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Some Approaches to Complexity Reduction: Application to - PowerPoint PPT Presentation

Oct 24, 2022 •208 likes •1.71k views

Some Approaches to Complexity Reduction: Application to Computational Chemistry Yvon Maday, Laboratoire Jacques-Louis Lions Universit Pierre et Marie Curie, Paris, Roscoff, Institut Universitaire de France and January 2020 ICODE workshop on

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This defines X 1 = Span { u ( µ 1 ) } There is the notion of orthogonal projection over X 1 = Span { µ 1 } : Π X 1 µ 2 is determined as max µ k u ( µ ) − Π X 1 [ u ( µ ) k This defines X 2 = Span { u ( µ 1 ) , u ( µ 1 ) } µ 3 is determined as max µ k u ( µ ) − Π X 2 [ u ( µ ) k proven to be close to optimal

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µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments

  10. <latexit sha1_base64="tzgXEMTARup4IKe9u1o/cyvdQtQ=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">AC2nicjVHLSsNAFD3Gd31VxZWb0EZQFEnd6LoxqWCtYVGSpKOdWheJDNiKW7ciVt/wK1+kPQP9C+8M03B6ITkpw5954zc+/1koBnwrYHY8b4xOTU9MxsYW5+YXGpuLxynsUy9VnNj4M4bXhuxgIesZrgImCNJGVu6AWs7nWPVLx+zdKMx9GZ6CXsInQ7Eb/kviuIahXLEtu3rT4jila3R3TCeWZRVaxbK9a+tl/gSVHJSrJWf7cVDtncTFVzhoI4YPiRAMEQThAC4yepqowEZC3AX6xKWEuI4z3KJAWklZjDJcYrv07dCumbMR7ZVnptU+nRLQm5LSxAZpYspLCavTB2X2lmxv3n3tae6W4/+Xu4VEitwRexfulHmf3WqFoFLHOgaONWUaEZV5+cuUndF3dz8VJUgh4Q4hdsUTwn7Wjnqs6k1ma5d9dbV8TedqVi19/NciXd1Sxpw5fs4f4Lzvd0K4VOa9CGawbrKGT5rmPKo5xghp59/GEZ7wYjnFn3BsPw1RjLNes4syHj8AWnmYvQ=</latexit> <latexit sha1_base64="rHF0abh+71EBtN7lt/MqzAqRho=">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</latexit> <latexit sha1_base64="2ghF3uPcfuKDZN1SH9XMHcW8Mo=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="vqrdpqStU5GzB3uiEnstyZkX5zE=">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</latexit> Framework u ( x, t ; µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments or b) possibly a mathematical model for the behaviour of the phenomenon, depending on the parameter µ

  11. <latexit sha1_base64="tzgXEMTARup4IKe9u1o/cyvdQtQ=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="rHF0abh+71EBtN7lt/MqzAqRho=">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</latexit> <latexit sha1_base64="2ghF3uPcfuKDZN1SH9XMHcW8Mo=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="vqrdpqStU5GzB3uiEnstyZkX5zE=">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</latexit> Framework u ( x, t ; µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments AND b) possibly a mathematical model for the behaviour of the phenomenon, depending on the parameter µ

  12. <latexit sha1_base64="2ghF3uPcfuKDZN1SH9XMHcW8Mo=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="vqrdpqStU5GzB3uiEnstyZkX5zE=">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</latexit> <latexit sha1_base64="tzgXEMTARup4IKe9u1o/cyvdQtQ=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="rHF0abh+71EBtN7lt/MqzAqRho=">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</latexit> Framework Possibly polluted with errors and randomness u ( x, t ; µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments AND b) possibly a mathematical model for the behaviour of the phenomenon, depending on the parameter µ

  13. <latexit sha1_base64="2ghF3uPcfuKDZN1SH9XMHcW8Mo=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="S+sGdEU40Rdmz6g68nGPFyvcjw=">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</latexit> <latexit sha1_base64="vqrdpqStU5GzB3uiEnstyZkX5zE=">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</latexit> <latexit sha1_base64="tzgXEMTARup4IKe9u1o/cyvdQtQ=">AC2nicjVHLSsNAFD2Nr1pfUXHlJrQVFEtJ3eiy6MZlBfsAW0ISRx2aF8lELMWNO3HrB+hWP0j8A/0L70xT8IHohCRnzr3nzNx7ncjiTDN15w2MTk1PZOfLczNLywu6csrSRMY5c13dAL45jJ8zjAWsKLjzWiWJm+47H2k7/QMblyxOeBgci0HEer59HvAz7tqCKEtfK5fTzSuLVwxh9StG10+3yuWCpZfMqmW8RPUMlCqF7vbDwAaof6CLk4RwkUKHwBGEPNhJ6TlCDiYi4HobExYS4ijNco0DalLIYZdjE9ul7TruTjA1oLz0TpXbpFI/emJQGNkgTUl5MWJ5mqHiqnCX7m/dQecq7DejvZF4+sQIXxP6lG2f+VydrETjDnqBU02RYmR1buaSq7ImxufqhLkEBEn8SnFY8KuUo7bChNomqXvbV/E1lSlbu3Sw3xbu8JQ249n2cP0Frp1ojfEST3sdo5bGOIjZpnruo4xANMl7iEc84VnrajfarXY3StVymWYVX5Z2/wEB2Jc3</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">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</latexit> <latexit sha1_base64="j1pQvRUbV7OCaYQRxUkAWyWMy1M=">AC2nicjVHLSsNAFD3Gd31VxZWb0EZQFEnd6LoxqWCtYVGSpKOdWheJDNiKW7ciVt/wK1+kPQP9C+8M03B6ITkpw5954zc+/1koBnwrYHY8b4xOTU9MxsYW5+YXGpuLxynsUy9VnNj4M4bXhuxgIesZrgImCNJGVu6AWs7nWPVLx+zdKMx9GZ6CXsInQ7Eb/kviuIahXLEtu3rT4jila3R3TCeWZRVaxbK9a+tl/gSVHJSrJWf7cVDtncTFVzhoI4YPiRAMEQThAC4yepqowEZC3AX6xKWEuI4z3KJAWklZjDJcYrv07dCumbMR7ZVnptU+nRLQm5LSxAZpYspLCavTB2X2lmxv3n3tae6W4/+Xu4VEitwRexfulHmf3WqFoFLHOgaONWUaEZV5+cuUndF3dz8VJUgh4Q4hdsUTwn7Wjnqs6k1ma5d9dbV8TedqVi19/NciXd1Sxpw5fs4f4Lzvd0K4VOa9CGawbrKGT5rmPKo5xghp59/GEZ7wYjnFn3BsPw1RjLNes4syHj8AWnmYvQ=</latexit> <latexit sha1_base64="rHF0abh+71EBtN7lt/MqzAqRho=">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</latexit> Framework Possibly polluted with errors and randomness u ( x, t ; µ ) In order to determine : what do we have at end ? u ( x i , t k , µ ) a) possibly measures, either pointwize R ϕ i,k ( x, t ) u ( x, t, µ ) dxdt or moments Possibly inaccurate and suffering from bias AND b) possibly a mathematical model for the behaviour of the phenomenon, depending on the parameter µ

  14. Let us assume that we have such a space Z N

  15. Let us assume that we have such a space Z N and a model

  16. Reduced basis method : approximation of a PDE With such a Z N … Perform a Galerkin approximation With domain decomposition : Reduced basis element method Much to say : off-line, on-line 2 books : J. Hesthaven; G. Rozza; B. Stamm & A. Quarteroni, F. Negri, A. Manzoni

  17. Reduced basis method : approximation of a PDE With such a Z N … Perform a Galerkin approximation With domain decomposition : Reduced basis element method Much to say : off-line, on-line for on-line efficiency for non linear problems : a fundamental ingredient is …

  18. EIM/GEIM

  19. EIM/GEIM Reconstruction from data .. only

  20. EIM/GEIM Reconstruction from data .. only and a background space Z N

  21. EIM/GEIM The Empirical Interpolation Method (EIM) proposed in 2004 with M. Barrault, N. C. Nguyen and A. T. Patera This approach allows to determine an “empirical” optimal set of interpola- tion points and/or set of interpolating functions. In 2013, with Olga Mula, we have generalized it (GEIM) to include more general output from the functions we want to interpolate : not only pointwize values but also some moments.

  22. <latexit sha1_base64="bHIb43CPYZaks7LkAzvD5AE64=">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</latexit> recursive (greedy) definition of the functions and the interpolation points if I n − 1 is defined by n − 1 X I n − 1 ( u ) = α i ζ i i =1 so that I n − 1 ( u )( x j ) = u ( x j ) then µ n = argmax µ k u ( µ ) − I n − 1 ( u ( µ )) k and x n = argmax x | u ( x ; µ n ) − I n − 1 ( u ( µ n ))( x ) |

  23. The algorithm tells you what points to choose in order to interpolate with functions in

  24. <latexit sha1_base64="LTgKANiVO+0AaLHQuihcNr28kPY=">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</latexit> GEIM recursive (greedy) definition of the functions and the interpolation points if J n − 1 is defined by n − 1 X J n − 1 ( u ) = α i ζ i i =1 so that σ j [ J n − 1 ( u )] = σ j [ u ] then µ n = argmax µ k u ( µ ) − J n − 1 ( u ( µ )) k and σ n = argmax σ | σ [ u ( µ n ) − J n − 1 ( u ( µ n ))] |

  25. Formula suggests that Λ n plays an important role in the result and it is therefore impor- tant to discuss its behavior as n increases. First of all, Λ n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that Λ n = 1 /β n , where h ϕ, σ i X , X 0 β n = inf sup k ϕ k X k σ k X 0 . ϕ ∈ X n σ ∈ Span { σ 0 ,...,σ n − 1 }

  26. Formula suggests that Λ n plays an important role in the result and it is therefore impor- tant to discuss its behavior as n increases. First of all, Λ n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that Λ n = 1 /β n , where h ϕ, σ i X , X 0 β n = inf sup k ϕ k X k σ k X 0 . ϕ ∈ X n σ ∈ Span { σ 0 ,...,σ n − 1 } GEIM interpreted as an oblic projection …

  27. Formula suggests that Λ n plays an important role in the result and it is therefore impor- tant to discuss its behavior as n increases. First of all, Λ n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that Λ n = 1 /β n , where h ϕ, σ i X , X 0 β n = inf sup k ϕ k X k σ k X 0 . ϕ ∈ X n σ ∈ Span { σ 0 ,...,σ n − 1 } the greedy approach seeks in some sense to minimise

  28. Formula suggests that Λ n plays an important role in the result and it is therefore impor- tant to discuss its behavior as n increases. First of all, Λ n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that Λ n = 1 /β n , where h ϕ, σ i X , X 0 β n = inf sup k ϕ k X k σ k X 0 . ϕ ∈ X n σ ∈ Span { σ 0 ,...,σ n − 1 } optimal placement of the sensors

  29. Formula suggests that Λ n plays an important role in the result and it is therefore impor- tant to discuss its behavior as n increases. First of all, Λ n depends both on the choices of the interpolating functions and interpolation points. We have proven (YM-Mula-Patera-Yano) that Λ n = 1 /β n , where h ϕ, σ i X , X 0 β n = inf sup k ϕ k X k σ k X 0 . ϕ ∈ X n σ ∈ Span { σ 0 ,...,σ n − 1 } the greedy approach seeks in some sense to minimise optimal placement of the sensors

  31. 35 Lebesgue constant for EIM — polynomial degree 12 40

  32. Which allows to use the frame “weak greedy” of the papers by • P. Binev, A. Cohen, W. Dahmen, R.A. DeVore, G. Petrova, and P. Woj- taszczyk, • and R. A. DeVore, G. Petrova, and P. Wojtaszczyk, to analyse the convergence properties of our algorithm In a nutshell, in the case where we have a Hilbert framework, our result states that Theorem (with O. Mula and G. Turinici) If (Λ n ) ∞ n =1 is a monotonically increasing sequence then i) if d n ≤ C 0 n − α for any n ≥ 1, then τ n ≤ C 0 ˜ β n n − α , with ˜ β n := 2 3 α +1 Λ 2 if n ≥ 2 . n , ii) if d n ≤ C 0 e − c 1 n α for n ≥ 1 and C 0 ≥ 1, then τ n ≤ C 0 ˜ β n e − c 2 n − α , with √ ˜ β n := 2Λ n , if n ≥ 2 .

  33. Which allows to use the frame “weak greedy” of the papers by • P. Binev, A. Cohen, W. Dahmen, R.A. DeVore, G. Petrova, and P. Woj- taszczyk, • and R. A. DeVore, G. Petrova, and P. Wojtaszczyk, to analyse the convergence properties of our algorithm In a nutshell, in the case where we have a Hilbert framework, our result states that Theorem (with O. Mula and G. Turinici) If (Λ n ) ∞ n =1 is a monotonically increasing sequence then i) if d n ≤ C 0 n − α for any n ≥ 1, then τ n ≤ C 0 ˜ β n n − α , with ˜ β n := 2 3 α +1 Λ 2 if n ≥ 2 . n , Kolmogorov n-width actual deviation ii) if d n ≤ C 0 e − c 1 n α for n ≥ 1 and C 0 ≥ 1, then τ n ≤ C 0 ˜ β n e − c 2 n − α , with √ ˜ β n := 2Λ n , if n ≥ 2 .

  36. An application

  37. Electronic Schrödinger equation

  38. Electronic Schrödinger equation It is well recognized that one of the major difficulty in quantum chemistry is the correlation arising from the mutual repulsion of electrons. The singularity in V ee at r i = r j leads to slow convergence with increase of basis set

  39. Electronic Schrödinger equation It is well recognized that one of the major difficulty in quantum chemistry is the correlation arising from the mutual repulsion of electrons. The singularity in V ee at r i = r j leads to slow convergence with increase of basis set The proposed idea is to change the interaction

  40. Avoid the singularity V ee joint work with E. Polack, J. Karwowski and A. Savin. erf( xr ) r 1 /r x = 2 x = 1 / 2

  41. N erf( x | r i − r j | ) X Avoid the singularity V ee V ee ( x ) = | r i − r j | i,j =1 H ( x ) = T + V ne + V ee ( x ) The idea is thus to approximate this simpler system for finite values of and derive the energy E( ) or other quantities like excited states. Then the idea is to extrapolate at infinity H ( x )Ψ( x ) = E ( x )Ψ( x )

  42. N erf( x | r i − r j | ) X Avoid the singularity V ee V ee ( x ) = | r i − r j | i,j =1 H ( x ) = T + V ne + V ee ( x ) The idea is thus to approximate this simpler system for finite values of and derive the energy E( ) or other quantities like excited states. Then the idea is to extrapolate at infinity H ( x )Ψ( x ) = E ( x )Ψ( x ) Interpolation and extrapolation is a classical problem in approximation

  43. Interpolation and extrapolation is a classical problem in approximation

  44. Interpolation and extrapolation is a classical problem in approximation Classical ! but what is the model? what is the interpolant system?

  45. Interpolation and extrapolation is a classical problem in approximation Classical ! but what is the model? what is the interpolant system? Due to the behavior of E ( x ) for large x , namely proportional to x − 2 , and the linear behavior with x when it approaches zero, we choose as basis (1 + ax 2 ) − 1 , with a ∈ [1 , 50].

  46. Interpolation and extrapolation is a classical problem in approximation Classical ! but what is the model? what is the interpolant system? Due to the behavior of E ( x ) for large x , namely proportional to x − 2 , and the linear behavior with x when it approaches zero, we choose as basis (1 + ax 2 ) − 1 , with a ∈ [1 , 50]. Classical ? but what are the interpolation nodes?

  47. Interpolation and extrapolation is a classical problem in approximation Classical ! but what is the model? what is the interpolant system? Due to the behavior of E ( x ) for large x , namely proportional to x − 2 , and the linear behavior with x when it approaches zero, we choose as basis (1 + ax 2 ) − 1 , with a ∈ [1 , 50]. Classical ? but what are the interpolation nodes? The interpolation/extrapolation nodes are chosen by a greedy procedure between 0 and a maximum value x 0 . This one is chosen so that the computation of E ( x ) is “easy” for 0 < x ≤ x 0 .

  48. Results : General behavior on the hydrogen molecule Errors (in hartree) made by using extrapolation method, to approximate the total electronic energy of the hydrogen molecule using an increasingly in size basis set and associated set of interpolation points, as a function of the largest interpolation point used, µ 0 . The yellow background covers the region where the error is smaller than chemical accuracy (1 kcal/mol).

  49. Results : General behavior on the hydrogen molecule Errors (in hartree) made by using extrapolation method, to approximate the total electronic energy of the hydrogen molecule using an increasingly in size basis set and associated set of interpolation points, as a function of the largest interpolation point used, µ 0 . The yellow background covers the region where the error is smaller than chemical accuracy (1 kcal/mol).

  50. Results : other examples Errors (in hartree) made by using extrapolation method, to approximate the total electronic energy of the hydrogen molecule using an increasingly in size basis set and associated set of interpolation points, as a function of the largest interpolation point used, µ 0 . The yellow background covers the region where the error is smaller than chemical accuracy (1 kcal/mol).

  51. Results : empirical error bars

  52. Remember the mollifier for different values of 3 1 /r µ = 1 2 erf( µr ) /r µ = 2 erf( xr ) µ = 1 / 2 r 1 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 r

  53. Remember the mollifier for different values of 3 1 /r µ = 1 2 erf( µr ) /r µ = 2 erf( xr ) µ = 1 / 2 r 1 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 r For x = 1 or x = 2 the solutions are more easy to compute.. requires a smaller basis set

  54. What if the data are polluted with noise

  55. We want now to use the fact that In the previous approaches, we have mainly used the fact that X N has good approximation properties

  56. This is the part of X 1 of interest

  57. This is the part of X 2 of interest

  58. And this is actually where we should be looking at X 1 ∩ X 2

  59. How can we do this ?

  61. Remember the recursive formula

  62. Remember the recursive formula h i u ( x M ,µ ) −I M − 1 [ u ( .,µ )]( x M ) I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ [ u ( ., µ M ) −I M − 1 [ u ( ., µ M )] u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M )

  63. Remember the recursive formula h i u ( x M ,µ ) −I M − 1 [ u ( .,µ )]( x M ) I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ [ u ( ., µ M ) −I M − 1 [ u ( ., µ M )] u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) That we better rewrite as

  64. Remember the recursive formula h i u ( x M ,µ ) −I M − 1 [ u ( .,µ )]( x M ) I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ [ u ( ., µ M ) −I M − 1 [ u ( ., µ M )] u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) That we better rewrite as h i [ u ( .,µ M ) −I M − 1 [ u ( .,µ M )] I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ u ( x M , µ ) −I M − 1 [ u ( ., µ )]( x M ) u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M )

  65. Remember the recursive formula h i u ( x M ,µ ) −I M − 1 [ u ( .,µ )]( x M ) I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ [ u ( ., µ M ) −I M − 1 [ u ( ., µ M )] u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) That we better rewrite as h i [ u ( .,µ M ) −I M − 1 [ u ( .,µ M )] I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ u ( x M , µ ) −I M − 1 [ u ( ., µ )]( x M ) u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) and let us introduce

  66. Remember the recursive formula h i u ( x M ,µ ) −I M − 1 [ u ( .,µ )]( x M ) I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ [ u ( ., µ M ) −I M − 1 [ u ( ., µ M )] u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) That we better rewrite as h i [ u ( .,µ M ) −I M − 1 [ u ( .,µ M )] I M [ u ( ., µ )] = I M − 1 [ u ( ., µ )]+ u ( x M , µ ) −I M − 1 [ u ( ., µ )]( x M ) u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M ) and let us introduce [ u ( .,µ M ) −I M − 1 [ u ( .,µ M )] q M = u ( x M ,µ M ) −I M − 1 [ u ( .,µ M )]( x M )

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