By Constantin Cranganu, Henri Luchian, Mihaela Elena Breaban

This ebook offers a number of clever techniques for tackling and fixing hard sensible difficulties dealing with these within the petroleum geosciences and petroleum undefined. Written by way of skilled teachers, this e-book bargains state of the art operating examples and offers the reader with publicity to the newest advancements within the box of clever tools utilized to grease and fuel study, exploration and creation. It additionally analyzes the strengths and weaknesses of every strategy awarded utilizing benchmarking, when additionally emphasizing crucial parameters resembling robustness, accuracy, velocity of convergence, desktop time, overlearning and the position of normalization. The clever methods provided comprise man made neural networks, fuzzy common sense, energetic studying process, genetic algorithms and aid vector machines, among others.

Integration, dealing with facts of gigantic measurement and uncertainty, and working with possibility administration are between the most important concerns in petroleum geosciences. the issues we need to clear up during this area have gotten too advanced to depend upon a unmarried self-discipline for powerful suggestions and the prices linked to bad predictions (e.g. dry holes) bring up. as a result, there's a have to determine a brand new process geared toward right integration of disciplines (such as petroleum engineering, geology, geophysics and geochemistry), facts fusion, possibility relief and uncertainty administration. those clever strategies can be utilized for uncertainty research, danger evaluate, facts fusion and mining, information research and interpretation, and information discovery, from diversified info resembling 3-D seismic, geological information, good logging, and creation info. This e-book is meant for petroleum scientists, information miners, info scientists and pros and post-graduate scholars eager about petroleum industry.

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**Example text**

Suppose that the statement holds for n, and let t1 ; . ; tn ; tnþ1 be n þ 1 numbers P such that nþ1 i¼1 ti ¼ 1. We have f ðt1 x1 þ Á Á Á þ tnÀ1 xnÀ1 þ tn xn þ tnþ1 xnþ1 Þ tn xn þ tnþ1 xnþ1 : ¼ f t1 x1 þ Á Á Á þ tnÀ1 xnÀ1 þ ðtn þ tnþ1 Þ tn þ tnþ1 By the inductive hypothesis, we can write f ðt1 x1 þ Á Á Á þ tnÀ1 xnÀ1 þ tn xn þ tnþ1 xnþ1 Þ tn xn þ tnþ1 xnþ1 6 t1 f ðx1 Þ þ Á Á Á þ tnÀ1 f ðxnÀ1 Þ þ ðtn þ tnþ1 Þf : tn þ tnþ1 Next, by the convexity of f , we have tn xn þ tnþ1 xnþ1 tn tnþ1 f ðxn Þ þ f ðxnþ1 Þ: 6 f tn þ tnþ1 tn þ tnþ1 tn þ tnþ1 Combining this inequality with the previous inequality gives the desired conclusion.

Optimization problems There is an informal conjecture stating that anything we are doing, we optimize something; or, as Clerc put it in (2006), iterative optimization is as old as life itself. While each of these two statements may be the subject of subtle philosophical debates, it is true that many problems can be stated as optimization problems. Finding the average of n real numbers is an optimization problem (ﬁnd the number a which minimizes the sum of its distances absolute values of the differences to each of the given numbers); the same goes for decisionmaking problems, for machine learning ones, and many others.

Xn 2 I. As before, for t1 ¼ Á Á Á ¼ tn ¼ 1n, we have ðx1 þ Á Á Á þ xn Þ ln n x1 þ Á Á Á þ xn X 6 xi ln xi : n i¼1 Applying this inequalities to xi ¼ jBjSji j, where p is a partition of S given by p ¼ fB1 ; . ; Bn g, we have ln n> À n X jBi j i¼1 jSj ln jBi j : jSj P The quantity À ni¼1 jBjSji j ln jBjSji j is the Shannon entropy of p. Its maximum value ln n is obtained when the blocks of p have equal size. 1 Ha ðpÞ ¼ HðpÞ. In other words, Shannon’s entropy is a limit case of the Ha -entropy. Let p; r be two partitions of a set S, where p ¼ fB1 ; .