symplectic pontryagin approximations in optimal control theory2006In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 40, no 1, p. 149-173Article in journal (Refereed)
    Abstract [en]

     Many inverse problems for differential equations can be formulated as optimal control problems. It is well known that inverse problems often need to be regularized to obtain good approximations. This work presents a systematic method to regularize and to establish error estimates for approximations to some control problems in high dimension, based on symplectic approximation of the Hamiltonian system for the control problem. In particular the work derives error estimates and constructs regularizations for numerical approximations to optimally controlled ordinary differential equations in R-d, with non smooth control. Though optimal controls in general become non smooth, viscosity solutions to the corresponding Hamilton-Jacobi-Bellman equation provide good theoretical foundation, but poor computational efficiency in high dimensions. The computational method here uses the adjoint variable and works efficiently also for high dimensional problems with d >> 1. Controls can be discontinuous due to a lack of regularity in the Hamiltonian or due to colliding backward paths, i.e. shocks. The error analysis, for both these cases, is based on consistency with the Hamilton-Jacobi-Bellman equation, in the viscosity solution sense, and a discrete Pontryagin principle: the bi-characteristic Hamiltonian ODE system is solved with a C-2 approximate Hamiltonian. The error analysis leads to estimates useful also in high dimensions since the bounds depend on the Lipschitz norms of the Hamiltonian and the gradient of the value function but not on d explicitly. Applications to inverse implied volatility estimation, in mathematical finance, and to a topology optimization problem are presented. An advantage with the Pontryagin based method is that the Newton method can be applied to efficiently solve the discrete nonlinear Hamiltonian system, with a sparse Jacobian that can be calculated explicitly.

  • 37. Szepessy, Anders
    An existence result for scalar conservation laws using measure valued solutions1989In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 14, no 10, p. 1329-1350Article in journal (Refereed)
  • 38.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Atomistic and continuum models for phase change dynamics2006In: Proceedings of the International Congress of Mathematicians, 2006, p. 1563-1582Conference paper (Refereed)
    Abstract [en]

    The dynamics of dendritic growth of a crystal in an undercooled melt is determined by macroscopic diffusion-convection of heat and capillary forces acting on length scales compared to the nanometer width of the solid-liquid interface. Its modeling is useful for instance in processing techniques based on casting. The phase field method is widely used to study evolution of such microstructures of phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen�Cahn/Ginzburg�Landau equation modeling the dynamics of an order parameter determining the solid and liquid phases, including also stochastic fluctuations to obtain the qualitative correct result of dendritic side branching. This lecture presents some ideas to derive stochastic phase field models from atomistic formulations by coarse-graining molecular dynamics and kinetic Monte Carlo methods.

  • 39. Szepessy, Anders
    Convergence of a shock-capturing streamline diffusion finite element method for a conservation law in two space dimensions1989In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, p. 527-545Article in journal (Refereed)
  • 40.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Convergence of a streamline diffusion finite element method for a conservation law with boundary conditions1991In: RAIRO Modelling and numercial analysis modelisations, Vol. 25, no 5, p. 749-783Article in journal (Refereed)
    Abstract [en]

    A higher order accurate shock-capturing streamline diffusion finite element method for general scalar conservation laws is analysed; convergence towards the unique solution is proved for several space dimensions with initial and boundary conditions, using a uniqueness theorem for measure valued solutions. Furthermore, some numerical results are given.

  • 41.
    Szepessy, Anders
    KTH, Superseded Departments (pre-2005), Numerical Analysis and Computer Science, NADA.
    Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions1991In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 25, no 6, p. 749-783Article in journal (Refereed)
    Abstract [en]

    A higher order accurate shock-capturing streamline diffusion finite element method for general scalar conservation laws is analysed; convergence towards the unique solution is proved for several space dimensions with initial and boundary conditions, using a uniqueness theorem for measure valued solutions. Furthermore, some numerical results are given.

  • 42.
    Szepessy, Anders
    KTH, Superseded Departments, Mathematics.
    Dynamics and stability of a weak detonation wave1999In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 202, no 3, p. 547-569Article in journal (Refereed)
    Abstract [en]

    One dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable, i.e. initially perturbed waves tend asymptotically to translated weak detonation waves. This model system was derived as the low Math number limit of the one component reactive Navier-Stokes equations by Majda and Roytburd [SIAM J. Sci. Stat. Comput. 43, 1086-1118 (1983)], and its weak detonation waves have been numerically observed as stable. The analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave, The shock translation solves a nonlinear integral equation, obtained by Green's function techniques, and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder. The inverse operator of the convolution and detailed properties of the traveling wave reduce, by monotonicity, the remainder to a small L-1 perturbation.

  • 43.
    Szepessy, Anders
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    High frequency asymptotics for 2d viscous shocks2000In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 49, no 4, p. 1623-1671Article in journal (Refereed)
    Abstract [en]

    High frequency perturbations u of a weak planar two dimensional isentropic slightly viscous shock are shown to give a geometric optics expansion u = e(if/epsilon)(u(0) + epsilonu(1)), where the gradients of the phase functions delf solve eikonal equations related to Riemann problems for classical and over-compressive shocks. The main result is a rigorous expansion of the linearized equations describing in detail the perturbations and their refraction patterns caused by the viscous shock, when the wave length epsilon is large compared to the width of the shock. The expansion is based on the phases constructed from heteroclinic orbits related to the Riemann problems and on the principal term with L-2-norm //e(if)/(epsilon)u(0)//(L2) = O(1) obtained from the transport equations generated by the phases. The estimate of the remainder, //e(if/epsilon)epsilonu(1)//(L2) much less than 1, holds up to time 1 for sufficiently weak shocks and is proven by weighted energy estimates based on shock compressibility and relaxation techniques.

  • 44.
    Szepessy, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Langevin molecular dynamics derived from Ehrenfest dynamics2011In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 21, no 11, p. 2289-2334Article in journal (Refereed)
    Abstract [en]

    Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a KacZwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M -1/2) friction and o(M-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuationdissipation relation.

  • 45. Szepessy, Anders
    Measure valued solutions of scalar conservation laws with boundary conditions1989In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 107, no 2, p. 181-193Article in journal (Refereed)
    Abstract [en]

    We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L, weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.

  • 46.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    On shock wave stability1992In: / [ed] V. Bo, F. Bampi and G. Toscani, 1992Conference paper (Refereed)
  • 47.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    On the stability of Broadwell shocks1993Conference paper (Refereed)
    Abstract [en]

    Introduction We study the asymptotic behavior of perturbations of traveling shock waves of the Broadwell model, Gamma @ @t + @ @x Delta f + = f 2 0 Gamma f + f Gamma ; x 2 R I 1 ; t ? 0; @ @t f 0 = Gamma 1 2 (f 2 0 Gamma f + f Gamma ); (1:1a) Gamma @ @t Gamma @ @x Delta f Gamma = f 2 0 Gamma f + f Gamma ; with initial data (1:1b) (f + ; f 0 ; f Gamma )(Delta; 0) = f I (Delta) ;<F13.54

  • 48.
    Szepessy, Anders
    KTH, Superseded Departments (pre-2005), Numerical Analysis and Computer Science, NADA.
    On the stability of finite element methods for shock waves1992In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 45, no 8, p. 923-946Article in journal (Refereed)
    Abstract [en]

    this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form (1:1) u t + f(u) x = u xx ; x 2 R I ; t ? 0 ; u 2 R I u(\Delta; 0) = u 0 (\Delta) : We treat systems which are strictly hyperbolic. Such systems possess a smooth travelling wave solution, which is called a viscous p-shock wave solution, u(x; t) = OE(x \Gamma oet) x!\Sigma1 OE(x) = u \Sigma ; provided that the shock strength ffl j ju + \Gamma u \Gamma j is small [19], the constant states u \Sigma and the wave speed oe are related by the Rankine-Hugoniot condition (1:3a) f(u \Gamma ) \Gamma f(u+ ) = oe(u \Gamma \Gamma u+ )

  • 49. Szepessy, Anders
    et al.
    Johnson, Claes
    On the convergence of a finite element method for a nonlinear hyperbolic conservation law1987In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 49, no 180, p. 427-444Article in journal (Refereed)
  • 50.
    Szepessy, Anders
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, G. E.
    Adaptive weak approximation of stochastic differential equations2001In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 54, no 10, p. 1169-1214Article in journal (Refereed)
    Abstract [en]

    Adaptive time-stepping methods based on the Monte Carlo Euler method for weak approximation of Ito stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading-order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps.

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