Uncertainty quanti cation (UQ) is a multidisciplinary area, that deals with quantitative characterization and reduction of uncertainties in applications. It is essential to certify the quality of numerical and experi- mental analyses of physical systems. The present manuscript aims to provide the reader with an introductory view about modeling and quanti cation of uncertainties in physical systems. In this sense, the text presents some funda- mental concepts in UQ, a brief review of probability basics notions, discusses, through a simplistic example, the fundamental aspects of probabilistic model- ing of uncertainties in a physical system, and explains what is the uncertainty propagation problem. Key words: Uncertainty quanti cation, stochastic modeling of uncertain- ties, probabilistic approach 1 An Introductory Overview on UQ Typically, highly complex engineering projects use both numerical simula- tions and experimental tests on prototypes to specify a certain system or component with desired characteristics. These two tools are used in a simi- lar way by scientists to investigate physical phenomena of interest. However, none of these approaches provides a response that is an exact reproduction of the physical system behaviour, because computational model and test rig are subject to uncertainties, which are intrinsic to modeling process (lack of knowledge on the physics) and model parameters (measurement inaccuracies, manufacturing variabilities, etc). NUMERICO - Nucleus of Modeling and Experimentation with Computers Universidade do Estado do Rio de Janeiro e-mail: americo@ime.uerj.br 1 2 Americo Cunha Jr In order to improve the reliability level of numerical results and exper- imental data, it is necessary to quantify the underlying uncertainties. The cautious experimentalists have been doing this for many decades, leading to a high level competence in what concerns the speci cation of the level of uncertainty in an experiment. It is worth remembering that an experiment that does not specify the level of uncertainty is not well seen by the tech- nical/scienti c community. On the other hand, just recently the numerical community has begun to pay attention to the need of speci ng the level of con dence for computer simulations. Uncertainty quanti cation (UQ) is a multidisciplinary area that deals with quantitative characterization and the reduction of uncertainties in applica- tions. One reason that UQ has gained such popularity over the last years, in numerical world, is due to several books on the subject have recently emerged [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. To motivate its study, we present three important scenarios where UQ is an essential tool: Decision making: Some risk decisions, which negative result can cause catas- trophic failure or huge nancial costs, need to be well analysed before a nal opinion by the responsible party. The possible variabilities that generate un- certain scenarios need to be taken into account in the analysis. The evaluation of these uncertain scenarios has the task of assisting the responsible party to minimize the chances of a wrong decision. Brie y, and in this context, UQ is essential to provide the necessary certi cation for a risk decision. Model validation: Experimental data are widely used to check the accuracy of a computational model which is used to emulate a real system. Although this procedure is already being used by scientists and engineers for many decades, there is still no universally accepted criteria to ensure the model quality. However, it is known that any robust criteria of model validation must take into account the simulation and experiment uncertainties. Robust design: An increasingly frequent requirement in several projects is the robust design of a component which consists make a speci c device low sensitive to variation on its properties. In a very simplistic way, we can summarize UQ objectives as (i) add error bars to experiments and simulations , and (ii) de ne a precise notion of the validated model . The rst objective is illustrated in Figure 1(a), which shows the comparison bewteen a simulation result with experimental data, and in Figure 1(b), that presents the previous graph with the inclusion of an envelope of reliability around the simulation. As careful experimentalists, which use error bars for a long time, UQ mainly focuses on \error bars for simulations". Moreover, a possible notion of validated model is illustrated in Figure 2, where experiment and simulation are compared, and the computational model is considered acceptable if the admissible range for the experimen- Modeling and Quanti cation of Physical Systems ... 3 0123456−3−2−10123 system input system response simulation experiment (a) 0123456−3−2−10123 system input system response simulation experiment confidence band (b) Fig. 1 (a) Comparison between simulation and experimental data, without an envolepe of reliability for the simulation, and (b) including this envelope. tal value (de ned by the point and its error bar) is contained within the reliability envelope around the simulation. simulation experimental OPTIMAL simulation experimental VIOLATION Fig. 2 Illustration of a possible notion of validated model. 4 Americo Cunha Jr This chapter is organised into six sections. Besides this introduction, there is a presentation of some fundamental concepts of UQ in section 2; a brief re- view on probability theory basics in section 3; an exposure of the fundamental aspects of probabilistic modeling of uncertainties, through a simplistic exam- ple, in section 4; the presentation of the uncertainty propagation problem in section 5; and the nal remarks in section 6. It is noteworthy that many of the ideas that are presented in this manuscript are very in uenced by courses taught by the author's doctoral supervisor, Prof. Christian Soize [13], [14] and [15]. Lectures of Prof. Gi- anluca Iaccarino, Prof. Alireza Doostan, and collaborators were also very inspiring [16], [17] and [18]. 2 Some Fundamental Concepts on UQ This section introduce some fundamental notions in the context of UQ. 2.1 Errors and Uncertainties Unfortunately, until the present date, there is still no consensus in UQ litera- ture about the notions of errors and uncertainties. This manuscript presents the de nitions we think make more sense, introduced by [19]. Let's start with three conceptual ideas that will be relevant to the stochas- tic modeling of physical systems: designed system ,real system and computa- tional model . A schematic illustration of these concepts is shown in Figure 3. manufacturing process (variabilities)mathematical modeling (model uncertainty)designed system real system (real parameters) (uncertain system)computational model real input real responsemodel input + model parameters (data uncertainty)model response Fig. 3 Schematic representation of the relationship between the designed system, the real system and the computational model [19]. Modeling and Quanti cation of Physical Systems ... 5 Designed system: The designed system consists of an idealized project for a physical system. It is de ned by the shape and geometric dimensions, material properties, connection types between components (boundary conditions), and many other parameters. This ideal system can be as simple as a beam or as complex as an aircraft [19]. Real system: The real system is constructed through a manufacturing process taking the designed system as reference. In contrast to the designed system, the real system is never known exactly, as the manufacturing process intro- duces some variabilities in the system geometric dimensions, on its materials properties, etc. No matter how controlled the construction process is, these deviations from the conceptual project are impossible to eliminate, since any manufacturing process is subjected to nite accuracy. Thus, the real system is uncertain with respect to the designed system [19]. Computational model: In order to analyze the real system behaviour, a com- putational model should be used as predictive tool. The construction of this computational model initially performs a physical analysis of the designed system, identi es the associated physical phenomena and makes hypotheses and simpli cations about its behaviour. The identi ed physical phenomena are then translated into equations in a mathematical formulation stage. Using the appropriate numerical methods, the model equations are then discretized and the resulting discrete system of equations is solved, providing an approx- imation to the computational model response. This approximate response is then used to predict the real system behaviour [19]. Numerical errors: The response obtained with the computational model is, in fact, an approximation to the model equation's true solution. Inaccuracies, intrinsic to the discretization process, are introduced in this step giving rise tonumerical errors [19]. Other source of errors are: (i) the nite precision arithmetic that is used to perform the calculations, and (ii) possible bugs in the computer code implementation of the computational model. Uncertainties on the data: The computational model is supplied with model input and parameters, which are (not exact) emulations of the real system input and parameters, respectively. Thus, it is uncertain with respect to the real system. The discrepancy between the real system and computational model supplied information is called data uncertainties [19], [4]. Uncertainties on the model: In the conception of the computational model, considerations made may or may not be in agreement with reality, which should introduce additional inaccuracies known as model uncertainties . This source of uncertainty is essentially due to lack of knowledge about the phe- nomenon of interest and, usually, is the largest source of inaccuracy in com- putational model response [19], [4]. 6 Americo Cunha Jr Naturally, uncertainties a ect the response of a computational model, but they should not be considered errors because they are physical in nature. Errors are purely mathematical in nature and can be controlled and reduced to a negligible level if the numerical methods and algorithms used are well known by the analyst [19], [4]. This di erentiation is summarized in Figure 4. errors uncertainties ww6=ww mathematical physical nature nature Fig. 4 The di erence between errors and uncertainties. 2.2 Veri cation and Validation Today veri cation and validation , also called V&V, are two concepts of fun- damental importance for any carefully done work in UQ. Early works advo- cating in favor of these ideas, and showing their importance, date back to the late 1990s and early 2000s [20], [21], [22], [23]. The impact on the nu- merical simulation community was not immediate, but has been continuously growing over the years, conquering a prominent space in the last ten years, especially after the publication of Oberkampf and Roy's book [24]. These notions are well characterized in terms of two questions: Veri cation: Are we solving the equation right? Validation: Are we solving the right equation? Although extremely simplistic, the above \de nitions" communicate, di- rectly and objectively, the key ideas behind the two concepts. Veri cation is a task whose goal is to make sure that the model equation's solution is being calculated correctly. In other words, it is to check if the computational implementation has no critical bug and the numerical method works well. It is an exercise in mathematics. Meanwhile, validation is a task which aims Modeling and Quanti cation of Physical Systems ... 7 to check if the model equations provide an adequate representation of the physical phenomenon/system of interest. The proper way to do this \valida- tion check up" is through a direct comparison of the model responses with experimental data carefully obtained from the real system. It is an exercise in physics. In Figure 5 the reader can see a schematic representation of the di erence between the two notions. solution model veri cation validation ww6=ww mathematics physics Fig. 5 The di erence between veri cation and validation. An example in V&V: A skydiver jumps vertically in free fall, from a heli- copter that is stopped in ight, from a height of y0= 2000mwith velocity v0= 0m=s. Such situation is illustrated in Figure 6. Imagine we want to know the skydiver height in every moment of the fall. To do this we develop a (toy) model where the falling man is idealized as point mass m= 70kg, under the action of gravity g= 9:81m=s2. The height at time tis denoted byy(t). y gy0 Fig. 6 V&V example: a skydiver in free fall from an initial height y0. The skydiver's height at time tcan be determined through the following initial value problem (IVP) 8 Americo Cunha Jr my(t) +mg= 0; (1) _y(0) =v0; y(0) =y0; where the upper dot is an abbreviation for a time derivative, i.e., _:=d=dt. This toy model is obtained from Newton's 2nd law of motion and considers the weight as the only force acting on the skydiver body. Imagine that we have developed a computer code to integrate this IVP using a standard 4-th order Runge-Kutta method [25]. The model response obtained with this computer code is shown in Figure 7. 0 5 10 15 2005001000150020002500 time heigth (m) solution verification simulation Fig. 7 Response obtained with the toy model. To check accuracy of the numerical method and its implementation we have at our disposal the analytical (reference) solution of the IVP, given by y(t) =1 2gt2+v0t+y0: (2) In Figure 8(a) we can see the comparison between toy model response (solid blue curve |) and the reference solution (dashed red curve - - -). We note that both curves are in excellent agreement, but if we look at Figure 8(b), which shows the di erence between numerical and analytical solutions, it is evident the e ectiveness of the numerical method and the robustness of its implementation become ever clearer. Here the veri cation was made taking as reference the real solution of the model equation. In the most frequent case, the model equations solution is not known. In such a situation, the veri cation task can be performed, for instance, using the method of manufactured solutions [26], [27], [24], [28]. Modeling and Quanti cation of Physical Systems ... 9 0 5 10 15 2005001000150020002500 time heigth (m) solution verification simulation analytical (a) 0 5 10 15 2000.20.40.60.81x 10−12 time absolute error solution verification (b) Fig. 8 (a) Solution veri cation: comparison between toy model response and reference solution; (b) absolute error of Runge-Kutta method approximation. Now let's turn our attention to model validation, and compare simulation results with experimental data, such as shown in Figure 9(a). We note that the simulation is completely in disagreement with experimental observations. In other words, the model does not provide an adequate representation of the real system behaviour. The toy model above take into account the gravitational force which at- tracts the skydiver toward the ground, but neglects air resistance e ects. This is the major reason for the observed discrepancy, the model de ciency (model uncertainty). If the air drag force e ects are included, the improved model below is obtained 10 Americo Cunha Jr 0 5 10 15 2005001000150020002500 time heigth (m) model validation toy model experiment (a) 0 5 10 15 2005001000150020002500 time heigth (m) model validation improved model toy model experiment (b) Fig. 9 (a) Model validation: comparison between experimental data and the toy model, (b) comparison between experimental data, the toy model, and the improved model. my(t) +mg1 2ACD _y(t)2= 0; (3) _y(0) =v0; y(0) =y0; whereis the air mass density, Ais the cross-sectional area of the falling body, andCDis the (dimensionless) drag coecient. With this new model, a better agreement between simulation and experi- ment is expected. In Figure 9(b) the reader can see the comparison between experimental data and the responses of both models, where we note that the improved model provides more plausible results. An important message, implicit in this example, is that epistemic uncer- tainties can be reduced by increasing the actual knowledge about the phe- nomenon/system of interest [22], [24]. Modeling and Quanti cation of Physical Systems ... 11 2.3 Two Approaches to Model Uncertainties Being uncertainties in physical system the focus of stochastic modeling, two approaches are found in the scienti c literature to deal with then: probabilis- tic, and non-probabilistic . Probabilistic approach: This approach uses probability theory to model the physical system uncertainties as random mathematical objects. This ap- proach is well-developed and very consistent from the mathematical foun- dations point of view. For this reason there is a consensus among the experts that it is preferable whenever possible to use it [4]. Non-probabilistic approach: This approach uses techniques such as inter- val analysis, fuzzy sets, imprecise probabilities, evidence theory, probability bounds analysis, fuzzy probabilities, etc. In general these techniques are less suitable for problems in high stochastic dimension. Usually they are applied only when the probabilistic approach can not be used [4]. Because of their aleatory nature, data uncertainties are, quite naturally, well represented in a probabilistic environment. Thus, the parametric proba- bilistic approach is an appropriate method to describe this class of uncertain- ties. This procedure consists in describe the computational model random parameters as random objects (random variables, random vectors, random processes and/or random elds) and then consistently construct their joint probability distribution. Consequently, the model response becomes aleatory, and starts to be modeled by another random object, depending on the nature of the model equations. The model response is calculated using a stochastic solver. For further details, we recommend [29], [30], [19], [4] and [31]. When model uncertainties are the focus of analysis, the non-probabilistic techniques receive more attention. Since the origin of this type of uncertainty is epistemic (lack of knowledge), it is not naturally described in a proba- bilistic setting. More details on non-probabilistic techniques can be seen in [32], [33], [34]. However, the use of probability theory for model uncertain- ties is still possible through a methodology called nonparametric probabilistic approach . This method, which also take into account the data uncertainty, was proposed in [35], and describes the mathematical operators in the com- putational model (not the parameters) as random objects. The probability distribution of these objects are then constructed in a consistent way, us- ing the Principle of Maximum Entropy. The methodology lumps the model level of uncertainty into a single parameter, which can be identi ed by solv- ing a parameter identi cation problem when (enough) experimental data is available. An overview of this technique can be seen in [19] and [31]. Ageneralized probabilistic approach describing model and data uncertain- ties on di erent probability spaces, with some advantages, is presented in [36] and [37]. 12 Americo Cunha Jr 3 A Brief on Probability Theory This section presents a brief review of probability basic concepts. Such expo- sition is elementary, being insucient for a solid understanding of the theory. Our objective is only to equip the reader with basic probabilistic vocabu- lary necessary to understand UQ scienti c literature. For deeper studies on probability theory, we recommend the