| 1. |
- Babuska, I., et al.
(författare)
-
A systematic approach to model validation based on Bayesian updates and prediction related rejection criteria
- 2008
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 197:29-32, s. 2517-2539
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 2. |
- Babuska, I., et al.
(författare)
-
Formulation of the static frame problem
- 2008
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 197:29-32, s. 2496-2499
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 3. |
- Babuska, I., et al.
(författare)
-
Galerkin finite element approximations of stochastic elliptic partial differential equations
- 2004
-
Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 42:2, s. 800-825
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 4. |
- Babuska, I., et al.
(författare)
-
Reliability of computational science
- 2007
-
Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 23:4, s. 753-784
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
(författare)
-
Solving elliptic boundary value problems with uncertain coefficients by the finite element method : the stochastic formulation
- 2005
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 194:16-dec, s. 1251-1294
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 6. |
- Babuska, I., et al.
(författare)
-
Solving stochastic partial differential equations based on the experimental data
- 2003
-
Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 13:3, s. 415-444
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 7. |
- Babuška, I., et al.
(författare)
-
Static frame challenge problem : Summary
- 2008
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825. ; 197:29-32, s. 2572-2577
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 8. |
- Babuska, I., et al.
(författare)
-
Worst case scenario analysis for elliptic problems with uncertainty
- 2005
-
Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 101:2, s. 185-219
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 9. |
- Oden, J. T., et al.
(författare)
-
Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty
- 2005
-
Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 194:05-feb, s. 195-204
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
