In the last lecture we talked about static deterministic linear inverse problem which are well posed why it is well posed because we did not find any difficulty in the solution process that is largely because the matrix H is well well is full rank because the matrix H is full rank H transpose H is symmetric positive definite H H H transpose H H transpose H both of them are full rank matrices therefore we could define the generalized inverse of H both in the underdetermined case and the overdetermined case we did not have any problem we sailed very smoothly by solving the normal the normal equations. So it makes sense to ask ourselves the question what happens if H is such that is not a full rank and that corresponds to ill posed problem. So ill posed version of the static deterministic linear inverse problem is what we are going to be looking at and we will also talk about what is called imperfect model along the way. So Z is equal to H of X is the linear square problem H is the matrix which is m by n in module 3.1 we already talked about the well posed problem linearly squares the well posed problem essentially essentially banks and the assumption that H is a full rank that means the rank of H is minimum of m and n. So when the overdetermined case the rank of H is n in the underdetermined case the rank of H is m. So we have considered both the cases under one rule namely the rule of H being a matrix of full rank. Now we are considering the complementary case where what happens when H is rank deficient. So what does it mean when the rank of H is not equal to the minimum of m and n but is less than the minimum of m and n. So in the overdetermined case it is not n but less than n in the underdetermined case it is not m but it is less than m. Such problems are called ill posed problems. So well posed versus ill posed largely determined by the properties of the matrix H please remember H Z is equal to H of X defines the static model the properties of the model is defined by the matrix H. So we considered one aspect of the properties of the model now we are considering another aspect of the properties of the model full rank versus rank deficient. In the when the H matrix is rank deficient the Gramian matrices H transpose H and H transpose are symmetric but they are not positive definite but they are singular that means the determines are 0. That means at least one of the Eigen values is 0 if at least one of the Eigen values is 0 it is not positive definite is only positive semi definite. So positive semi-definiteness of H transpose H and H H transpose leads to singularity and that leads to the fact I cannot simply now compute H transpose H inverse or H transpose inverse I cannot compute them because they are all singular I cannot do I cannot I cannot compute these inverses readily these do not do not exist. So when these do not exist I cannot follow the principles that we have developed in the last lecture. So this calls for newer methods to be able to handle and we are going to be talking about some of these newer techniques. To reinforce further when H is rank deficient when H is rank deficient we cannot use the formula for the generalized inverse of H this is the generalized inverse of H for the over-determined case this is the generalized formula for the generalized inverse for the over-determined case this is the formula for the under-determined case these generalized inverses that are used in computing XLS now cannot be done now cannot be computed because they cannot be computed I cannot use the old pathway to compute the solution. But the theory of generalized inverse still tells you while I cannot compute the generalized inverse using these formulas that is other ways of computing H plus I can still compute H plus but using the method called singular value decomposition. We will talk lot more about singular value decomposition in one of the modules coming later and using this one can solve even though one can solve using singular value decomposition at this juncture we are not going to use this we are simply looking for an another alternate formulation of linearly square problems. So, let me clarify where we are we have a linearly square problem Z is equal to H of X H is rank deficient I cannot use the method that I have already described thus far rank deficient cases can be handled by one of the methods called singular value decomposition we still have not developed the method of singular value decomposition now we will do it later. So, while we will have occasion to revisit the solution of this problem at the time when we describe the linear singular value decomposition at this time I am still interested in solving this. So, I am seeking an alternate method alternate to singular value decomposition that method is called method of regularization this method of regularization was introduced by a Russian mathematician called Tykanov this method is meant to get around the rank deficiency of H is a very simple elegant method to approximate solutions of ill post problems rank deficient problems ill post rank deficient I am using this anonymously. So, what is Tykanov method? Tykanov method is a modification of the method that we use in solving the under determined case in the under determined case we want to we seek to minimize the norm of H but we require Z minus H to be a strong constraint and we use Lagrangian multiplier what did Tykanov say Tykanov said you still consider a factor that corresponds to the square of the norm of x alpha is a parameter Z minus H of x norm. So, this is square of the norm this is square of the norm alpha is a kind of a penalty parameter. So, this is what is called a penalty function approach. So, what is that we are looking for I am looking for a solution f of a solution that minimize f of x the solution that minimize f of x is such that alpha times the square of the norm must be as small as possible and Z must be as close to H of x as possible but not exactly 0. So, the addition of alpha x square term to the traditional sum of square criterion helps to avoid the challenges resulting from rank deficiency. Now, please understand in the over determined problem we f of x was essentially this term for the over determined case for the over determined case we essentially use f of x to be this. Now, to that f of x I am adding this term by adding this term this is a kind of a penalty term. So, we mix several like other and concoct a new objective function and our problem is to be able to minimize this f of x and let us talk about the impact of this after we solve the minimization problem. So, rewriting f of x as in equation 2 we can readily compute the gradient and equate the gradient to 0 if you equate the gradient to 0 I get a solution for XLS by setting this to 0 I get the solution to be like this. So, you can see least square solution is a function of alpha H transpose H plus alpha I inverse H transpose Z when you set alpha is equal to 0 if this becomes a solution of the over determined case. So, you can think of this as a generalization of the concept of solution for the over determined case. So, the over determined case I consider only this term we added that. So, the alpha term is the one that is added to this. Now, look at this now H transpose H by itself is singular, but I am adding alpha times an identity matrix. So, this is called diagonal perturbation this is this changes only the diagonal elements of the matrix H transpose H. So, by adding a diagonal perturbation to a singular matrix I can make the whole matrix non-singular if the matrix is non-singular I can compute the inverse if you can compute the inverse I have the solution. So, what is this we are not solving the original problem we are solving a modified problem the modification is obtained by adding a diagonal perturbation to the Gramian H transpose H. So, since H transpose H is singular by adding a diagonal perturbation alpha I to H transpose H we ensure that H transpose H plus alpha is non-singular when this is non-singular I can compute the inverse if I can compute the inverse I have an expression for the least square solution. So, the expression for the least square solution for this ill-posed problem is given by is given by the equation 3 in slide 4 equation 3 in slide 4 now you may ask a question how do I pick that alpha in fact I am going to give an intuitive fill the alpha that you need to choose must be the smallest alpha that will make the matrix H transpose H non-singular this matrix by this matrix by itself is singular I am going to add a perturbation I would like to require this whole matrix to be non-singular. So, I want to ask myself the question what is the least alpha that I should use in order to render the resulting matrix to be non-singular such an alpha I always exist it can be proven so this is a very nice generalization. So, when H is a full rank we simply set alpha is equal to 0 when H is rank deficient you pick the least alpha that will make this matrix non-singular. So, once you have picked the least alpha that will make it non-singular I can solve the linear least square problem I have a least square solution the existence of such alpha is guaranteed by a theorem in matrix theory called Gershgorin circle theorem using the Gershgorin circle theorem you one can estimate the least value of alpha one can estimate the least value of alpha that is needed to be rendered this is non-singular such a thing exists it is a very simple result. So, what is that we are trying to do by formulating the problem as a penalty function problem we can even solve an ill-posed problem nicely. Now, I am going to talk about the use of matrix identity I hope you all remember that we have talked about several different matrix identities when we dealt with the module on matrices you have very well known matrix identity takes this following shape given by equation 4 this identity is very well known in matrix theory. Now, I am going to start with this well known identity I am going to specialize A is equal to H B is equal to I D inverse is alpha I why I would like to be able to use this identity in the Takanov solution to see what is the relation between the under determine or the over determine case that is what our aim is this identity with this substitution now becomes this using this identity this left hand side is equal to the right hand side. So, this left hand side becomes this the right hand side becomes that the right hand side can be simplified and that becomes this. So, look at look at look at this now H transpose H plus alpha I inverse H transpose is equal to H transpose alpha I plus H H transpose inverse these two problems these two matrices are equal that is the essence of equation 5 from here now I can do I can see the lots of lots of little things if I said alpha is equal to 0 this becomes a solution for the over the left hand side becomes a solution of the over determine system the right hand side becomes a solution of the under determine system. So, Takanov by introducing this penalty factor alpha was able to unify the solution for the over determine under determine case by invoking to this very well known matrix identity. So, that is the beauty of the solution of Takanov. So, Takanov solution is important in two ways one it helps to solve the ill post problem another one it helps to unify in trying to define the relation between under determine over determine.  Takanov has specialized in solving inverse problems of various type he has written a marvelous book that deals with regularization methods for solving inverse problem. This is one of the simplest of the methods of regularization that is often used in solving linear least square ill post problems. So, I have already talked about the unifies approach when alpha is 0 6 leads to optimal solution for the full rank problem when m is greater than n when alpha is 0 in 7 that leads to the optimal solution for the full rank problem and under determine case. So, all the previous solution can be obtained as special cases from this unified approach and hence the importance of of of of Takanov of of contributions. Now, I am going to talk about the role of model perfect versus imperfect under static constraint. So, models are static models can be perfect model can be imperfect the same goes no model is perfect, but some models are useful often one assumes the model is perfect imperfections in a model comes from various directions. The imperfect imperfections from the model can come from incomplete physics not complete physics sorry the imperfections in the model can come from incomplete physics or wrong parameterization or any combinations thereof or other reasons irrespective of whether the model whether is perfect or not in the over determine case the model is always in the problem is always inconsistent. In the sense that we saw in the previous lecture over determine systems are always inconsistent in the under determine case the choice of the method depends on whether the model is perfect or not. So, if the model is perfect we can formulate wrong way the model is imperfect we can format the other way. So, you should have a good feel for how good the model is. So, that brings us to the notion of different ways of formulating the solution of least square problems when the model is perfect when the model is imperfect that gives rise to a new way of looking at it called strong constraint versus weak constraint formulation. When M is less than M the model and the model is perfect. So, I am considering I am considering an under determine case there are less observation than number of parameters and also the model is perfect and that is the assumption we already made. We never question about the veracity of the model until now only now we are trying to ask the question is the model perfect if the model is not perfect there is one way if the model is perfect the other way. If the model is perfect I would like to be able to enforce the model equation strictly that gives rise to strong constraint the model constraint is strong using Lagrangian multiplier. This version of using Lagrangian multiplier when you believe the model is perfect is called strong constraint formulation we utilize this strong constraint formulation in trying to bring uniqueness to the under determine system. If the model on the other hand is not perfect it is pointless to enforce it strictly why we know the model is not perfect. So, why would you even enforce something that you know you are not sure about. So, in this case we still want to respect the model equation, but not strictly, but only approximately this ability to require the model equation to be satisfied not perfectly, but very closely is the concept of weak constraint formulation strong constraint weak constraint. Strong constraint is intimately related to strong constraint formulations are formulated as Lagrangian multiplier problem weak constraint problem are formulated as penalty function which we have already seen in the optimization module. So, I am now going to quickly illustrate a version of the strong constraint formulation let z is equal to hfx I am considering the under determine case assume h is a full rank recall there are infinitely many solutions we seek the unique solution that minimizes the following cost functional. So, model is a constraint among all the solutions in the model I want to find that solution that minimizes the cost function the model has infinitely many solutions among the infinitely many solutions I am seeking a solution that minimizes this cost function. So, this cost function the generalization of the cost function that we have utilized already in the analysis of under determine system in the under determine case what it is that we did we would like to be able to minimum find x whose norm is minimum, but in here I would like to be able to find an x such that it minimizes jfx which is a general quadratic function. A strong constraint formulation for this a strong constraint formulation is to build a Lagrangian the Lagrangian is jfx plus lambda transpose z minus hfx I am now computing the derivative or I am so the gradient of L with respect to x and lambda I get two sets of I get two sets of equations I get two sets of equations I simultaneously solve these two systems I get the solution lambda s s refers to strong solution x f s refers to strong solution the strong solution both lambda and x f s are given by the equation 12 which can be easily solved by which can easily obtained by solving 10 and 11 by setting b is equal to 0 c is equal to 0 and a is equal to I we get the well known solution for the under determine case which we have already seen. So, you can see this is a kind of a generalization of what we have done in the under determine case. Now let us consider the weak constraint formulation in this case I am going to build a penalty function pfx that is equal to jfx plus alpha by 2 z minus hfx transpose z minus hfx the necessary condition for a minimum is given by the gradient of pf alpha must be 0 the gradient of alpha for this is given by the right hand side equate this to 0 and solving this you get the solution x 1 x is equal to x 1 alpha plus x 2 alpha x 1 alpha has taken this form x 2 alpha has taken this form. So, the solution has two components both of which depends on alpha the sum of 17 and 18 is 16. So, 16 is the solution that minimizes the penalty function earlier we talked about the relation between weak solution and the strong solution I know the form of the weak solution I know the form of the strong solution. We also saw the weak solution tends to the song solution as the penalty parameter alpha tends to infinity. Now I am going to show that result in here to this end we use the Sherman-Marrison Woodbury formula if I use the Sherman-Marrison Woodbury formula you can see the relation on the left hand side this inverse applying the Sherman-Marrison Woodbury formula becomes this this inverse applying the Sherman-Marrison Woodbury formula becomes this we have already talked about the Sherman-Marrison Woodbury formula we also have given a proof of the Sherman-Woodbury formula the section on matrices. So multiplying both sides of 20 on the left by epsilon x transpose and simplifying in fact I would like to refer the reader to details in chapter 17 of the book by Louis Laxmere Achanandal the our textbook we obtain the matrix identity which is given by this relation 21 I know there is a lot of computational and checking is there but I am sure I am checking is a homework problem I am hitting on all the major concepts once you have the major concepts you can really follow these to be able to verify by computing various things now by setting these we can readily see this formula is given by this we can then we can see x1 star is the limit of x1 alpha which becomes this. So look at this now when alpha goes to infinity the right hand side does not have any dependence on alpha. So this is the limit of one of the components of the solution x1 alpha in the limit likewise you can compute the solution x2 alpha and using this identity again this matrix identity we have already described using this matrix identity which becomes this as alpha tends to infinity this becomes this this is independent of alpha in the limit therefore x2 star is equal to the limit of x2 alpha which is given by that therefore x1 star plus x2 star is given by this equation if you simplify that that becomes the strong solution as is in 12. So we have demonstrated that the weak solution in the limit as alpha tends to infinity becomes a strong solution in other words as the penalty parameter alpha increases with that band the weak solution converges to the strong solution. So where do I use the penalty function formulation you believe in the model but you also know the model is not perfect so whenever you have to use the model as a constraint use the weak constraint you know the model you assume the model is perfect in that case you assume I am sorry you use the model as a strong constraint. So which method do you use that depends on how strongly you believe on the goodness of the model on the goodness of the model. Therefore weak we have now described how to take care of many very many special cases. So I would like to summarize by observing the following the linear static deterministic inverse problem can be broadly divided into well posed and ill posed well posed problems are very straight forward ill posed problems in principle can cause headache. There are multiple ways of solving ill posed problem one way would be to use tycoon of regularization as we have talked about the second way would be to use a singular value decomposition we have not done that we will wait until that is done to revisit this issue. We also then incorporated the notion of perfect and imperfect models. So you can now consider quite a variety of situation model is perfect model is static model is perfect imperfect model is deterministic model is linear. So you have linearity perfectness or imperfectness you have well posed or ill posed over determined or under determined. So you can consider a quite a variety of formulation of the even simple linearly square problem that brings about the beauty that underlie the notion of simple linear least square methods in the context of linear inverse problems. I would like to very strongly recommend that you all work out exercises these are simple extensions of the methods that we have solved I am particularly referring to example 7.3 in here I am considering a matrix whose columns are 111, 111 plus epsilon when epsilon is 0 the two columns are linearly dependent when epsilon is not 0 mathematically it is linearly independent. Therefore the Eigen values of h transpose h are functions of epsilon. So what is that I would like you to do h is given by this epsilon is greater than 0 epsilon is small compute h transpose h compute the Eigen values of h transpose h plot the variation of Eigen value using a mat lab as epsilon varies from minus 1 to plus 1. So you can see what is the impact of the rank. So when epsilon is 0 the rank this is the rank deficient when epsilon is not 0 it is not it is strictly rank 2. So but when epsilon is small even though mathematically it is rank 2 it is it can still cause problems and how it creates problems one can understand by computing the Eigen values. Once you compute the Eigen values you remember we can compute the condition number. Therefore by computing the condition number for h transpose h we can infer how ill posed or how well posed the problems are. So that is the measure by which we can compute the degree of ill posedness. So this problems 7.3 is an important exercise and there are few other problems which are routine which follow the directions of the development. And this module follows from chapter 5 and the following report. We recently computed a report in 2014 on the convergence of a class of week solution to the strong solution of an equality constraint minimization problem. I am sorry this must be we must remove the point there. A direct proof using matrix identities is a technical report from the School of Computer Science University of Oklahoma. With this I think we have provided a broad overview of the richness of the linearly square problem static version thereof. Thank you.


I've always had as chairman of the Commission for European Affairs excellent relations with the Irish Parliament and we've had very, very strong relations. So it's very, very something, I'm very, very pleased to be here. And I'll try to be short in my speech because I would like to answer questions and ask questions. What is the situation before the European elections we have? I guess that Europe, it was very easy to explain Europe, I would say up to 15 years ago, Europe was symbol of peace. And everybody agreed on that and it was in France, of course the reconciliation with the Germans, it was a sort of magnificent project of reconciliating countries which had done the war once, one against the other. And then it was the symbol of the reunification of Europe. It was the symbol of the fall of the communism, the fall of the wall. And so that all the people understood, all the people agreed. But I would say that when you speak now to the young generation, they haven't lived the war, they haven't, the fall of the communism is something which is behind them. And so it's more difficult to explain Europe, it's more difficult to explain the interest of Europe. And there's a sort of, in the population, there's a sort of reaction towards Europe. If there is unemployment, it's the fault of Europe. If there is difficulties, if there are difficulties in the everyday life, it's the fault of Europe, which is not, of course, because Europe is not really directly concerned by unemployment. It's, of course, the policies of the countries which have to be studied. But, and there's also this reflex of the government, and I would say it can be the right, centre-right government as well as the socialist government, to accuse Brussels. So easy to accuse Brussels, because you don't know who is Brussels. And actually, even now, every day I see ministers who come in the French Parliament and who accuse Brussels, whereas actually they were at the Council of Ministers, they approved the decision at the Council of Ministers, and once they are in the Parliament, they accuse Brussels, they accuse themselves, I mean. And this is now, and this was true also under our governments. And so we've created the reflex against Brussels, against the technocracy, against the, whereas it's not, it's not Brussels who decided of the measures. It was, it was the ministers, the Council of Ministers. And when there's a good news coming from Europe, then you forget to say it's Europe who helped us. I remember Jacques Chirac, my President of the Republic, when we got this fantastic future project of Haïter, in the south of France, he said it's the victory of France. And he completely forgot to say it was the victory of Europe too, because Europe was very, Europe decided to support France in this project. And actually, if you remember, Spain wanted this project to be in Spain, and they decided that they would accept that it would be in France. And so there was a big support of Europe. And then when we have good news, we forget to say it's, because of Europe we have this good news. So the mood in France, it was the case 10 years ago. The mood in France is not as it was when the European Parliament was created. And it's a sort of, I wouldn't say aggressivity, but sort of indifference, which is not good, of course. And UMP, my party, is going to make a very, very active campaign, because we've got to show all the people that Europe is absolutely essential for us. I'm going to try and be objective, although it's not easy when you are in the opposition. But I would say I'm very concerned. I'm going to give you my feeling. I'm very concerned about the Franco-German relation during this year and a half. To say the truth, when Sarkozy was in power, the Franco-German relation was not easy every day. But there was a good relation. I mean, there was discussion, but there was a good relation. When François Hollande arrived in power, I would say his first reaction was to do the contrary of Sarkozy, and so to have a sort of conflict position with Germany. When he said it on the French TV, I have a tension, a friendly tension with the chancellor, which in diplomatic terms means I have a real conflict relation. And after that, it was horrible sentences pronounced by ministers, by the president of the National Assembly saying Mrs. Merkel is Bismarck, she wants to train over Europe. And in fact, what happened is that the economic situation of Germany was so much better as the economic situation of France that, in fact, it is difficult to have a good relation when the situation is so different. There was about 180 billion of surplus Germany, and there was 80 million deficit in France, to say that, but all the employment is 5% in Germany, it is 10.9% in France. And so France was weakened first by its economic situation. And the position of Hollande was to have a sort of confrontation with Mrs. Merkel. And so he tried to have relations with Spain, relations with Italy, insisted on growth, and I know this position was appreciated in Ireland, of course, but the problem is that he didn't have the strength of maintaining his position and that, in fact, he accepted what Mrs. Merkel wanted Europe to do. You remember his position during the campaign, he said, I'll never ratify the treaty, which had been negotiated by Nicolas Sarkozy and Angela Merkel while he ratified it. He decided to ask his parliament to ratify it. He said he would like Euro bonds to be created. He decided to leave this project. And during the budget discussion, he wanted the budget to be more important. And actually there was a sort of coalition between Angela Merkel and Cameron, which we had never seen before. And actually the budget was diminished, and it was diminished. We hadn't seen that since the beginning of the European construction. Such a diminishing of the budget. We are going to have a double challenge for UMP. I'm talking about my position. We are going to have, of course, to criticize the government. It's the main task of the opposition. But we'll have to criticize also the extremes. The extreme right, the extreme left. You don't have this problem here in Ireland, because you are reasonable people. All the parties are in the center. I would like to have that in France. We have two parties at the extreme left and the extreme right, which are going to make a lot of positions very populist, very nationalist. They propose to get out of Europe, get out of Europe, have frontiers around France, and going back to the 18th century, actually. Malthusianism. And this, of course, is not appreciated by the French people. And we're going to attack them on that, because the Front National has no proposal. Nothing in its program. I've read the program. I've tried to read the program. But there's no proposal in their program. The only proposal, which they do, is getting out of the Europe, getting out of Europe, and establish frontiers around France, so that all the products of other countries can't come to France and France can't export. I mean, it's completely foolish. So we are going to attack that as we are going to attack the French government, but we are going to attack, of course, the position of the extremes. And the extreme right and the extreme left, it's funny, but it's really evident, have the same program on Europe. It's exactly the same program on Europe. As I think Europe is at a crossroad, we've got to go much further on European issues. We've created the common policy on agriculture, and I think we've got to go much further. We've got to do a common policy on energy. We've got to do a common policy on industry. We've got to do a common policy on immigration. And we've got to have a cooperation on defense and foreign affairs. So we think it is very important that without the creation of new treaties, we can do it without treaties to develop those common policies which have to be developed and which have to be done. And we've got, of course, to have a more competitive Europe and flexible so that we have growth and employment. We have to, of course, control the companies which practice the social dumping in the UE. We've got to also defend Europe against companies or products which come from China, which come from India, which come from the States also. And they have a position on giving privilege to their national companies, helping their companies in public markets, but I don't see why Europe would not do the same if it was the case, so that these countries respect us. We've got to have a bi-European act, as the Americans have there, a small business act. And we've got to have Europe of culture. I'm here in a country where culture is very, very important, and I know we share the same interest in culture, and this is to be done with, of course, the youth, the young people with the Erasmus, et cetera. We've got to have a clear definition of what we want in Europe. We've decided to make Europe with more and more countries, but this situation makes it more and more difficult to, to dirige, dirige, to lead. It's more and more difficult to lead. The more and more we were around the table, the more and more it is difficult. When you discuss with ministers who go to the Council of Ministers, they say it's more and more difficult to work because you have the Minister of Malta, who speaks for half an hour, and then you have 28 countries discussing like this. How do you want to work? And the, the function, the functioning of the Commission is not at all the functioning one, what it was in the beginning, because you know that in the Commission normally there's a Commissioner who presents his report, and then the Commission is not a Council of Ministers. The Commission is, has to vote on the report. And each Commissioner is important in this vote. But when you get more and more numerous, when you get 28, then there's not this discussion within the Commission. So the Commissioner comes with his report, and it is most frequently adopted without any thinking of each of the Commissioners. And so the Commission has not the same role it had when it was the law, when the President was the law. And so we've got to define the future of Europe. We can't go on saying we're going to make Ukraine get into Europe, we're going to make Moldavia get into Europe. The people, I'm telling you about France, but they need to know what is the Europe we want to come to build. And there I think we've got to be very, very clear. And to say that the Balkans is all, are of course European countries. But after the Balkans, we stop it. We've got to be very clear, because if we go on integrating other countries, what I said in the beginning, to build a political Europe, we won't be able to do it, because if we are very, very numerous, the political Europe is not a possibility we can reach. And we have a very clear position on Turkey. We think Turkey is not, has not to be integrated in Europe. We think it has to have a partonaria privilégie, privileged relation, partonaria with Europe but not be in Europe, because it would change completely the possibility of building this European, of political building this political Europe. I haven't spoken about changing the institutions, because I think we've done it with Lisbon. The only thing I propose is to have, which is a proposal of Hollande and Mrs. Merkel, to have a stable president of the eurozone. Because you know, I think it's very important that we have a stable president. For the moment it changes regularly. And the eurozone is very important and we have to have a stable president. And we have to have, and that's a thing I tried during my presidency of the Commission, to have very strong relations with the European Parliament. The national parliament have to have very strong relations with the European Parliament. And I had very, very important relations with the European Parliament. We even had meetings of my commission, with the commission of the European Parliament and this was very, very important. Because in a country like ours, we have 72 European parliaments. We are elected, the European members of parliament are elected in regions. There's one region which begins from Clermont-Ferrand, Lyon, up to Corsica. How do you want 10 members of parliament to be only well-known in this region? It's so enormous. And so that's what I think I convinced my party. It's very important that the national members of parliament get into the campaign. Because they have their constituencies. The constituencies are 100,000 people. And even, I think that the national parliament in France must be much more concerned in European affairs. And so, I don't know if it has been brief. I don't know how long I took. It should finish. Yeah, more or less. Thank you very much. What I want to tell you is that I'm really convinced that Europe is very, very important for our citizens, for your citizens. It's a main issue. But we are going to have difficult elections in France. And instead of continuing, I will ask you to ask questions about the elections. And I will answer you.


I want to make a bold statement that it is time for this generation to unlearn the wrong principles acquired from society and to learn and apply God's true principle that will lead to happy marriages, that will lead to fruitful marriages, that will lead to established marriages, that will lead to sustainable marriages, if you permit.


So, this is a very visually I will take you to the entire spectrum of extreme and far lateral approaches to the skull basin I mean at the end of the presentation I hope that you will understand the differences between the nuances of different types of approaches as well as you will be able to do all of them without any problems I mean there is nothing to do it actually it is just the name which is more important.  So, what you are doing is you either go just anterior laterally or completely laterally and reach anterior to the spinal cord from a lateral aspect which means that you will have to either drill the so you have to drill the condyles there. So, just to drill the condyles you may or you may not I will just tell you when you need to drill it and when you do not need to drill it, but this way you can actually go to the anterior part of the tumor and so the basic thing is that none of them this is from you know theory these are all cases which have been operated and this is completely visual. So, there are no written words there right except of course just to give me a hint that I should not forget those things.   So, first I encourage you you know give you tell you the word the anatomy you already know then take you through the complex anatomy. Now, we come to some interesting anatomy you know a bit of this you know that this is the occipital condyle right you know that the hypoglossal canal is in the middle of it. So, from here to here this. So, it will be like this and it will be two-thirds and one-third where is the juggler process juggler process is this piece of bone right. So, this is the juggler foreman and this piece of bone is the juggler process right. So, just these three things which I esteem on ok is not relevant, but juggler process. So, this is very very important and I will tell you why it is important you will understand in a minute right. So, this you all got you located this yes all of you now look at it from the inside. So, this is the condyle this is the hypoglossal this is the where the hypoglossal now comes out and the other important thing is that this is juggler foreman and this is the juggler tubercular. So, there was a juggler process from outside and a juggler tubercular from inside. Now, you can imagine that if you actually rather than coming posteriorly like this if you come like this drilling the lateral one-third of the condyle staying posterior to the hypoglossal nerve then you get an access to the clivus and to the anterior foreman magnum. The only problem is that you have this juggler tubercle which sometimes comes in your way and causes a problem. I will show you real examples of then you know what are the different approaches what are the different approaches to this. So, first is just a far lateral approach far lateral approach means do not touch the condyle just go lateral like this and go anterior, but obviously it means that you cannot go to anterior you can maybe go up to this point, but you cannot go anteriorly that is a far lateral approach. The second is a transcondyler approach what is a transcondyler approach you go like this drill the lateral one-third of the condyle and reach the anterior part. So, one-third of the condyle no instability because the rest of the condyle is intact that is a transcondyler.  So, here by the supra condyler we mean that you actually move like this to the clivers without touching the condyles. So, there is a supra condyler, transcondyler, paracondyler and far lateral is that clear to everyone. So, that is the that is all I mean that is all there is to it now I just show you examples. So, you know you can do it actually. So, now let us go take examples. So, one the first is vortible artery mobilization. So, you will do an extreme lateral transcondyler approach with intra and extradural vortible artery mobilization when would you actually require this let us take this example. So, extensive anterior dural based humors reaching up to bilateral vertebral arteries meaning a meningoma obviously, but it means that it is attached it is not you cannot mobilize it a neurofibroma on the other hand you can you know decompress and mobilize but a meningoma stuck there you cannot mobilize. So, there that is when you want to go more anterior and how do you go more anterior I will just tell you. So, we have a clinical situation where there is a huge dural tail extensive causing cervical medullary compression both vertebral arteries within this meningoma right. Now, obviously I mean somebody says oh I can do it with an endoscope I can do it with a far lateral approach I can do it with a retro sigma approach acceptable I am not saying that I am just telling you the nuances now it is up to you to decide which approach you would like to take.  So, that is how it is. So, this is a thing. So, what is the position? Now, the position is something there is some very small nuance that you really need to know about the position of this patient dead lateral. Why is dead lateral position important? Because if you rotate the neck even a little then you are the relative position of the foreman transorcerium or C1 and C2 changes and therefore, your vertebral artery changes you get it.  But supposing you want to do a mastoidectomy and all that. So, you would prefer this supposing you want to go up to the midline then you know you though there are different nuances to that, but you can use any of these flaps. Then coming on to this so, you made this and there are three important landmarks the one is external occipital protrusions the second is a C1 posterior arch. So, posterior arch in the posterior tubercle you cannot palpate right now, but then of course, when you are actually dissecting you can actually feel I am sure or C2 spinous process which you can actually prepare and the transverse process of C1. So, this if you go laterally right now you cannot do it, but this can be palpated during surgery. So, these are three important things. So, two structures in the midline external occipital protrusions C2 and the third is C1 transverse process. And then of course, you have different muscles which you need to reflect do not go into details you can reflect them all at one times and it actually helps you in the less atrophy it is better to remove them all in one piece rather than you know to do a piecemeal detection. This is just for your anatomical principles and then what you need to do is to come to the suboccipital triangle. Now what is the suboccipital triangle consists of three muscles at the level of the suboccipital region. So, one is attached to the transverse process going almost up to the midline second from the midline to another midline here inferior midline and the third is from the midline to the transverse process here and the three muscles are supia inferior oblique and rectus corpitis posterior major.  I mean you hardly are able to identify these muscles in real life situation, but you must understand the concept and what is the concept? The concept is that you palpate the transverse process of the C1 right do not worry about these muscles you just process the not the transverse the posterior arch of C1 just palpate which you can. So, go from the posterior to the posterior you just palpate it like that right once you palpate it then where is the vertebral artery in relation to that on the superior border lateral one-third. So, that is all there is the significance of this triangle the significance of the triangle you just do not worry about the muscles palpate the bones palpate the bones palpate the C1 arch and at the lateral one-third you have the vertebral artery right. So, a real life situation this is lateral mastoid this is the external occipital protuberance here and once you have retracted all your muscles in one piece you will have a triangle and do not worry about the triangle the only thing is you palpate the C1 arch here posterior to C1 right. So, this is just what you need to do is you can do it easily everybody has done it. So, it is not it is not something very difficult then what happens is this is the arch of C1 the hint that you are actually going to the vertebral artery is by two nerves the C1 nerve and the C2 nerve. So, you can you will clearly see the C2 nerve you will clearly see the C1 nerve and the other thing is the superior border of C1 which will take you to the vertebral artery is that clear you will clearly see this I will show you. So, what happens is when you are actually dissecting it you see the this is the posterior arch of C1 this is the edge of C1 you can actually feel it going there dissect along the bones you can see C2 going up here and vertebral artery will be somewhere here which you right now do not know where it is right and this of course is the mastoid. So, this is how it will actually look but it is not difficult at all just a simple thing we have all done it did the section we have all done right. Now, what happens is you drill it from the foremen transversarium I will tell you the operative steps in the next one I just wanted to do the first one is just mobilization. So, what is the significance of this the significance of this is that when you have taken out the vertebral artery from the foremen transversarium of C1 and sometimes also of C2 then you see and you have retracted it downwards then what you get here this portion is the occipital condyle. So, this is the sub occipital this is the dura this is the spinal dura and you can feel the condyle here and drill it. So, what you have done is you mobilize the vertebral artery which was like flat like this and coming like this and then going intradural here rather than doing that you have mobilized it downwards and medially and therefore, you get access to this portion once you drill this portion then what happens is you are working significantly anterior to the spinal dura you can well imagine this is that clear to everyone you understand. So, this is the significance of vertebral artery mobilization right and once you have mobilized you have drilled it and you open the dura you see this is the spinal cord and the cervical medullary junction and this is the tumor right. So, you have directly reached the interface between the tumor and the cervical medullary junction without touching the spinal cord otherwise you would have had to go from this aspect and you imagine that the spinal cord is between you and the tumor here is that clear is that clear to everyone yeah. So, just the small thing and after the tumor is removed you leave a dural cuff here and this is the cervical medullary junction you see the color has changed. So, this is the pink color which is in the foreman transversarium this is a red color which is on the C1 arch these are natural there is a dural cuff here and this becomes intradural here and this is the cervical medullary junction you can remove the entire thing. So, you can imagine you can see the contralateral vertebral artery also you can actually see the contralateral vertebral artery also right. So, this is the post operative. So, now what you have done is this is the condyle preop and what you have done is you have drilled a little bit of the condyle here and actually gone anterior to the core right see here you have drilled and gone anterior. So, now we go to the actual operative steps right. So, this is this huge tumor that you see now this is this is a significant tumor right and this is all anterior to the cervical medullary junction. Now, what are the operative steps? So, up to this point everybody can reach up to this point everybody can reach this is the suboccipital region you can see the vertebral artery you can see the C1 nerve root going like this and this is the C1 posterior arch just here do not use monopolar just follow it and use a periosteal disector. So, once you will go along the entire border you will definitely see part of the vertebral artery and you can see the C1 clearly. Now, what happens is the vertebral artery goes like this and then it goes through the fermatransversary and down and here there are lots of paravertible venous plexus do not let them deter you because it is very simple to actually dissect it from there what you can do is you can leave a little bit of tissue all around the vertebral artery. So, that it does not get thrombosed and then go on dissecting coagulating dissecting coagulating do not actually come to the vertebral artery take a lot of tissue along with it. So, that it does not get thrombosed just a small trick you understand and when you are beginning your surgery for this work from the side of the non-dominant vertebral artery. When you work from the side of the non-dominant vertebral artery if it is a midline lesion then the advantage is that even if something happens to the vertebral artery is not a problem at all yes. So, this is just small tricks to you know help you. So, now the next step is you have actually dissected out the entire thing you have traced it right up to its entry into the formant transversarium here. Now, it is very simple to just expose this and drill this part and using a diamond drill it is very simple to do it there is no problem at all right just do it.  So, you have seen this going like this going like this and this is where the condyle is this is where the condyle is okay. So, once you drill this condyle then this takes you anterior to the cord here right. So, you have drilled this and this so now it has become completely mobile. So, this is where the dura is there the posterior has been removed. So, it was horizontal like this and going vertical like that now all you have done is to mobilize it and turn it downwards when you have done it what do you get? You get this whole area here which you can drill and once you drill that you reach anterior to the cervical medullary junction is that clear? And then you open the dura once you open the dura what do you get? You get a lot of low cranial nerves and you get the 11th cranial nerve the accessory now spinal accessory now. So, this is the cervical medullary junction this is the cerebellum this is where the tumor is and this is the erecnoid over the tumor right. So, imagine because you have drilled this much of bone you are actually seeing from here otherwise you would have seen from here you understand the difference are you getting the difference? So, this is how it is the difference between going that way and this way and then when you open the erecnoid you can see the tumor anterior to the cord here and then of course with intertumoral decompression and preserving all the cranial nerves you can work anteriorly and go on mobilizing till you remove the tumor then it is possible to just coagulate the anterior margin and see the opposite vertebral artery. You can actually see the opposite vertebral artery mobilize the entire tumor here and this is so this is the cerebellum this is the cervical medullary junction this is the area which was created by the tumor and then finally you have the postoperative where you actually have been able to go like this to the anterior part of the cord simply because you have mobilized the vertebral artery right. So, now you have this it is not always necessary to mobilize the vertebral artery sometimes supposing you have a tumor like this which is a recurrent tumor where the vertebral artery was already compromised then all you need to do is this extensive tumor you take the same approach do not worry about the vertebral artery just take out the entire part. So, it is not necessary that every time you need to mobilize the vertebral artery you need to do all these things and so here you find that the contralateral artery was well preserved and the ipsilateral artery was already had already been ligated and therefore what you needed to do was complete removal just by this approach without without actually bothering about the vertebral artery in this case right. Now transcondental approach without vertebral artery mobilization so you have this patient who had the subrecnoid hemorrhage and intraventricular hemorrhage and when you see this angiogram you find that there is an aneurysm which is quite low down almost at the level of the condyle and this is arising from the pica right. So, there is a pica aneurysm so this is the condyle there is a c1 arch and there is this aneurysm lighted the foremen magma right. So, this is this aneurysm pica and why was not why was not this embolized you will understand when we did an angiogram what did we find we found that this is the basilar artery this is the vertebral artery right and so there was a dissection in the basilar artery so the basilar artery was not filling this is the dominant pica and there was a dissecting aneurysm of the pica here and this pica was bio hemispheric so the dissecting aneurysm was arising from a bio hemispheric pica and there was this solitary basilar artery which was supplying both the posterior cerebral arteries arising from the vertebral artery is that clear? So, they said no question we cannot go into it right. Now, this is quite lateral this is situated almost at the level of the condyle here this is a bio hemispheric pica and this is actually a dissecting aneurysm here and basilar artery is filling up very very little right. Now, this is quite lateral dissecting aneurysm so here again the same approach. So, this is superior this is inferior this is C 1 arch like I told you but you will always get this you will always see vertebral artery just superior to the C 1 arch here right. Then drill the C 1 arch right there is no need to mobilize the vertebral artery there is no need to because it is quite low down and so there is no need for a all you needed was a this vertebral artery is quite low down what you need to do is to drill this part of the condyle without mobilizing of the vertebral artery. Once you have drilled this then open the dura and you see that this is a part of the cerebellum this is a part of the foremen magnum right. This is the vertebral artery this is the dominant pica and this is the aneurysm.  So, this is the dominant pica here. So, just see the entire you have the entire pica in your view then you just secure it and this aneurysm was clipped and the part of it was removed. So, imagine that you have actually reached quite anterior to the brainstem just by this time you have not even mobilized the you have not even drilled the condyle right. Now, when would a only pure far lateral suffice there is no need to drill the condyle there is no need to for vertebral artery mobilization just far lateral approach. So, you have the cystic lesion you have the cystic lesion which is anterior which is a cystic lesion. However, because it is mainly cystic you see that this is exactly occupying the same space with the other tumors were done doing but the point is that this is cystic. So, it is easily decompressible right and once it is decompressible. Therefore, what you do is the same thing you went by a lateral approach but this is the dura this is the cranial part and open the dura and here you can see again all the nerves and the C1 and all these nerves and this is the cerebellum here but this cystic tumor could easily be removed by not drilling anything. So, just decompression and the entire thing comes out. So, this is a far lateral approach right. So, you can understand that sometimes you will need if it is higher up tumor if the vertebral artery is coming in your way then you need to mobilize it bring it down right. On the other hand if there is a lot of gap between the condyle and the vertebral artery then you do not need to mobilize the vertebral artery you can just drill the condyle without mobilizing the and sometimes when you have a cystic tumor which is quite low down there is no need for anything you can just go by a lateral approach. And now this is another pica aneurysm. So, this is another pica aneurysm and this is the this is the pica right. So, this is from the vertebral artery and this is quite lateral. So, this is the midline this is the midline and when you assess the from the midline this is quite lateral. So, this is almost reaching up to the cerebellum pontine angle or the cerebellum medullary cistern. So, here there is no need to drill anything there is no need to mobilize anything what you can just do is this is the cerebellum this is the cerebellum these are the tonsils just retract the tonsils what you see are the low cranios you can see the distal pica right and you can see the aneurysm here. And what is this structure? I told you about this structure this is the jugular tubercle. So, you can see the jugular tubercle. So, sometimes you will need to drill the jugular tubercle to get proximal control. So, here you are not getting proximal control. So, if I had drilled all this area drill the condyle then I would have come anterior like you saw in the previous case here I have not done that. So, all I am getting is a distal vessel and I can see the aneurysm here and this is partially embedded into the jugular tubercle right. In this case I could apply a clip, but then it is dangerous because you do not have proximal control here. When would you not use any of these approaches at all far lateral extreme lateral or anything you have this classical case where you have the significant you know like classical you I will go lateral lateral lateral and then what happened was there is this extradural component. So, there is this huge extradural component means this will give you space a lot of space. So, when would you not use any of these far lateral approaches when it is there is an extradural component, when there is a dumbbell component which will give you enough space when there is a significant lateral component to the lesion. If it is a neurofibroma because a small neurofibroma can easily be mobilized even with a simple approach and if it is cystic lesion like you saw that neurointricist there is no need for any of these approaches no need for it you can just go lateral and remove it right. Of course, we have all these problems the most important problem is low cranial palaces because you know they are working between the cranial so you have to be very very careful about low cranial palaces and the other thing is venous bleeding. So, you must always have some fibrin glue and it is very simple just put a little surgicell with the fibrin glue it will completely stop your bleeding. There is a lot of paravortable venous plexus which may sometimes bleed and instability usually does not occur provided you have not drilled too much of the coxipodial condyle. So, there is no need to do it I mean it is just superfluous to actually do stability there is no need to do it. So, this gives you direct access to the anterior cervical medullary junction you must have it in your armamentarium I do not say that you need to mobilize it in every case. So, just in case you feel that it is anteriorly blaze meningioma which is completely adherent to the dura right and encasing both the vertebral arteries with no lateral component that is when you actually need to mobilize the vertebral artery right. And of course, even the contralateral vertebral artery can be mobilized using this approach is that clear. Thank you.