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  • 1. Babuska, I.
    et al.
    Liu, K. M.
    Tempone, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Solving stochastic partial differential equations based on the experimental data2003In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 13, no 3, p. 415-444Article in journal (Refereed)
    Abstract [en]

    We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, omega) is expressed by a finite number N-KL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable N-KL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, omega) is completely known. When a random function a(x, omega) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

  • 2. Babuska, I.
    et al.
    Nobile, F.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    A systematic approach to model validation based on Bayesian updates and prediction related rejection criteria2008In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 197, no 29-32, p. 2517-2539Article in journal (Refereed)
    Abstract [en]

    This work describes a solution to the validation challenge problem posed at the SANDIA Validation Challenge Workshop, May 21-23, 2006, NM. It presents and applies a general methodology to it. The solution entails several standard steps, namely selecting and fitting several models to the available prior information and then sequentially rejecting those which do not perform satisfactorily in the validation and accreditation experiments. The rejection procedures are based on Bayesian updates, where the prior density is related to the current candidate model and the posterior density is obtained by conditioning on the validation and accreditation experiments. The result of the analysis is the computation of the failure probability as well as a quantification of the confidence in the computation, depending on the amount of available experimental data.

  • 3. Babuska, I.
    et al.
    Nobile, F.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Formulation of the static frame problem2008In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 197, no 29-32, p. 2496-2499Article in journal (Refereed)
    Abstract [en]

    This report describes a static framework validation challenge problem used in the SANDIA Validation Challenge Workshop, May 21-23, 2006. The challenge problem has clear engineering character, is simple to state and allows many different approaches to solve it. The regulatory assessment problem is to estimate the probability of a given vertical displacement to exceed a prescribed threshold.

  • 4. Babuska, I.
    et al.
    Nobile, F.
    Tempone, Raul
    Reliability of computational science2007In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 23, no 4, p. 753-784Article in journal (Refereed)
    Abstract [en]

    Today's computers allow us to simulate large, complex physical problems. Many times the mathematical models describing such problems are based on a relatively small amount of available information such as experimental measurements. The question arises whether the computed data could be used as the basis for decision in critical engineering, economic, and medicine applications. The representative list of engineering accidents occurred in the past years and their reasons illustrate the question. The paper describes a general framework for verification and validation (V&V) which deals with this question. The framework is then applied to an illustrative engineering problem, in which the basis for decision is a specific quantity of interest, namely the probability that the quantity does not exceed a given value. The V&V framework is applied and explained in detail. The result of the analysis is the computation of the failure probability as well as a quantification of the confidence in the computation, depending on the amount of available experimental data.

  • 5. Babuska, I.
    et al.
    Nobile, F.
    Tempone, Raul
    Worst case scenario analysis for elliptic problems with uncertainty2005In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 101, no 2, p. 185-219Article in journal (Refereed)
    Abstract [en]

    This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.

  • 6. Babuska, I.
    et al.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Galerkin finite element approximations of stochastic elliptic partial differential equations2004In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, no 2, p. 800-825Article in journal (Refereed)
    Abstract [en]

    We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

  • 7. Babuska, I.
    et al.
    Tempone, Raul
    Zouraris, G. E.
    Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation2005In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 194, no 16-dec, p. 1251-1294Article in journal (Refereed)
    Abstract [en]

    This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y, x), y is an element of Gamma, x is an element of D, where D subset of R-d, d = 1, 2, 3, and Gamma is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Gamma x D. A tensor product finite element method, of h-version in D and k-, or, p-version in Gamma, is proposed for the approximation of it. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Gamma, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.

  • 8. Babuska, Ivo
    et al.
    Nobile, Fabio
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    A stochastic collocation method for elliptic partial differential equations with random input data2007In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 45, no 3, p. 1005-1034Article in journal (Refereed)
    Abstract [en]

    In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 ( 2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

  • 9. Babuška, I.
    et al.
    Tempone, Raúl
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
    Static frame challenge problem: Summary2008In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, Vol. 197, no 29-32, p. 2572-2577Article in journal (Refereed)
    Abstract [en]

    This paper summarizes five solutions to the static frame validation challenge problem. The main goal is to highlight the different approaches present at each stage of the solution process. These include, among others, the description of the elastic properties of the frame's material and their calibration, the use of the validation and accreditation experiments for eventual model rejection and the final statement on the desired regulatory compliance, together with its reliability depending on the amount of available experimental data. It is shown that different methodologies lead to significantly different results. Finally, the conclusions highlight the main findings.

  • 10. 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)
  • 11.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Tembine, Hamidou
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tempone, Raul
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A stochastic maximum principle for risk-sensitive mean-field-type control2014In: Proceedings of the IEEE Conference on Decision and Control, IEEE conference proceedings, 2014, no February, p. 3481-3486Conference paper (Refereed)
    Abstract [en]

    In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle for optimal control of stochastic differential equations of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng's type stochastic maximum principle is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type under linear stochastic dynamics with exponential quadratic cost function. Explicit characterizations are given for both mean-field free and mean-field risk-sensitive models.

  • 12.
    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.

  • 13. Gren, I. M.
    et al.
    Destouni, G.
    Tempone, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Cost effective policies for alternative distributions of stochastic water pollution2002In: Journal of Environmental Management, ISSN 0301-4797, E-ISSN 1095-8630, Vol. 66, no 2, p. 145-157Article in journal (Refereed)
    Abstract [en]

    This study investigates the role for cost effective coastal water management with regard to different assumptions of probability distributions (normal and lognormal) of pollutant transports to coastal waters. The analytical results indicate a difference in costs for a given probability of achieving a certain pollutant load target whether a normal or lognormal distribution is assumed. For low standard deviations and confidence intervals, the normal distribution implies a lower cost while the opposite is true for relatively high standard deviations and confidence intervals. The associated cost effective charges and permit prices are higher for lognormal distributions than for normal distributions at relatively high confidence intervals and probabilities of achieving the target. An application to Himmerfjarden-an estuary south of Stockholm, Sweden-shows that the minimum costs of achieving a 50 per cent reduction in nitrogen load to the coast varies more for a lognormal than normal probability distribution. At high coefficient of variation and chosen probability of achieving the target, the minimum cost under a lognormal assumption can be three times as high as for a normal distribution.

  • 14. Hills, Richard G.
    et al.
    Pilch, Martin
    Dowding, Kevin J.
    Redhorse, John
    Paez, Thomas L.
    Babuska, Ivo
    Tempone, Raúl
    Validation Challenge Workshop2008In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 197, no 29-32, p. 2375-2380Article in journal (Refereed)
    Abstract [en]

    This special issue presents the results of the Sandia organized Model Validation Challenge Workshop, held May 2006. The workshop brought together researchers from different fields to present various approaches to model validation, and focused on the methodological elements of model validation rather than on model building. Three problems were defined in the disciplines of structural statics, structural dynamics, and heat transfer, all with a uniform structure. The workshop was specifically designed to investigate the relative merits of different approaches to hierarchal model validation through application to these problems. This paper describes a hierarchal approach in the challenge problems, presents the uniform conceptual framework that was used for the challenge problem definitions, and provides an overview of the organization of this special issue.

  • 15.
    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.

  • 16.
    Kiessling, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tempone, Raúl
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Computable error estimates of a finite difference scheme for option pricing in exponential Lévy modelsManuscript (preprint) (Other academic)
    Abstract [en]

    Option prices in exponential L´evy models solve certain partial integrodifferential equations (PIDEs). This work focuses on a finite difference scheme that issuitable for solving such PIDEs. The scheme was introduced in [Cont and Voltchkova, SIAM J. Numer. Anal., 43(4):1596–1626, 2005]. The main results of this work are new estimates of the dominating error terms, namely the time and space discretization errors. In addition, the leading order terms of the error estimates are determined in computable form. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschtitz continuous as in previous works.

    If the underlying Lévy process has infinite jump activity, then the jumps smallerthan some ε> 0 are approximated by diffusion. The resulting diffusion approximationerror is also estimated, with leading order term in computable form, as well as its effecton the space and time discretization errors. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the parameter ε.

  • 17. Kiessling, Jonas
    et al.
    Tempone, Raúl
    Diffusion approximation of Lévy processes with a view towards finance2011In: Monte Carlo Methods and Applications, ISSN 0929-9629, Vol. 17, no 1, p. 11-45Article in journal (Refereed)
    Abstract [en]

    Let the (log-)prices of a collection of securities be given by a d –dimensional L´evy process Xt having infinite activity and a smooth density. The value of a European contract with pay off g(x) maturing at T is determined by E[g(XT )]. Let ¯XT be a finite activity approximation to XT , where diffusion is introduced to approximate jumps smaller than a given truncation level ! > 0. The main result of this work is a derivation of an error expansion for the resulting model error, E[g(XT )g( ¯XT )], with computable leading order term. Our estimate depends both on the choice of truncation level ! and the contract payo ff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error. Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.

  • 18.
    Kiessling, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tempone, Raúl
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Diffusion approximation of Lévy processes with a view towardsfinanceManuscript (preprint) (Other academic)
    Abstract [en]

    Let the (log-)prices of a collection of securities be given by a d–dimensional Lévy process Xt having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(XT )]. Let ¯XT be a finite activity approximation to XT , where diffusion is introduced to approximate jumps smaller than a given truncation level ε > 0. The main result of this work is a derivationof an error expansion for the resulting model error, E[g(XT )−g( ¯XT )], with computable leading order term. Our estimate depends both on the choice of truncation level ε and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error.

    Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.

  • 19. 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.

  • 20. 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.

  • 21.
    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)
  • 22.
    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.

  • 23. Nobile, F.
    et al.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Webster, C. G.
    A sparse grid stochastic collocation method for partial differential equations with random input data2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, p. 2309-2345Article in journal (Refereed)
    Abstract [en]

    This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coeffcients and forcing terms ( input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using L-q norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem ( number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.

  • 24. Nobile, F.
    et al.
    Tempone, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Webster, C. G.
    An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data2008In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 46, no 5, p. 2411-2442Article in journal (Refereed)
    Abstract [en]

    This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms ( input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen-Loeve truncations of "smooth" random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates ( sub) exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

  • 25. Oden, J. T.
    et al.
    Babuska, I.
    Nobile, F.
    Feng, Y. S.
    Tempone, Raul
    Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty2005In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 194, no 05-feb, p. 195-204Article in journal (Refereed)
    Abstract [en]

    The reliability of computer predictions of physical events depends on several factors: the mathematical model of the event, the numerical approximation of the model, and the random nature of data characterizing the model. This paper addresses the mathematical theories, algorithms, and results aimed at estimating and controlling modeling error, numerical approximation error, and error due to randomness in material coefficients and loads. A posteriori error estimates are derived and applications to problems in solid mechanics are presented.

  • 26.
    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.

  • 27.
    Tempone Olariaga, Raul
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Numerical Complexity Analysis of Weak Approximation of Stochastic Differential Equations2002Doctoral thesis, comprehensive summary (Other scientific)
    Abstract [en]

    The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods.

    The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling.

    The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70.

  • 28.
    Tempone Olariaga, Raúl
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Weak approximation of ItÔ stochastic differential equations and related adaptive algorithms2000Licentiate thesis, comprehensive summary (Other scientific)
1 - 28 of 28
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