Jackie Krafft

Francesco Quatraro

The Dynamics of Technological Knowledge: From Linearity to Recombination

Introduction

Innovation and technological knowledge have long attracted the interest of scholars in economics. Most of the attention has been paid by the pioneers in the economics of innovation on the economic effects of the introduction of new technological knowledge as well as on the structural conditions better triggering innovative performances. This has paved the way to an empirically grounded research tradition which has initially considered knowledge as an homogeneous stock, as if it were the outcome of a quite uniform and fluid process of accumulation made possible by R&D investments, the same way as capital stock. This made it possible to include knowledge capital stock within an extended production function framework, as an additional input to labour and fixed capital [START_REF] Griliches | Issues in Assessing the Contribution of Research and Development to Productivity Growth[END_REF][START_REF] Mansfield | Basic research and productivity increase in manufacturing[END_REF].

The focus therein was on the empirical assessment of the impact of technological knowledge on economic performances. Yet, very little was known about how new knowledge is brought about and, consequently, about how to provide a representation of knowledge that could be meaningful also from the epistemological viewpoint. Technology was mostly a black box, which begun to be explored in depth with a significant lapse of time. The idea progressively arose that knowledge was something more than the mere outcome of a linear accumulation process. Indeed such an idea was grounded on theoretical reflections on the nature of knowledge creation processes, with a particular emphasis on the concept of search and on the institutions involved in the production of new technologies [START_REF] Nelson | The role of knowledge in R&D efficiency[END_REF]1986;[START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF]Rosenberg, 1982).

Drawing on insightful intuitions of [START_REF] Schumpeter | The Theory of Economic Development[END_REF][START_REF] Schumpeter | Capitalism, Socialism and Democracy[END_REF] and [START_REF] Usher | History of Mechanical Inventions[END_REF], an increasingly share of scholars in the economics of innovation has recently elaborated theoretical approaches wherein the process of knowledge production is viewed as the outcome of a recombination process, according to which innovations stem either from the combination of brand new components or from the combination of existing components in new ways (Weitzmann, 1998;Kauffman, 1993). These theoretical efforts are in turn complemented by a well-defined cognitive approach to innovation as well as by the increasing availability of historical accounts and sectoral studies on the dynamics of technological knowledge [START_REF] Vincenti | What Engineers Know and How They Know It[END_REF][START_REF] Nightingale | A cognitive model of innovation[END_REF][START_REF] Katila | Something old, something new: A longitudinal study of search behaviour and new product introduction[END_REF][START_REF] Fleming | Recombinant uncertainty in technological search[END_REF][START_REF] Van Der Bergh | Optimal diversity: Increasing returns versus recombinant innovation[END_REF].

Such framework has been largely used to build empirical studies aimed at investigating the dynamics of knowledge from the viewpoint of the complex systems approach. Knowledge was indeed seen as a set of elements connected by a network of relationships, the architecture of which affects its performances. However, despite the emergence of these new lines of inquiry in the economics of knowledge, only a few efforts can be found in literature attempting to analyze their empirical consequences, with respect to i) the identification of the relevant properties that better proximate the concept of recombinant knowledge, and hence provide a more sensible representation of knowledge on the one hand; ii) the operational translation of such properties, as well as the identification of the most appropriate analytical tools on the other hand. Moreover, such approaches are also characterized by an important theoretical limit, according to which the architecture of knowledge structure is stable over time, i.e. complexity exogenous rather than endogenous.

This chapter aims at providing an original review of the main theoretical approaches to technological knowledge, both implicit and explicit, and of their empirical counterparts in the field of economics of innovation. While there are in the literature interesting contributions aiming at assessing the relative goodness of the different proxies used in empirical analysis of innovation (see for example [START_REF] Kleinknecht | The non-trivial choice between innovation indicators[END_REF], there is a lack of efforts explicitly directed towards synthesis of theoretical and empirical issues in a historical perspective.

In this direction, we will go through the most recent debates on the dynamics of knowledge by proposing new methodologies to identifying relevant properties of knowledge that are consistent with the recombinant knowledge concept and allow for its grafting in the complex system dynamics approach in a fairly different way from the extant literature.

In particular, such methodologies are well suited to reconcile two different aspects of the analysis of the complex dynamics of technology, that is the view of technology as an artefact and as an act (Arthur, 2009;Lane et al. 2009). Indeed, by proposing that knowledge is the outcome of a collective process of recombination, we may argue that technological knowledge itself is characterized by an internal structure emerging out of a complex dynamics that is strictly connected to the dynamics affecting the formation and evolution of technology coalitions [START_REF] David | The endogenous formation of scientific research coalitions[END_REF].

The chapter is organized as follows. Section 2 provides an overview of the main different approaches to technological knowledge, both in empirical and theoretical terms. Section 3 lays down the basic ingredients of complex system dynamics and establishes the linkages with knowledge dynamics. In Section 4 we discuss the operational implications of knowledge understood as a complex system, by proposing a set of indicators that may fit this framework. Section 5 provides the conclusion and establish an agenda for future research.

Technological Knowledge: From Knowledge Capital Stock to Complex Knowledge

Knowledge capital stock and the linear model

The importance of creativity for the production of goods and wealth is not a recent discovery within economics. The earlier treatment can indeed be found already in Adam Smith"s first four books of the Wealth of Nations. After more than a century, Alfred Marshall elaborated upon Adam Smith"s contribution, by proposing a former systemic account of the role of knowledge in the production process. In particular, Marshall made it very clear both in Industry and Trade (1919) and in the Principles of Economics (1920) that knowledge is a key input in the production process and the main engine of economic growth.

Despite the venerable origins of the interest in technological knowledge within the field of economics, the former attempts to provide empirical accounts of the dynamics and the effects of innovation appeared only in the late 1950s. The studies by [START_REF] Griliches | Issues in Assessing the Contribution of Research and Development to Productivity Growth[END_REF] and [START_REF] Mansfield | Technical change and the rate of imitation[END_REF] on the diffusion of innovation can be viewed as the earlier empirical efforts in this sense. However, very little was known at that time about knowledge and in particular about its production and exploitation. The earlier empirical works in which the word "knowledge" appeared to refer to a factor affecting the production of firms can be dated back to the late 1970s. Zvi Griliches turned out to be a pioneer in the field again. In his 1979 paper indeed he proposed the famous extended production function, which paved the way to a pretty wide body of empirical investigations. In such paper the traditional production function was extended so as to include an additional explanatory variable, as follows:

   i i i i K L C Y  (1)
Where C is the fixed capital stock, L stands for labour services and K is the knowledge capital used by firm i. Strangely enough, the empirical literature has generated a great deal of confusion on this contribution, as it is usually taken as key reference in papers using the so-called "knowledge production function" approach. We believe this is due to a basic misunderstanding. Indeed, Professor Griliches in his article made some step forward to give an empirical meaning to the K term. To this purpose he proposed the following relationship:

     , R B W G K  (2)
Where R is R&D expenditures and υ is a set of unobserved disturbances. The term W(B) is instead a lag polynomial describing the relative contribution of past and present R&D expenditures to the accumulated level of knowledge. Clearly, this representation is one more application of the distributed lag literature, which influenced Griliches to a great extent. Far from proposing a knowledge production function, this relationship simply was the formalization of the concept of knowledge capital stock, which the author subsequently used in his 1980 paper on the US productivity slowdown [START_REF] Griliches | R&D and the Productivity Slowdown[END_REF]. In a nutshell, the 1979 paper offered the formal basis to the application of the permanent inventory method to calculate the knowledge stock starting from R&D expenditures, which are then considered as a flow measure.

The specification of knowledge capital also called for a proper account of the effects of knowledge spillovers, i.e. knowledge borrowed or stolen from other firms or industries that can equally affect productivity of the observed firm or industry. Knowledge spillovers have been accommodated in an extended production function at the firm level by including a proxy for the aggregate stock of knowledge available within the industry firm i operates:

    a i i i i K K L C Y  (3)
Such equation enables to distinguish between the total effect of aggregate private knowledge and the total spillover effect. Since all private knowledge is supposed to spill over to some extent, the total effect of all private knowledge at the aggregate level is given by γ+μ [START_REF] Griliches | Issues in Assessing the Contribution of Research and Development to Productivity Growth[END_REF][START_REF] Griliches | The search for R&D spillovers[END_REF].

On the basis of the argument elaborated so far, we may provide some insights about the possible theoretical underpinnings to the concept of knowledge capital stock. Indeed, we lack an explicit theoretical reasoning on technological knowledge leading to its operationalization in terms of knowledge capital stock. A quote from [START_REF] Griliches | Distributed Lags: A Survey[END_REF] may be of some help here: "For example, let investments affect the level of patenting with a lag whose generating function is given by W 1 (z), let these new inventions be embodied in new investment with a lag W 2 (z) and let new investment affect total factor productivity with a lag W 3 (z); then the total lag distribution of productivity behind investment is given by

) ( ) ( ) ( ) ( 3 2 1 z W z W z W T W 
" (Griliches, 1967: p. 20).

It is clear that the application of lag generating functions to investments measures so as to get a stock implies an underlying sequential process that start with R&D investments to yield a proxy of cumulated knowledge that in turn is supposed to show some effects on economic performances. In this direction, we believe it would not be that inappropriate saying that knowledge capital stock implies a vision of knowledge accumulation as an outcome of a linear process like this one: science precedes technology development, which then comes to be adopted by firms, and finally affects production efficiency.

After all, Vannevar Bush"s report to the US president had long been the main reference text to students of science and technology. Therefore it"s likely that the articulation of the linear model he proposed has influenced the way scholars from other fields looked at technological knowledge as well. Moreover, Kline and Rosenberg"s critique came only in the 1980s, and so did many of the works that opened up a new view on knowledge and innovation providing the basis to the knowledge production function approach [START_REF] Bush | Science the endless frontier. Report to the president[END_REF][START_REF] Kline | An overview on innovation[END_REF][START_REF] Balconi | In defense of the linear model: An essay[END_REF] 1 .

Knowledge production function

The inclusion of knowledge capital stock within an extended production function approach allows economists to preserve the basic microeconomic assumptions about production sets out of which firms take their profit-maximizing choice. However, such approach assumes the existence of a separate R&D sector that is partly responsible of the change in the production technology, and hence of the shift of the production function [START_REF] Nelson | Production sets, Technological Knowledge, and R&D: Fragile and overworked constructs for the analysis of productivity growth[END_REF].

Because of this limitation, such a representation begun to be challenged mainly by evolutionary economists, who proposed to expand the view upon technological knowledge so as to account for it inherent compositeness. At the same time, scholars of science and technology started criticizing the linear model, by proposing an alternative view basically drawing upon systemic models of innovation based upon the interaction among different and yet complementary institutions involved in the complex business of knowledge production [START_REF] Kline | An overview on innovation[END_REF]Gibbons et al. 1992).

A couple of Dick Nelson"s contributions in the early 1980s provided a clear statement of the problems with the concept of knowledge capital stock, along with the theorization of a more articulated concept of knowledge, understood as a set of capabilities guiding the search processes undertaken by organizations performing R&D. Such capabilities may be themselves the outcome of R&D activities, and are likely to improve over time due to dynamic increasing returns stemming from learning by doing dynamics [START_REF] Nelson | Production sets, Technological Knowledge, and R&D: Fragile and overworked constructs for the analysis of productivity growth[END_REF][START_REF] Nelson | The role of knowledge in R&D efficiency[END_REF].

In this sense, such contributions may be viewed as pioneering in the attempt of opening the black box of technological knowledge so as to explicitly improve upon Griliches" and Mansfield"s former operationalizations. Moreover, they also proposed a more realistic view in which science and technology are far from being sharply differentiated.

There are a number of institutions producing knowledge, some of them are public while some others are private, and it is not possible to identify a one to one mapping from science to public institutions or from applied technology to private business firms. Scholars must acknowledge that different kinds of organizations take part in the process of knowledge production, like firms, research labs and universities [START_REF] Nelson | The role of knowledge in R&D efficiency[END_REF]1986).

This set of arguments has been well received mostly in the literature dealing with knowledge production at the aggregate level. In particular the literature on regional systems of innovation provided a fertile ground to develop the implications of this new view [START_REF] Cooke | The New Wave of Regional Innovation Networks: Analysis, Characteristics and Strategy[END_REF][START_REF] Cooke | Regional innovation systems: institutional and organisational dimensions[END_REF]. Regional economists translated the idea that knowledge is the result of the interaction of a number of complementary inputs provided by different research institutions, into the concept of knowledge production function. The differences with the concept of knowledge capital stock are clear. Knowledge is no longer the mere result of cumulated R&D spending subject to decreasing returns. The knowledge production function provides a mapping from knowledge inputs to knowledge outputs that appears as follows:

          ) log( ) log( ) log( ) log( t t t t Z U R K (3)
Where K stands for a measure of knowledge output, say patents, R stands for the industry R&D and U represents the university research, while Z includes a proxy for the concentration of a given type of activity [START_REF] Acs | Patents and innovation counts as measures of regional production of new knowledge[END_REF][START_REF] Fritsch | Measuring the quality of regional innovation systems: A knowledge production function approach[END_REF]. Equation (3) represents a production function, the arguments of which enter a multiplicative relationship, and hence are seen as complementary rather than substitute. The coefficients are in turn the elastiticities of knowledge output to knowledge inputs.

On a fairly similar ground, the localized technological change approach has stressed that the dynamics of knowledge production are characterized by the joint utilization of internal and external knowledge, both tacit and codified. Mechanisms of learning, socialization and recombination are considered as crucial in a context characterized by the production of knowledge by means of knowledge itself [START_REF] Antonelli | The microdynamics of technological change[END_REF].

The knowledge production function approach represents an improvement both from the theoretical and the empirical viewpoint, with respect to the concept of knowledge capital stock. It allows to gaining a better understanding of the interactive dynamics leading to the production of technological knowledge, by accounting for possible dynamic increasing returns stemming from learning dynamics as well as knowledge externalities. However, knowledge on the left hand side of the equation still is conceived as an homogeneous stock, and little is said about the intrinsic heterogeneity of knowledge base. In other words such representation still lacks proper cognitive models of knowledge production.

Complex knowledge and NK models

The development of the knowledge production approach inevitably leaves with a basic question as to what are the micro-founded mechanisms underlying knowledge production. In this respect, the interest in the cognitive mechanisms leading to production of new technological knowledge has recently emerged in the field of economics of innovation. This strand of analysis has moved from key concepts brought forward by [START_REF] Schumpeter | The Theory of Economic Development[END_REF][START_REF] Schumpeter | Capitalism, Socialism and Democracy[END_REF] and [START_REF] Usher | History of Mechanical Inventions[END_REF], and then elaborated upon the models proposed within evolutionary economics [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF].

In his seminal works, Schumpeter proposed to view innovation as the outcome of a recombination process. Most of innovations brought about in the economic system stem from the combinations of existing elements in new and previously untried ways. Such innovations appear to be mainly as incremental. Radical innovations stem instead from the combination of existing components with brand new ones.

The contributions by [START_REF] Weitzman | Hybridizing growth theory[END_REF]1998) represent the former, and very impressive, attempt to draw upon such assumptions. His recombinant growth approach provides a sophisticated analytical framework grafting a micro-founded theory of knowledge production within an endogenous growth model. The production of knowledge is seen as the outcome of an intentional effort aimed at reconfiguring existing within a genuine cumulative perspective. However, there is no particular focus on the constraints that the combination of different ideas may represent, especially when these ideas are technologically distant. The only limiting factor seems to be the bounded processing capacity of economic agents.

The recombinant knowledge approach is based on the following assumptions. The creation of new knowledge is represented as a search process across a set of alternative components that can be combined one another. However, within this framework a crucial role is played by the cognitive mechanisms underlying the search process aimed at exploring the knowledge space so as to identify the pieces that might possibly be combined together. The set of potentially combinable pieces turns out to be a subset of the whole knowledge space. Search is supposed to be local rather than global, while the degree of localness appears to be the outcome of cognitive, social and technological influences. The ability to engage in a search process within spaces that are distant from the original starting point is likely to generate breakthroughs stemming from the combination of brand new components [START_REF] Nightingale | A cognitive model of innovation[END_REF][START_REF] Fleming | Recombinant uncertainty in technological search[END_REF].

Incidentally, such an approach also enables to better qualify the distinction between exploration and exploitation formerly articulated by [START_REF] March | Exploration and exploitation in organizational learning[END_REF]. Most of the research in organization studies has usually seen search processes as ranging between two poles of a one-dimensional continuum, i.e. exploration and exploitation. The view of knowledge as an outcome of a recombination activity allows the introduction of two nested dimensions, defined according to degree to which agents decide to rely either on exploration or exploitation or on a combination of both. To this purpose concepts like search depth and search scope have been introduced. The former refers to degree to which agents intend to draw upon their prior knowledge, while the latter refers to the degree to which agent intend to rely on the exploration of new areas in the knowledge space [START_REF] Katila | Something old, something new: A longitudinal study of search behaviour and new product introduction[END_REF].

Recombination occurs only after agents have put much effort in searching within the knowledge space. This strand of literature posits that knowledge so obtained is complex, meaning that it comprises many elements that interact richly [START_REF] Simon | The Sciences of the Artificial[END_REF]Kauffman, 1993). This has paved to way to an increasing number of empirical works based on the NK model proposed by Kauffman, according to which the search process is conducted across a rugged landscape, where pieces of knowledge are located and which provides the context within which technologies interact.

The bulk of the focus is on the concept of interdependence among the pieces that are combined together, while complexity is defined as the relationship between the number of components and the degree of interdependence [START_REF] Fleming | Technology as a complex adaptive system: Evidence from patent data[END_REF][START_REF] Sorenson | Complexity, network and knowledge flows[END_REF]. Following the intuition on the importance of patent citations contained in the seminal paper by Manuel [START_REF] Trajtenberg | A Penny for Your Quotes: Patent Citations and the Value of Innovations[END_REF], the empirical implementation of the interdependence concept is based on the deployment of the information contained in patent documents, i.e. technological classes and citations to other patents. In particular, interdependence is considered as a powerful explanatory variable building upon the technological classes the patent is assigned to. The interdependence of a patent l is obtained in two steps. First of all one has to calculate the ease of recombination for each subclass i (E i ), defined as the count of subclasses j≠i previously combined with class i weighted by total number of patents assigned to class i:

    i i l i j E (4)
Then one can calculate the degree of interdependence of patent l (K l ) by inverting its average ease of recombination:

     l i i l E l i K (5)
This empirical approach allows for evaluating the relative probability of recombination of each technological class observed in the patent sample, and then to assign an average recombination score to a patent. The basic idea is that the more combinable are the classes contained within a patent, the lower the degree of interdependence, as the technology is susceptible to be developed in a larger number of directions. On the contrary, should the classes be hardly combinable, then a relatively low number of possible combinations is possible, for which the technology turns out to show a high degree of interdependence. Such measure of interdependence is in turn expected to explain differentials in usefulness of inventions as proxied by the flow of citations received by patents over time.

Such framework clearly has the merit to push the economic discussion about technological knowledge beyond the conventional vision considering it as a sort of black box. It sheds light on the possibility to further qualify knowledge as proxied by patents, by better exploiting the information contained in patent documents. Moreover, it provides a former and innovative link between knowledge and complexity.

However, the notion of complexity used therein seems to be constrained to a generic definition of an object the elements of which are characterized by a high degree of interaction. As an implication the empirical effort does not go beyond the count of classes and of patents assigned to classes. The NK models fail to identify knowledge as an emergent property of an adaptive complex system, characterized by an architecture that can influence the actions at the micro and meso levels as well as be influenced as a result of what happens at lower layers. This requires first to make it explicit a concept of knowledge structure and then to explore the different tools made available by different methodological approaches.

Summing up, the grafting of complexity theory into economic sciences has proved to be particularly fertile, especially for what concerns the economics of knowledge and innovation. The explicit reference to the NK-model by the recombinant knowledge literature provides a clear example in this respect.

Most NK-models are however affected by a severe limit, which constrains their usefulness. The complex system is characterized by a set of elements and the connections amongst them. The configuration of the linkages connecting the elements of the system is likely to affect agents" performances. The main problem here is that the architecture of the system is often considered as table over time rather than evolving [START_REF] Frenken | Technological innovation and complexity theory[END_REF]. This amounts to consider the degree of complexity of the system as exogenous, defined ex ante. The contribution by Fleming and Sorenson discussed above presents exactly this limitation, which makes it unsuitable to the analysis of the evolutionary and path dependent dynamics of technological change.

Endogenous Complexity and Technological Knowledge

The main issue to be considered now is that the architecture of a complex system may well change over time, and so may the structure of epistatic relationships. This may occur either due to a change in the relative weight of some elements in the system, these elements switching from a non-influential to an influential position, or by means of introduction of new elements within the system. This is in turn likely to alter the existing structure of relationships. Within this context, the pleiotropy represents the number of elements in the system that are affected by the appearance of new elements. It is clear that the higher the pleiotropy, the greater the change in the architecture of the system that the inclusion of new elements may engender.

The model of constructional selection by [START_REF] Altenberg | Evolving better representations through selective genome growth[END_REF][START_REF] Altenberg | Genome growth and the evolution of the genotype-phenotype map[END_REF] represents one of the few attempts to cope with the issue of changing architectures of complex systems. As noted by [START_REF] Frenken | Technological innovation and complexity theory[END_REF]2006), such class of models is well suited to investigate the evolution of technologies considered as artefacts made of interdependent elements [START_REF] Lane | Ontological uncertainty and innovation[END_REF].

The viewpoint of endogenous complexity makes the analysis of knowledge dynamics particularly appealing and challenging. Knowledge can indeed be represented as an emergent property stemming from multi-layered complex dynamics (see Figure 1). Knowledge is indeed the result of a collective effort of individuals who interact with one another, sharing their bits of knowledge by means of intentional acts of communication [START_REF] Antonelli | Localized technological change: Towards the economics of complexity[END_REF][START_REF] Saviotti | On the dynamics of generation and utilisation of knowledge: The local character of knowledge[END_REF]. In other words, the adoption of an endogenous complexity made possible by the recombination approach allows for the combination of the view on technology as an artefact with the view of technology as an act, i.e. as the product of collective actions involving agents with converging incentives and aligned interests (Arthur, 2009;Lane et al., 2009).

INSERT FIGURE 1 ABOUT HERE

The structure of the network of relationships amongst innovating agents represents therefore a crucial factor able to shape the ultimate outcome of knowledge production processes. Constructional selection matters, in that new institutions entering the network need first of all to choose with which incumbents they want to be linked with. The concept of preferential attachment applies to this situation. In a wide number of contexts, the new nodes in a network generally end up to link with those "old" nodes already characterized by a large number of connections [START_REF] Barabasi | Emergence of scaling in random networks[END_REF]. As a consequence, the entrance of new actors in the network is likely to reshape the relative weight of nodes, and hence modify the structure and the balance of relationships.

Collective knowledge so produced stems from the combination of bits of knowledge dispersed among innovating agents. Creativity refers to the ability of agents to combining together these small bits of knowledge so as to produce an original piece of technological knowledge. This in turn may be thought about as a collection of bits of knowledge linked one another. The knowledge base of a firm can be therefore imagined as a network in which the nodes are the small bits of knowledge and the links represent their actual combination in specific tokens. Knowledge in this sense turns out to be an emergent property of complex dynamics featuring the interdependent elements of the system, i.e. the bits of knowledge. This is a quite unexplored consequence of the collective character of knowledge production, which provides further richness to its dynamics. Such complex system may be represented as network the nodes of which are the smaller units of knowledge while the edges stand for their actual combination. Hence the knowledge base is characterized by a structure with its own architecture. This in turn may evolve over time, as an effect of the introduction of new small bits of knowledge and the consequent change in the relative weight of the nodes within the network. Indeed, like in the networks of innovators, new nodes will be attached to some existing nodes, the centrality of which will be altered. Learning dynamics and absorptive capacity represent a channel through which the topology of knowledge structure affects search behaviour at the level of agents networks. Indeed, agents move across the technology landscape in regions that are quite close to the area of their actual competences. Technological change is localized as an effect of the interactions between the complex dynamics at the knowledge and the agents" level. However the topology of knowledge structure is in turn shaped by the choices made by innovating agents as to which bits of knowledge combine together. A self-sustained process is likely to emerge, according to which the knowledge creation process tends more and more towards a local attractor in which they are locked in (see the chapter by Colombelli and von Tunzelmann in this book). This dynamics indeed makes preferential attachment work also at the knowledge level. Agents" search behaviour is indeed constrained by the topology of the knowledge structure. In this direction, those small bits of knowledge which have grown in importance are likely to exert a much stronger influence. This process is rooted in historical time, according to which the gradual sorting out of knowledge bits which have proved not to be so fertile, leaves the floor to few and more fertile bits. New bits of knowledge entering the knowledge base later on are likely to be linked to these few pillars.

Preferential attachment introduces a great deal of path dependence in system dynamics of technological knowledge. It amounts to articulate the concept of persistence beyond the rate of introduction of innovations, so as to apply it to the centrality of the specific smaller bits of knowledge which make the structure of the knowledge base.

Still, while this self-enforcing process is likely to trap the search process within a bounded area, the dynamics of technological communication at the agents" level as well as the capabilities to cope with search in areas that are far away from the competences of innovating agents are likely to introduce discontinuities in the evolutionary pattern. This amounts to introduce a wide variety of new bits of knowledge which are loosely related with those already existing in the knowledge base, so as to give rise to radically new combinations. The process of evolution, fed by learning dynamics and cumulativeness, leads to the gradual selection of the best combinations, which grow in centrality and hence begin to constrain agents" search behaviour.

Knowledge sharing and technological communication ensure therefore the emergence of new variety, which is more likely to occur in transition phases. At this stage a wide range of alternatives are viable, and multiple local attractors are likely to emerge from mutual influences between complex dynamics at the knowledge and the agents" layers.

Some Operational Methodologies

The outcome of considering endogenous complexity in technological knowledge is that the layout of knowledge structure appears to be both an outcome and a determinant of agents" search. This deserves further careful attention and more in depth analysis. In what follows we propose two alternatives methodologies which have been recently introduced, and are equally suitable to improve our empirical ability to measure the various facets of the evolution of the knowledge base, including the occurrence of path dependency and persistence phases as well as the emergence of variety and discontinuity phases.

Measures based on co-occurrence matrixes

The purpose of this first methodology consists in the exploration of the evolution of the properties of the knowledge base, with particular emphasis on the issues of variety, similarity and complementarity.

1) Variety can be measured by using the information entropy index. Entropy measures the degree of disorder or randomness of the system, so that systems characterized by high entropy is also be characterized by a high degree of uncertainty [START_REF] Saviotti | Information, variety and entropy in technoeconomic development[END_REF].

The information entropy has some interesting properties, and especially a property of multidimensional extension [START_REF] Frenken | Entropy Statistics as a Framework to Analyse Technological Evolution[END_REF]. Consider a pair of events (X l , Y j ), and the probability of co-occurrence of both of them p lj . A two dimensional total variety (TV) measure can be expressed as follows:

            l j lj lj p p Y X H TV 1 log ) , ( 2 (6)
If one considers p lj to be the probability that two technological classes l and j cooccur within the same patent, then the measure of multidimensional entropy focuses on the variety of co-occurrences of technological classes within regional patents applications.

Moreover, the total index can be decomposed in a "within" and a "between" part anytime the events to be investigated can be aggregated in a smaller numbers of subsets. Within-entropy measures the average degree of disorder or variety within the subsets, while between-entropy focuses on the subsets measuring the variety across them. [START_REF] Frenken | Related Variety, Unrelated Variety and Regional Economic Growth[END_REF] refer to between-and within-group entropy respectively as unrelated and related variety.

It can be easily shown that the decomposition theorem holds also for the multidimensional case. Hence if one allows lS g and jS z (g = 1,…,G; z = 1,…, Z), we can rewrite H(X,Y) as follows:

      G g Z z gz gz Q H P H TV 1 1 (7)
Where the first term of the right-hand-side is the between-entropy and the second term is the (weighted) within-entropy. In particular:

      G g Z z gz gz Q P P H UTV 1 1 2 1 log (8)      G g Z z gz gz H P RTV 1 1 (9)     g Z S l S j lj gz p P              g z S l S j gz lj gz lj gz P p P p H / 1 log 2
We can therefore refer to between-and within-entropy respectively as unrelated technological variety (UTV) and related technological variety (RTV), while total information entropy is referred to as general technological variety.

2) The similarity amongst different types of knowledge can be captured by a measure of cognitive distance. A useful index of distance can be derived from the measure of technological proximity originally proposed by [START_REF] Jaffe | Technological Opportunity and Spillovers of R&D: Evidence from Firms' Patents, Profits, and Market Value[END_REF][START_REF] Jaffe | Real Effects of Academic Research[END_REF], who investigated the proximity of firms" technological portfolios. Subsequently [START_REF] Breschi | Knowledge relatedness in firm technological diversification[END_REF] adapted the index in order to measure the proximity, or relatedness, between two technologies. The idea is that each firm is characterized by a vector V of the k technologies that occur in its patents. Knowledge similarity can first be calculated for a pair of technologies l and j as the angular separation or uncentred correlation of the vectors V lk and V jk . The similarity of technologies l and j can then be defined as follows:

       n k jk n k lk n k jk lk lj V V V V S 1 2 1 2 1 (10)
The idea underlying the calculation of this index is that two technologies j and l are similar to the extent that they co-occur with a third technology k. The cognitive distance between j and l is the complement of their index of the similarity:

lj lj S d  1 (11)
Once the index is calculated for all possible pairs, it needs to be aggregated at the industry level to obtain a synthetic index of technological distance. This can be done in two steps. First of all one can compute the weighted average distance of technology l, i.e. the average distance of l from all other technologies. Where P j is the number of patents in which the technology j is observed. Now the average cognitive distance at time t is obtained as follows:

    l l lit lit lit t P P WAD CD (13)
Complementarity: typically a firm needs to combine, or integrate, many different pieces of knowledge to produce a marketable output. In order to be competitive a firm not only needs to learn new 'external' knowledge. It also needs to learn how to combine it with other, new and old, pieces of knowledge. We can say that a knowledge base, in which different pieces of knowledge are well combined, or integrated, is a coherent knowledge base. Such technologies are by definition complementary in that they are jointly required to obtain a given outcome. We can now turn to calculate the coherence (R) of the knowledge base, defined as the average relatedness of any technology randomly chosen within the sector with respect to any other technology [START_REF] Nesta | Knowledge and productivity in the world"s largest manufacturing corporations[END_REF][START_REF] Nesta | Coherence of the Knowledge Base and the Firm's Innovative Performance: Evidence from the U.S. Pharmaceutical Industry[END_REF]2006;[START_REF] Nesta | Knowledge and productivity in the world"s largest manufacturing corporations[END_REF]. To yield the knowledge coherence index, a number of steps are required however. In what follows we describe how to obtain the index at the sector level. First of all, one should calculate the weighted average relatedness WAR l of technology l with respect to all other technologies present within the sector. Such a measure builds upon the measure of technological relatedness  lj (see [START_REF] Nesta | Coherence of the Knowledge Base and the Firm's Innovative Performance: Evidence from the U.S. Pharmaceutical Industry[END_REF]. Following [START_REF] Teece | Understanding Corporate Coherence: Theory and Evidence[END_REF], WAR l is defined as the degree to which technology l is related to all other technologies jl in the sector, weighted by patent count Finally the coherence of knowledge base within the sector is defined as weighted average of the WAR lt measure:

     j l l lt lt lt t P P WAR R (15)
It is worth stressing that such index implemented by analysing co-occurrences of technological classes within patent applications, measures the degree to which the services rendered by the co-occurring technologies are complementary one another. The relatedness measure τ lj indicates indeed that the utilization of technology l implies that of technology j in order to perform specific functions that are not reducible to their independent use.

Measures based on social network analysis

The starting point of this second methodology is to consider that a network may be defined as a graph made of nodes that are tied each other by one or more types of interdependency. Relationships among nodes are expressed by arcs, which in turn may be directed or undirected. Two nodes that are connected by a line are said to be adjacent to one another. Adjacency is therefore the graphical expression of the fact that two nodes are directly related or connected to one another. The points to which a particular point is adjacent are termed its neighbourhood. Points may be directly connected by a line, or they may be indirectly connected through a sequence of lines. It may be thought as a "walk" in which each point and each line are distinct. This is called path. The length of path is measured by the number of lines that constitute it. The distance between two points is the shortest path (the geodesic) that connects them.

One of the most widely used measures to describe a network is the density. It describes the general level of linkage among the points in a graph. The density of a network is therefore defined as the total number of actual lines, expressed as a proportion of the maximum possible number of lines:

2 / ) 1 (    n n l (16)
A network is complete when all the nodes are adjacent, and the measure of density attempts to summarize the overall distribution of lines in order to assess how far the network is from completion. Density depends upon two other important parameters of the network, i.e. the inclusiveness and the sum of the degree of its points. Inclusiveness can be defined as the share of network nodes that are not isolated, i.e. the share of nodes that are connected to at least another node. For example, in a network of 20 nodes with 5 isolated nodes the inclusiveness is 0.75. The more inclusive the graph, the more dense the network will be.

However some nodes will be more connected than other ones. The degree of a node is an important measure of centrality that refers to the total number of other points in its neighbourhood. Formally one can represent the degree by the following equation:

    v V s vs x v D ) ( ( 17
)
This measure is obviously biased by the network size. Therefore it is useful to use a standardized measure, which consists in dividing the degree measure by its maximum value as follows:

1 ) ( ) (   n v D v ND (18)
The higher the degree of the connected points in the network, the higher will be the density. For this reason the calculation of density needs to take into account both measures. It should compare the actual number lines present in the graph with the total number of lines that the graph would show if it were complete.

While the density describes the network as a whole, the measures of centrality refer to the relevance of the nodes belonging to the network. A point is locally central if it has a large number of connections with other points in its immediate environments, i.e. other points in its neighbourhood. Global centrality refers instead to the prominence of the node with respect to the overall structure of the network. Measures of global and of local centrality have a different meaning.

Measures of global centrality are expressed in terms of the distance among various points. Two of these measures, i.e. closeness and betweenness, are particularly important. The simplest notion of closeness is that calculated from the "sum distance", the sum of geodesic distances to all other points in the graph [START_REF] Sabidussi | The centrality index of a graph[END_REF]. After having calculated the matrix of distances among the nodes of the network, the sum distance is the row of column marginal value. A point with a low sum distance is close to a large number of other points, and so closeness can be seen as the reciprocal of the sum distance. Formally it can be expressed as follows:

    v V t G t v d v C ) , ( 1 ) ( (19)
Where the denominator represents the sum of the geodesic distance of the vertex v to all other points.

The betweenness measures the extent to which a particular point lies "between" the other points in the graph: a point with a relatively low degree may play an important intermediary role and so be very central to the network [START_REF] Freeman | Centrality in social networks: Conceptual clarification[END_REF]. The betweenness of a node measures how much it can play the part of a broker or gatekeeper in the network. Freeman"s approach is built upon the concept of local dependency. A point is dependent upon another if the paths which connect it to the other points pass through this point. Formally, let G be a graph with n vertices, then the betweenness is calculated as follows:

      t s V t v s st st v v B   ) ( ) ( (20)
Where st  is the number of shortest geodesic paths from s to t, and

) ( v st 
is the number of shortest geodesic paths from s to t passing through a vertex v.

The centrality measures discussed above, allow us to characterize each single network node. However, it is also possible to calculate the sector averages for all of the three indexes. In this direction, one must consider that each node corresponds to a technological class observed with a specific relative frequency, which must be taken into account when averaging out the centrality measures. We can then propose weighted average centrality measures as follows. Let Z(v) be one of the three centrality measures referred to the generic node v, the weighted average centrality at time t is:

   v v v P P v Z v Z ) ( ) ( (21)
Where P v is the number of patents in which the technology v is observed.

Conclusion: Avenues for Future Research

The chapter was intended to provide an original and creative review of the literature on the dynamics of technological knowledge. Table 1 provides a synthesis and a taxonomy of the different approaches to technological knowledge, as well as of their theoretical underpinnings and empirical consequences.

INSERT TABLE 1 ABOUT HERE

We argue that among the new developments on the theme, the investigation on endogenous complexity in technological knowledge is certainly the most promising advance. First it provides an accurate representation of how knowledge is created and diffused at the analytical level, and second it also benefits of an empirical value since it can be expressed by a wide range of indicators and measures. In particular, we claimed that such framework has a great potential in that it provides both the theoretical and empirical grounds to carry out an interdependent analysis of technology as an act and as an artefact. In this direction, the structure of technological knowledge is represented as a network the architecture of which is in turn influenced by the architecture of the network of innovation, and vice versa. It follows a never ending process of mutual influences that keeps the system constantly out of equilibrium (see the chapter by Antonelli in this book).

The notion of coalitions for innovation gains momentum in this context (David and Keely, 2002). They can be regarded as the product of spontaneous order, yet their emergence can be guided and designed by means of the intentional intervention of policy makers as well as corporate strategies. The successful introduction of an innovation may be regarded as the result of a hegemonic coalition, that is a coalition that has been able to design a group of complementary agents, coordinate their incentives and integrate their competences so as to of achieve hegemony in a given technological space. The design of coalitions for innovation is therefore likely to exert a great deal of influence on the direction of technology evolution, and hence on future developments of the knowledge space. Within non-ergodic systems, this is likely to favour the lock-in engendered by path dependent dynamics, unless the structures of the two nested networks change so much that a new hegemonic coalition emerge able to introduce a discontinuity in the technology space.

The implications of such approach are far reaching. One of the major domains of application so far has been the analysis of the technological basis of knowledge of firms, characterized by patent portfolios (see [START_REF] Nesta | Knowledge and productivity in the world"s largest manufacturing corporations[END_REF]. Further applications have been recently proposed in empirical studies dealing with the evolutionary patterns of development of knowledge intensive sectors, especially focused in the identification of the introduction of discontinuities and the periodicity of random screening and organized search stages (Krafft, Quatraro and[START_REF] Krafft | Evolution of the Knowledge Base in Knowledge Intensive Sectors[END_REF][START_REF] Krafft | The Evolution of Knowledge Base in Knowledge-Intensive Sectors: Social Network Analysis of Biotechnology[END_REF][START_REF] Antonelli | Recombinant knowledge and growth: The case of ICTs[END_REF].

A non exhaustive list of potential applications can be elaborated, and each element in this list can be considered as a major avenue of research to be explored in the future:

-industrial dynamics and evolution: the fact that, in an industry, knowledge can either come from a recombination of existing knowledge or from the creation of new knowledge has an impact on industrial evolution. Incumbents may play the role of efficient recombination of existing knowledge, but very often may also rely on new entrant firms on the creation of new knowledge. Depending on the share of combination of existing knowledge versus creation of entirely new knowledge, incumbents or new entrants may act as leaders in the industry.

-networks: in most industries, networks occur among firms, and appear more and more as a stable form of industrial organization. Endogenous complex knowledge allows an accurate mapping of the formation of networks, and their transformation over time. Moreover depending on preferential attachment characteristics of the agents within the network, it is possible to identify the centrality of some actors in the network at some point in time, and to predict how it may change over time with the entrance of new actors.

-geographical issues: the recent debates on knowledge cities, or the more traditional ones on learning regions, also can have a new echo based on the use of the analysis of complex knowledge. On this theme, the approach can provide new quantitative results on the importance of geography in the creation and recombination of knowledge. Especially it is possible to assess quantitatively how new actors bringing new pieces of knowledge may aggregate other actors already installed or not, and eventually how these new actors may gain over time some weight (or centrality) over older ones, shaping thus the technological characteristics of a region.

Table 1 -Taxonomy of the different approaches to technological knowledge

 1

	Early 1980s	Extended production	Knowledge capital stock	Homogeneous good	Linear model	Linear mode and top
		function	as explanatory variable			down process
						R&D and specialized
						institutions of
						knowledge
						Large, vertically related
						companies
						Internal financial
						markets
	Late 1980s	Knowledge production	Knowledge capital stock	Homogeneous good	Systemic interactions	Learning effects and
	Early 1990s	function	as dependent variable			bottom up process
						Markets for knowledge
						and strong IPR regimes
						Small, specialized firms
						Venture capital and
						IPOs
	Late 1990s	Exogeneous complexity Citations and ease of	Heterogeneous good	Emergent property of a	Search conducted
	Early 2000s		recombination		given architecture	across a rugged
						landscape
						Explain differentials in
						usefulness of inventions
	Early 2000s	Endogeneous	Technological classes	Heterogeneous good	Emergent property of a	Knowledge
		complexityt	and knowledge		changing architecture	discontinuities and
			structure			search strategies
						Stable innovation
						networks with large and
						small firms
						retain and reinvest and
						long term investors
						strategies

We do not intend to go into the debate on the virtues and drawbacks of the linear model. The work by[START_REF] Balconi | In defense of the linear model: An essay[END_REF] provides an excellent synthesis in this direction.


Introduction

Many of the oil and gas field development problems can be formulated as optimization problems. Ranging from production optimization to well placement optimization, the use of optimization methods becomes more and more popular. The objective function optimized, which is usually evaluated using a reservoir simulator, is defined in a way to evaluate the economics of the project (i.e., the NPV). It can also simply be defined as the cumulative oil production. The parameters of the problem encode the decision variables to be optimized (e.g., the position of the different wells in well placement optimization, or the bottom-hole pressures or the flow-rates in production optimization).

Many optimization methodologies were used in the literature to tackle field development problems. Approaches based on stochastic search algorithms were used such as genetic algorithms (Guyaguler & Horne, 2000;Guyaguler et al., 2000;[START_REF] Yeten | Optimization of nonconventional well type, location and trajectory[END_REF], simulated annealing [START_REF] Beckner | Field development planning using simulated annealing -optimal economic well scheduling and placement[END_REF][START_REF] Norrena | Automatic determination of well placement subject to geostatistical and economic constraints[END_REF], particle swarm optimization [START_REF] Onwunalu | Application of a particle swarm optimization algorithm for determining optimum well location and type[END_REF], CMA-ES [START_REF] Bouzarkouna | Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models[END_REF][START_REF] Bouzarkouna | Partially separated meta-models with evolution strategies for well placement optimization[END_REF]. Deterministic optimization methods were also used in some studies such as in [START_REF] Handels | Adjoint-based wellplacement optimization under production constraints[END_REF], [START_REF] Sarma | Efficient well placement optimization with gradient-based algorithms and adjoint models[END_REF], [START_REF] Forouzanfar | A two-stage well placement optimization method based on adjoint gradient[END_REF].

However, in the well placement problem, as well as in many other field development optimization problems, geological uncertainty is a key source of risk affecting the viability of field development projects, although still neglected in many research studies. The problem arises when we have multiple possible geological realizations of the reservoir. The multiple realizations are generated using geostatistical techniques and in general deemed equiprobable. Taking into account these several geological realizations adds an important challenge to the optimization of the objective function; in particular it leads to a large increase of the number of performed reservoir simulations (Usually, a single objective function evaluation requires a number of reservoir simulations equal to the number of considered realizations).

In this paper, we propose a new approach to handle geological uncertainty with a reduced number of reservoir simulations. The proposed approach is demonstrated on the well placement problem, but can be extended to other Geosciences problems such as production optimization. In particular, this paper addresses the problem of how to define the objective function when dealing with uncertainty for well placement and whether we should perform evaluations on all the possible realizations in order to define the objective function. Thus, we propose to use only a small number of reservoir simulations (typically one) for each well configuration, together with its neighborhood in order to estimate its corresponding objective function value instead of using multiple realizations.

We denote by    n f :

the objective function to optimize and by r N the number of geological realizations which are denoted by

r N i i R ,..., 1 ) (  .
For each well configuration proposed by the optimizer, we have r N possible values of the objective function, one for each realization where each will be denoted for a given well configuration x by ) , ( i R f x corresponding to a the realization i R .

This paper is structured as follows. Section 2 provides a detailed literature review for well placement optimization under geological uncertainty. Section 3 defines a new approach to handle geological uncertainty for well placement using the neighborhood, which is combined with the optimization method CMA-ES. In Section 4, we demonstrate the contribution of the proposed approach in capturing the geological uncertainty and in reducing the number of reservoir simulations on the synthetic benchmark reservoir case PUNQ-S3 [START_REF] Floris | Methods for quantifying the uncertainty of production forecasts: A comparative study[END_REF].

Optimization under uncertainty: a literature review

In the context of field development optimization under geological uncertainty, the measurable fitness values correspond to the values

r N i i R f ,..., 1 ) , (  x
which represent the evaluations on each of the realizations. Therefore, the objective function can be in general written as:

     r N i i r R f N f 1 ) , ( 1 ) ( x x
.

(1)

However due to the expensive computational effort required to evaluate the objective function over one realization R i , the expected objective function is in some cases approximated in a way to use a fewer number of samples instead of using all the realizations. Thus, one common way to approximate the expected objective function here is by averaging over a number of samples N s ≤ N r .

The problem of optimization under geological uncertainty shares many similarities with the problem of optimizing noisy functions. A function f is said to be noisy if the only measurable value of f on

n   x
is a random variable that can be written as ) ), F(f( z x where f is a time-invariant function and z is a noise often assumed to be normally distributed with a zero mean and variance σ 2 , and denoted by ) , 0 ( 2  

. The noise can be also defined differently (e.g., Cauchy distributed), and can be either additive or multiplicative. To optimize noisy functions, the objective function is usually estimated by the expected value defined as follows:

       dz z p z f ) ( ) ), F(f( ) ( x x , (2)
where p(z) is the probability density function of the noise. Thus, a common way to approximate the expected objective function is again by averaging over a finite number of random samples N s :

     s N i i s z N f 1 ) ), F(f( 1 ) ( x x , (3)
In the following, we briefly review the existing approaches often used in optimization under uncertainty. On the one hand we review the approaches defined by the optimization community mainly to cope with noise but that can be extended to the different field development optimization under geological uncertainty. On the other hand we review the approaches already applied in the petroleum community to cope with geological uncertainty.

Optimization community

This section summarizes the different ways to handle uncertainty within the evolutionary optimization community. A detailed overview of the existing approaches addressing uncertainties in evolutionary optimization is presented by [START_REF] Jin | Evolutionary optimization in uncertain environments -a survey[END_REF]. Let us suppose in this section that the function f to optimize is a noisy function. The approaches to handle uncertainty are mainly divided into two categories.

Explicit Averaging

Using mean of several samples for each individual

The simplest and the most common way to address the uncertainty issue is to define the objective function for each point by averaging over a number of samples (Eq. ( 3)). Increasing the sample size N s is equivalent to reducing the variance of estimating the objective function. In general, the objective function is defined using an averaged sum of a constant sample size. In this case, for each single evaluation of the expected objective function, one needs to evaluate the objective function on N s samples. In the context of costly objective functions, depending on the number of samples, there is a compromise between the computational cost of the optimization and the accuracy of the estimation of the objective function. Increasing (respectively, decreasing) the number of samples tends to improve (respectively, worsen) the accuracy of the estimated objective function, but on the other hand it tends also to increase (respectively, reduce) the computational cost of the optimization. The idea of using an adapted sample size during the optimization was first proposed by [START_REF] Aizawa | Dynamic control of genetic algorithms in a noisy environment[END_REF] and [START_REF] Aizawa | Scheduling of genetic algorithms in a noisy environment[END_REF]. In [START_REF] Aizawa | Scheduling of genetic algorithms in a noisy environment[END_REF], it is shown that adapting the number of samples performs better than using constant sample sizes, and it is suggested to increase the sample size with the generation number and to use a higher number of samples for individuals with higher estimated variance. Another way to adapt the sample size is based on an individual's probability to be among a number of the best individuals [START_REF] Stagge | Averaging efficiently in the presence of noise[END_REF]. Recently, another approach relying on the rank based selection operators was proposed by [START_REF] Hansen | A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion[END_REF]. In [START_REF] Heidrich-Meisner | Hoeffding and bernstein races for selecting policies in evolutionary direct policy search[END_REF], an adaptive uncertainty handling procedure is proposed, based on selection races.

Using the neighborhood for each individual

An alternative approach to defining the objective function as an averaged sum of a number of samples (constant or adapted) is to define the objective function using the neighborhood points already evaluated [START_REF] Pänke | Efficient search for robust solutions by means of evolutionary algorithms and fitness approximation[END_REF][START_REF] Branke | Efficient fitness estimation in noisy environments[END_REF][START_REF] Branke | Faster convergence by means of fitness estimation[END_REF][START_REF] Sano | Optimization of noisy fitness functions by means of genetic algorithms using history of search[END_REF][START_REF] Sano | Optimization of noisy fitness functions by means of genetic algorithms using history of search with test of estimation[END_REF]. The general idea has first been suggested by [START_REF] Branke | Creating robust solutions by means of evolutionary algorithms[END_REF] in which it is suggested to estimate the fitness as a weighted average of the neighborhood with a linearly decreasing weight function up to some fixed maximum distance. In [START_REF] Pänke | Efficient search for robust solutions by means of evolutionary algorithms and fitness approximation[END_REF], [START_REF] Branke | Efficient fitness estimation in noisy environments[END_REF] and [START_REF] Branke | Faster convergence by means of fitness estimation[END_REF], a locally weighted regression is used for estimation. This technique is shown to be a good solution to improve the accuracy of the estimated objective function without increasing the computational cost.

Implicit Averaging

When increasing the population size, the probability to obtain similar points is higher. Thus, a way to cope with noise is to simply increase the population size [START_REF] Fitzpatrick | Genetic algorithms in noisy environments[END_REF]. In this case, with a large population size, the influence of noise on a given point can be reduced due to the evaluations on other similar points. Conflicting conclusions [START_REF] Fitzpatrick | Genetic algorithms in noisy environments[END_REF][START_REF] Arnold | Efficiency and mutation strength adaptation of the (μ/μ i ,λ)-ES in a noisy environment[END_REF][START_REF] Arnold | Local performance of the (μ/μ i ,λ)-ES in a noisy environment[END_REF][START_REF] Beyer | Towards a theory of evolution strategies[END_REF] were shown in the literature when comparing explicit averaging and implicit averaging.

Petroleum community

Several studies in the literature have addressed the problem of optimization under geological uncertainty. Optimization under geological uncertainty in the petroleum community considers always a finite number of realizations r N and models the objective function following Eq. ( 1). In the following we briefly review the approaches to handle uncertainty in optimization within the petroleum community.

To the best of our knowledge, all the studies that consider a number r N of possible geological realizations use the approach "Using mean of several samples for each individual". Moreover, all the studies in the literature, except the approach proposed in [START_REF] Wang | Optimal Well Placement Under Uncertainty Using a Retrospective Optimization Framework[END_REF] that will be detailed later, perform r N reservoir simulations for every single objective function evaluation. Although sharing this common similarity, the proposed approaches introduce different formulations of the objective function.

In [START_REF] Schulze-Riegert | Well path design optimization under geological uncertainty: Application to a complex north sea field[END_REF][START_REF] Schulze-Riegert | Multiple-objective optimization applied to well path design under geological uncertainty[END_REF][START_REF] Onwunalu | Application of a particle swarm optimization algorithm for determining optimum well location and type[END_REF] and Chen (2010), the objective function is formulated as the expected value of the net present value over all the realizations, as shown in Eq. (1). Chen (2010) tackles the problem of closed-loop production optimization using the optimizer EnOpt (Chen & Oliver, 2009;Chen et al., 2009) which is applied to the geological model ensemble updated by either EnKF [START_REF] Evensen | Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics[END_REF] or EnRML [START_REF] Gu | An iterative ensemble Kalman filter for multiphase fluid flow data assimilation[END_REF]. In [START_REF] Yeten | Optimization of nonconventional well type, location and trajectory[END_REF], [START_REF] Aitokhuehi | Optimization of advanced well type and performance[END_REF] and [START_REF] Alhuthali | Optimizing smart well controls under geologic uncertainty[END_REF], multiple geostatistical realizations of the reservoir are considered in the formulation of the objective function:

   r R f N f r N i i r    1 ) , ( 1 ) ( x x , (4)
where   r is the risk factor and σ is the standard deviation of f on x over the realizations, defined as follows:

      r N i i r f R f N 1 2 ) ( ) , ( 1 x x  , (5)
where:

     r N i i r R f N f 1 ) , ( 1 ) ( x x . ( 6
)
The term  r in Eq. ( 4) is used to take into account the decision maker's attitude toward risk. A positive r indicates a risk-prone attitude, a negative r indicates a risk-averse attitude and an r = 0 indicates a risk-neutral attitude. This formulation is close to the formulations defined in [START_REF] Guyaguler | Uncertainty assessment of well placement optimization[END_REF] and [START_REF] Ozdogan | Optimization of well placement under time-dependent uncertainty[END_REF] using utility functions. In [START_REF] Artus | Optimization of nonconventional wells under uncertainty using statistical proxies[END_REF], a more general formulation of the objective function is defined as follows. A genetic algorithm is used, in which at each iteration only a predefined percentage of the individuals, chosen according to a set of scenario attributes, is simulated. For the simulated individuals, Artus et al. propose

x x x x f r f r f r f   , ( 7
) where the parameters r 10 , r 50 and r 90 are defined according to the decision maker's attitude toward risk. A risk-neutral attitude corresponds to the case where (r 10 , r 50 , r 90 ) = (0, 1, 0). However, a risk-averse investor tends to increase the value of r 10 , and a risk-prone investor tends to increase the value of r 90 . Another way to formulate the objective function under geological uncertainty is to optimize the worst case scenario using a min-max problem formulation. This approach is used in [START_REF] Alhuthali | Optimizing smart well controls under geologic uncertainty[END_REF] to optimize smart well controls.

The only approach in the literature selecting only a number of samples instead of all the realizations is defined in [START_REF] Wang | Optimal Well Placement Under Uncertainty Using a Retrospective Optimization Framework[END_REF]. The approach is based on the so-called retrospective optimization and divides the problem into a number of sub-problems, where the initial solution of the current subproblem is simply the returned solution from the previous sub-problem. Each point to be evaluated is approximated by the average over a number of realizations, where the number of selected realizations is increased from sub-problem to sub-problem. The approach implies then defining a sequence of samples. The example shown in [START_REF] Wang | Optimal Well Placement Under Uncertainty Using a Retrospective Optimization Framework[END_REF] considers a well placement problem on 104 permeability and porosity realizations and therefore defines sub-problems with a sequence {20,15,10,5} of iterations and a sequence {1,5,16,21,104} of sample sizes. Although Wang et al. suggest further testing of the overall framework to determine the appropriate sequence of sample sizes, an answer can be the work on adapting automatically the sample sizes already proposed in [START_REF] Stagge | Averaging efficiently in the presence of noise[END_REF] or in [START_REF] Hansen | A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion[END_REF] but still demanding in the number of objective function evaluations.

Well placement under uncertainty with CMA-ES using the neighborhood

This section proposes a new approach to handle geological uncertainty for well placement. The proposed approach focuses on reducing the uncertainty by using the objective function evaluations of ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012 already evaluated individuals in the neighborhood. In this section, we propose then to apply an approach based on using the neighborhood for each individual. The approach is based on the optimization algorithm Covariance Matrix Adaptation -Evolution Strategy (CMA-ES), but can be combined with any other optimization algorithm.

The optimization algorithm: CMA-ES

In the following, we describe the CMA-ES [START_REF] Hansen | Completely derandomized self-adaptation in evolution strategies[END_REF]. CMA-ES is a stochastic population-based optimization algorithm where at each iteration step g, a population of λ points is generated by sampling a multivariate normal distribution. The objective function values of the λ points are then evaluated and the parameters of the multivariate normal distribution are updated. More specifically, let (m g , g   ) be the sequence of mean values of the multivariate normal distribution and let (σ g , g   ) and (C g , g   ) be respectively the sequences of step-sizes and covariance matrices. The sampling of the λ points of the new population ( g i x ) i =1,…,λ can then be written as:

  ,..., 1 for ), , 0 (    i N g i g g g i C m x , (8)
where ) , 0 (

g i N C
are independent multivariate normal distributions with zero mean vector and covariance matrix C g . A more detailed overview of CMA-ES and its application on the well placement problem can be found in [START_REF] Bouzarkouna | Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models[END_REF].

Using the neighborhood approach

We define a CMA-ES optimizing an estimated fitness defined on a given point using a weighted average of a small number of evaluations on the considered point and a number of evaluations already performed on the neighborhood (up to some fixed maximum distance) with a decreasing weight function depending on the Mahalanobis distance -between the considered point and the neighbor point-with respect to the covariance matrix C defined by CMA-ES. Although considering a Mahalanobis distance with respect to σ 2 C is suspected to be a better choice (since we are using a fixed maximum distance to select the neighbors), it has not been tested in this paper.

Let us consider a well placement optimization problem with a number of wells (producers and/or injectors) to be placed. Let us denote by n the dimension of the problem, i.e., the number of parameters needed to encode the wells to be placed. The wells to be placed are parameterized using real numbers encoding its coordinates. Without loss of generality, we will consider in the sequel the NPV as the objective function that we aim to optimize, unless otherwise explicitly stated. Thus, we want to find a vector of parameter

n R   max, p such that:   ) ( NPV max ) ( NPV max, p p p R R R  , (9)
where NPV R is the averaged sum of the NPVs of a given well configuration represented by a vector of parameters p over all the realizations:

   r N i i R r R R N 1 ) , ( NPV 1 ) ( NPV p p . ( 10
)
In the proposed approach, we define a so-called estimated objective function that will be optimized instead of the true objective function NPV R defined in Eq (10). The estimated function will be denoted in the sequel by NPV E . Thus in the proposed approach, contrary to what is shown in Eq (9), we will try to find the vector of parameter

n E   max, p such that:   ) ( NPV max ) ( NPV max, p p p E E E  . ( 11
)
The simplest case, in which solving Eq. ( 9) is equivalent to solving Eq. ( 11), is when NPV E is a monotonic transformation of NPV R . However in this work, we do not aim to define an estimated objective function NPV E such that we can prove that Eq. ( 11) is equivalent to Eq. ( 9). Our aim is that by solving Eq. ( 11), we can propose good points with high NPV R values (see below for the definition of NPV E ).

ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012

To optimize NPV E , we propose to use the CMA-ES optimizer. During the optimization process, we build a database -called also training set-in which after every performed reservoir simulation for a given point x on a realization R, we store the point x together with its corresponding evaluation NPV( x , R).

It remains now to define the estimated objective function NPV E for a given point (well configuration) denoted by a vector of parameters p : 1. At the beginning of the optimization and until reaching a given number sim N of performed reservoir simulations, we define a number of reservoir simulations

   1 1 1 ) , ( NPV 1 ) ( NPV s i N i j s E R N p p . ( 12
)
In this case, the evaluation of NPV E requires a number1 s N of reservoir simulations. 2. If more than sim N reservoir simulations are performed, we perform the following steps.

We begin by defining a number of reservoir simulations 2 x the selected points 1 . The distance used for this purpose is the Mahalanobis distance with respect to the current covariance matrix C of CMA-ES defined for two given points

n   1 z and n   2 z by ) ( ) ( ) , ( 2 1 1 2 1 2 1 z z C z z z z C     T d .
The estimated objective function on p reads as follows:

                  n s i N i i i i N i j i E R p R p S 1 1 ) , ( NPV ) , ( NPV 1 ) ( NPV 2 x p p , ( 13
)
where

1  i p , 2 2 max ) ) , ( ( 1 ~          d d p i p x C and       n s N i i N i i p p S
N N   2 max , , r N N   2 sim , 1 1  s N and 1 2  s N
. A users' choice is the maximum distance of Biarritz, France, 10-13 September 2012 selection for the neighborhood max d , and which is a problem-dependent constant. An investigation of the impact of the choice of max d will be briefly shown in the next section through some examples.

An estimated standard deviation can also be included in the formulation of the estimated objective function NPV E . In this case, the estimated objective function, which will not be tested in this paper, can be formulated as follows:

) ( ) ( ) ( NPV p p p E E E r m    , ( 14
)
where r is a constant, E m is defined as follows:

                  n s i N i i i i N i j i E R p R p S m 1 1 ) , ( NPV ) , ( NPV 1 ) ( 2 x p p , ( 15
)
and E

 is defined as follows:

                                n s i N i E i i i N i E j i E m R p m R p S 1 2 1 2 ) ( ) , ( NPV ) ( ) , ( NPV 1 ) ( 2 p x p p p  . ( 16
)

Application of CMA-ES using the neighborhood approach on the PUNQ-S3 case

In this section, we apply the CMA-ES using the neighborhood approach -that we will call in the sequel the "using the neighborhood" approach-on the well placement problem on the benchmark reservoir case PUNQ-S3 [START_REF] Floris | Methods for quantifying the uncertainty of production forecasts: A comparative study[END_REF]. The model grid contains 19 cells in the x-direction, 28 cells in the y-direction and 5 cells in the z-direction. The cell sizes are 180m in the x and y directions and 18m in the z-direction. The elevation and the geometry of the field are shown in Fig. 1.

We consider 20 geological realizations that will be again denoted by (R i ) i=1,…,20 . Each realization defines one possible porosity map and one possible permeability map. In these examples, the number of realization N r is then equal to 20. We plan to drill two wells: one unilateral injector and one unilateral producer. The dimension of the problem is then equal to 12(= 6×2) corresponding to the number of parameters defining the coordinates of the wells to be drilled. The used parameterization is the same as defined by [START_REF] Bouzarkouna | Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models[END_REF]. In all the following applications, we use CMA-ES as an optimization algorithm with a population size equal to 40.

Figure 1 Elevation (in meters) and geometry of the PUNQ-S3 test case.

ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012

Figure 2 The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the mean of samples" approach. The best mean value of the NPV over the 20 possible realizations, i.e., NPV R is shown. Three runs are performed.

(a) (b) (c)
Figure 3 The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the neighborhood" approach, for three independent runs in (a), (b) and (c). The evolutions of the best estimated objective function, i.e., NPV E are drawn with green lines. The evaluations on the true objective function over the 20 possible realizations, i.e., NPV R are depicted with red crosses. The maximum distance of selection for the neighborhood max d is equal to 4000.

ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012

As a reference approach, we perform three independent runs in which we optimize the objective function NPV R as defined in Eq. ( 10). In this reference approach, we perform for each well configuration to be evaluated 20 reservoir simulations. The reference approach will be called in the sequel the "using the mean of samples" approach. Fig. 2 shows the evolution of the best mean value of NPV R , i.e., the NPV over the 20 possible realizations, for the three performed runs. The "using the mean of samples" approach is shown to be able to reach a mean value of NPV R equal to $9×10 9 using 15200 reservoir simulations. It is able also to reach a mean value of NPV R equal to $9.3×10 9 using 31200 reservoir simulations and a mean value of NPV R equal to $9.5×10 9 using 44400 reservoir simulations.

To evaluate the "using the neighborhood" approach, we use typical values of the parameters sim N ,

1 s N , 2 s N and max , n N
as defined in the previous section, i.e.,

r n N N   2 max , , r N N   2 sim , 1 1  s N and 1 2  s N
. We begin by choosing the maximum distance of selection for the neighborhood max d equal to 4000. Fig. 3 shows the evolution of the optimization process for three independent runs of CMA-ES with the "using the neighborhood" approach. The evolutions of the best estimated objective function, i.e., NPV E are drawn with green lines. During the optimization process, each new overall best point found on NPV E , is evaluated on NPV R . The evaluations performed on NPV R are depicted with red crosses. Fig. 3 shows that when optimizing NPV E , we are able to propose good points according to NPV R (points with an NPV R greater than $9×10 9 ). Moreover, NPV R tends to increase with an increasing number of performed reservoir simulations. Fig. 3(c) shows a particular run in which the best NPV E value found at the first generation is equal to $9.7×10 9 . This value is calculated according to Eq. ( 12), and thus calculated using only one single reservoir simulation (with one single random realization). Indeed, with a single reservoir simulation to evaluate one point, the estimated objective function cannot in general propose a good point according to NPV R . Consequently, the best point found at the first generation according to NPV E has a "bad" NPV R value equal to $5.8×10 9 . Thus, the optimization process does not propose for 112 iterations a new overall best point to be evaluated on NPV R . The performance of this run can be avoided either by evaluating more often points using NPV R2 or simply by using more reservoir simulations for each point to be evaluated at the beginning of the optimization, i.e., choosing 2 1  s N .

(a) (b)

Figure 4 The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the neighborhood" approach for eight independent runs. (a) shows the evolution of the evaluations on NPV R . (b) shows the evolution of the best found evaluation on NPV R . The maximum distance of selection for the neighborhood max d is equal to 4000.

Figure 5

The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the mean of samples" approach and the "using the neighborhood" approach. The evolution of the best found evaluation on NPV R for the "using the neighborhood" approach is drawn with red lines. The evolution of the best found evaluation on NPV R for the "using the mean of samples" approach is drawn with blue lines. Three independent runs are performed for each approach. For the "using the neighborhood" approach, the maximum distance of selection for the neighborhood max d is equal to 4000.

(a) (b)

Figure 6 The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the neighborhood" approach for four independent runs. ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012

Figure 9 The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using one realization" approach. The best mean value of the NPV over the 20 possible realizations, i.e., NPV R is shown. Three runs are performed.

We show in Fig. 4 the performance of eight independent runs of CMA-ES with the "using the neighborhood" approach. Fig. 4(a) shows the evolution of the evaluations performed on NPV R . The evaluated points correspond to the best overall points found during the optimization process of NPV E . Fig. 4(b) shows the evolution of the best evaluation performed on NPV R . Seven runs out of the eight performed runs (88%) are able to reach an NPV R value greater than to $9×10 9 , using a mean number of reservoir simulations equal to 2851. Consequently the reduction of the number of reservoir simulations to reach an NPV R greater than to $9×10 9 when using the "using the neighborhood" approach compared to the "using the mean of samples" approach is equal to 81%. Six runs out of eight performed runs (75%) are able to reach a value of NPV R greater than to $9.3×10 9 , using a mean number of reservoir simulations equal to 4307, which offers a reduction of the number of reservoir simulations when comparing to the "using the mean of samples" approach equal to 86%. However, only two runs out of the eight performed runs (25%) are able to reach a value of NPV R greater than to $9.5×10 9 . The mean number of reservoir simulations required to reach this value is 6160. Consequently the reduction of the number of reservoir simulations to reach an NPV R greater than to $9.5×10 9 when comparing to the "using the mean of samples" approach is again equal to 86%. Three runs of CMA-ES with the "using the neighborhood" approach are shown together with the three performed runs of CMA-ES with the "using the mean of samples" approach in Fig. 5. Results show that although the "using the neighborhood" approach does not guarantee finding the best values of NPV R found by the "using the mean of samples" approach when comparing with the "using the mean of samples" approach, the number of reservoir simulations is reduced significantly by more than 81%.

The impact of the choice of the maximum distance of selection for the neighborhood max d is shown in Figs. 6 and7. Comparing the results in Figs. 6, 4 and 7 (with max d = 3000, 4000 and 6000) shows that the approach is not very sensitive to the choice of max d . In the sequel, we compare the "using the neighborhood" approach with another approach in which the estimated objective function to be optimized is equal to an evaluation on a randomly chosen realization. This approach is called the "using one realization" approach. In this approach, we also evaluate on NPV R only the overall new best points found on the estimated objective function. Figs. 8 and9 show the evolution of the optimization process for three independent runs of CMA-ES with the "using one realization" approach. In Fig. 8, the evolutions of the best estimated objective function are again drawn with green lines. When comparing the "using the neighborhood" and the "using one realization" approaches through Figs. 3 and8, it is clear that contrary to the "using the neighborhood" ECMOR XIII -13 th European Conference on the Mathematics of Oil Recovery Biarritz, France, 10-13 September 2012 approach which is shown to be able to capture the geological uncertainty, the "using one realization" approach is shown to be not able to propose good points with high NPV R . The three performed runs with the "using one realization" approach are not able to reach an NPV R value greater than $9×10 9 .

Summary and discussions

In this paper, we have defined a new optimization approach under geological uncertainty with a reduced number of reservoir simulations. The approach uses the objective function evaluations of already simulated well configurations in the neighborhood of each well configuration. The proposed approach can be combined with any optimization algorithm. In this work, we show an application using the state-of-the-art stochastic optimizer CMA-ES on the well placement problem.

On the benchmark reservoir case PUNQ-S3, the proposed approach is shown to be able to capture the geological uncertainty while using only a reduced number of reservoir simulations. More specifically, the proposed approach is able to reduce significantly the number of reservoir simulations by more than 80% compared to the reference approach, i.e., the approach using all the possible realizations for each well configuration. This work can be extended and enhanced by numerous means, mainly by using an adaptive strategy to define the parameters of the algorithm, although we suspect that the used values of the parameters defining the number of performed reservoir simulations can be a good choice.

The definition of the distance metric has a major influence on the neighbourhood selection, and thus on the definition of the objective function. Another improvement can then be achieved by using a Mahalanobis distance with respect to σ 2 C (instead of C).

  set. The estimated objective function on the point p reads as follows:

  p from the training set. Here, we select only the points with a distance less or equal to a given fixed distance of selection denoted by max d . We denote by n N the number of selected points and by  

Figure 7 Figure 8

 78 Figure 6The evolution of the well placement optimization process on the PUNQ-S3 case using CMA-ES with the "using the neighborhood" approach for four independent runs. (a) shows the evolution of the evaluations on NPV R . (b) shows the evolution of the best found evaluation on NPV R . The maximum distance of selection for the neighborhood max d is equal to 3000.

  to perform again r

					N reservoir simulations for each well configuration x in order to
	evaluate the values of	, f x (	i R	)	on all realizations and then to derive the cumulative distribution
	function cdf{f} on x. From this distribution, the values of	f	( 10 x	)	,	f	( 50 x	)	and	f	( 90 x	)	are determined.
	The value 10
	The values	f	( 50 x	)	and	f	( 90 x	)	are defined in a way similar to	f	( 10 x	)	. The objective
	function is then formulated as follows:
				(		)	10	10	(	)	50	50	(	)	90	90	(	)

f on x denotes the value of f on x corresponding to a probability of 0.1, i.e., there is a probability 0.1 that the value of f on x will be less than 10 f on x. The value 10 f on x can be written as cdf{f} -1 (0.1).

For each selected point x i for the training set, we have a corresponding evaluation on a given realization. For the sake of notation simplicity we will denote the corresponding stored evaluation by NPV(x i , R i ) although it is not necessarily evaluated on realization R i .

For example, one can evaluate the best found point according to NPV E at each iteration on NPV R

Acknowledgements

The authors would like to thank Nikolaus Hansen and Christian Igel for their insightful comments on this work.