Uncertain input data problems and the worst scenario method

Hlaváček, Ivan

doi:10.1007/s10492-007-0010-9

Cited by 41 publications

(53 citation statements)

References 32 publications

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“…This functional is chosen with respect to the aim of interest/expert decision. Although we have quite a lot of freedom in the definition of the criterion, certain continuity conditions must be fulfilled, for details see [3]. In our case we introduce…”

Section: Worst Scenario Methodsmentioning

confidence: 99%

“…On the other hand, the method does not require any probabilistic information on the data distribution and another pros is the relative simplicity and wide applicability of the method. For comprehensive guide on the method we refer [3] and the references therein. This monograph contains also clues to other approaches to problems with uncertainties.…”

Section: Remarksmentioning

confidence: 99%

“…To solve such a problem, we adopt a deterministic method called the worst scenario method introduced by H l a váč e k , see [3]. The main idea consists in defining a functional over a suitable set of admissible data serving as a criterion that evaluates some state/physical quantity from certain point of view.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On a solution of monotone type problems with uncertain inputs

Nechvátal

2011

Tatra Mountains Mathematical Publications

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Section: Worst Scenario Methodsmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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On a solution of monotone type problems with uncertain inputs

Nechvátal

2011

Tatra Mountains Mathematical Publications

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“…• [8] (Hlaváček et al, 2004) is a book focusing on the effects of uncertainty in the input data of the solution. It is a mathematical book addressing the worst scenario approach.…”

Section: A Selected Literature On Vandvmentioning

confidence: 99%

Reliability of computational science

Babuška

Nobile

Tempone

2007

Numerical Methods Partial

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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.

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“…Such a verification is the main purpose of a posteriori error estimation methods. Several approaches for deriving various a posteriori estimates for elliptic problems for errors measured in global (energy) norms ( [1], [4], [5], [17], [20], [39], [46], [47], [50]), or in terms of various local quantities ( [6], [12], [19], [25], [40], [45]) have been suggested (see also references in the above mentioned works). However, most of the estimates proposed there strongly use the fact that the computed solutions are true finite element (FE) approximations which, in fact, rarely happens in real computations, e.g., due to quadrature rules, forcibly stopped iterative processes, various round-off errors, or even bugs in computer codes.…”

Section: Introductionmentioning

confidence: 99%

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

Karátson

Korotov²

2009

Appl Math

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The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed in an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in course of the calculations.

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Uncertain input data problems and the worst scenario method

Cited by 41 publications

References 32 publications

On a solution of monotone type problems with uncertain inputs

On a solution of monotone type problems with uncertain inputs

Reliability of computational science

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

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