# High Dimensional Model Representation Approximation of an Evolution Operator with a First Order Partial Differential Operator Argument.

код для вставкиСкачатьAppl. Num. Anal. Comp. Math. 1, No. 1, 280 ? 289 (2004) / DOI 10.1002/anac.200310025 High Dimensional Model Representation Approximation of an Evolution Operator with a First Order Partial Differential Operator Argument I?rem Yaman?1 and Metin Demiralp??1 1 Computational Science and Engineering Program, Informatics Institute, I?stanbul Technical University, Maslak, 34469, I?stanbul, Turkey Received 15 November 2003, revised 30 November 2003, accepted 2 December 2003 Published online 15 March 2004 Key words Lie algebraic exponential operator, High Dimensional Model Representation Subject classi?cation 17B65 (In?nite-dimensional Lie algebras), 62H99 (Multivariate analysis), 14Q15 (Higher-dimensional variates), In this work, a novel approximation scheme based on a recently developed representation, called the High Dimensional Model Representation (HDMR), is proposed to approximate evolution operators. The approximation is not developed at the operator level, instead the effect of the operator on an arbitrary function is approximated. The purpose of the work is to investigate how HDMR can be applied for the approximation of evolution operator and its ef?ciency. Although the approximation can be constructed for any order of multivariance, only the constant and univariate components of HDMR are considered in this work. c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Almost all types of evolution phenomena encountered in natural and applied sciences can be characterized by so?called evolution operators. These operators depend on some variables de?ning the state of the system under consideration and a tracing parameter which characterizes how the system?s state evolves. The state variables generally correspond to the positions of some reference points and are suf?cient to describe the system precisely, hence they may be considered as spatial variables. The internal structure of the evolution operator is, of course, determinatory of the behavior of the system?s evolution and for the sense of mathematical modelling it may result in mathematical complexity in such a way that the resulting equations may not be handled easily. For this reason, it is better to con?ne ourselves to simpler cases, one of which is the case where the operator is exponential and its argument is of Lie type with ?rst order partial differential operator of state variables. The coef?cients depend on the same variables through second degree multinomials. Therefore we consider the following evolution operator Q ? etL , L? m ?i (x1 , и и и , xN ) i=1 ? ?xi (1) where, L is the Lie Operator we have mentioned above. It is composed of partial derivatives with respect to independent variables xi and coef?cient functions ?i of these variables. Although t is basically a mathematically general tracing parameter of evolution, as we mentioned above, it can be thought of as the time variable. If g is assumed to be an in?nitely many times differentiable function of independent variables x1 , и и и , xN then we can consider a spatially and temporally varying function [ 1 ? 4 ] which satis?es the equation f (x1 , и и и , xN , t) = Qg(x1 , и и и , xN ) ? ?? (2) Corresponding author: e-mail: [email protected], Phone: +90 212 285 70 77, Fax: +90 212 285 70 73 e-mail: [email protected], Phone: +90 212 285 70 82, Fax: +90 212 285 70 73 c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 281 Temporal differentiation of both sides of this equation yields the following partial differential equation ? ? ? ? ?1 (x1 , и и и , xN ) ? и и и ? ?N (x1 , и и и , xN ) f (x1 , и и и , xN , t) = 0 ?t ?x1 ?xN (3) The initial condition which accompanies the equation (3) is considered to be f (x1 , и и и , xN , 0) = g(x1 , и и и , xN ) (4) Now we are ready to consider the application of HDMR to approximate the solution of the above equation and its accompanying initial condition. 2 High Dimensional Model Representation Approach The High Dimensional Model Representation (HDMR) of a given function f (x1 , ..., xN , t) is constructed as the following multivariate ordering expansion [ 5 ]. f (x1 , и и и , xN , t) = f0 (t) + N i=1 N fi (xi , t) + fi1 i2 (xi1 , xi2 , t) + и и и + f1иииN (x1 , и и и , xN , t) (5) i1 ,i2 =1 i1 <i2 Here, t is considered just a parameter not an HDMR variable. The last term of the above expansion is composed of just a single function which depends on all independent variables. Since we want to obtain an approximation to the original function rather than ?nding an exact solution, it will suf?ce to truncate this expansion at the ?rst few terms. The functions at the right hand side of the equation (5) can be considered as the orthogonal components of the original function. Orthogonality comes from the vanishing conditions which are imposed below. They are de?ned via an inner product over the space of square integrable functions and therefore the right hand side components of the above equation are also assumed to be square integrable functions [ 6 ? 12 ]. bis ais dxis Wis (xis )fi1 i2 ...iN (xi1 , ..., xiN ) = 0, 1?s?N (6) These vanishing conditions imply that the HDMR components are orthogonal to each other through the inner product which is de?ned as the integral of the inner product arguments under the weight formed as the product of the above univariate weight functions. For brevity, the univariate weight functions are chosen to be normalized as below. bis dxis Wis (xis ) = 1 (7) ais Substituting the form given by (5) into equation (3) the following equation is obtained. N N df0 (t) ? ? + fi (xi , t) ? ?i (x1 , и и и , xN ) fi (xi , t)+ dt ?t i=1 ?xi i=1 N N N ? ? + fi1 i2 (xi1 , xi2 , t) ? ?i (x1 , и и и , xN ) fi1 i2 (xi1 , xi2 , t) + и и и = 0 ?t i ,i ?xi i ,i i=1 1 2 i1 <i2 (8) 1 2 i1 <i2 2.1 Zeroth Order HDMR Approximation Multiplying both sides of equation (8) with the product of the univariate weight functions which are chosen as 1 Wi (xi ) = a?b for all independent HDMR variables and integrating over the interval [a, b] and using vanishing properties of the components; the following formula is obtained. c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 282 I?. Yaman and M. Demiralp: High Dimensional Model Representation... ? ? b b N N df0 (t) ? ? ? иии dxj Wj (xj )? ?i (x1 , и и и , xN ) fi (xi , t) dt ?xi a a j=1 i=1 ? ? b b N N N ? ? ? иии dxj Wj (xj )? ?i (x1 , и и и , xN ) fi1 i2 (xi1 , xi2 , t) = 0 ?xi i ,i a a j=1 i=1 (9) 1 2 i1 <i2 a and b are assumed to be positive constants. In this work it is assumed that the ?i functions have the following second degree multinomial form. (i) ?i (x1 , и и и , xN ) = c + N (i) bj xj + N ?1 N j=1 k=j+1 (i) 2ajk xj xk + N j=1 (i) ajj xj 2 (10) j=1 Insertion of this equality into the equation (9) enables us to write N (i) N (i) (i) i?1 df0 (t) ? ? (t) + 2a x + 2a x a ?i (t)+ i j j ii ji ij i=1 j=1 j=i+1 dt N N ?1 (i) (i) N j=1 a ?i (t) + R(t) = 0 k=j+1 2a j=1 xj jj xj + jk xk j=i where b ?i (t) = j=i dxi Wi (xi )xi 2 a ?i (t) = ?i (t) = xi = b a b dxi Wi (xi )xi dxi Wi (xi ) a b k=i ? fi (xi , t) ?xi ? fi (xi , t) ?xi ? fi (xi , t) ?xi (11) (12) (13) (14) dxi Wi (xi )xi (15) dxi Wi (xi )x2i (16) a and xi = a b R function, which is de?ned in equation (11), represents the contribution of bivariate HDMR components. The initial condition which accompanies equation (11) is obtained by applying HDMR to the initial condition of (4). f0 (0) = g0 (17) where g0 is the constant HDMR component of g(x1 , и и и , xN ). We can approximately solve HDMR equations by omitting high variate terms. If we de?ne zeroth order approximation as the case where only constant contributions are retained then the partial differential equation, (0) involving the function f0 becomes (0) df0 (t) =0 dt (18) Using equality (17), the solution of equation (18) is obtained as (0) f0 (t) = g0 (19) We have evaluated the Zeroth Order HDMR Approximation of f function. Our next step is to obtain the First Order HDMR Approximation. c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 283 2.2 First Order HDMR Approximation To introduce univariate HDMR components, both sides of equation (8) are integrated (N-1) times excluding the integration over the variable xk (where 1 ? k ? N ). This gives the following equations for univariate terms after 0 (t) neglecting bivariate and higher terms and replacing dfdt with its counterpart in (11). ? ? (1) (k) (k) (k) (1) (k) (k) (k) fk (xk , t)? A0 + A1 xk + A2 x2k f (xk , t) = B0 (t)+B1 (t)xk +B2 (t)x2k (20) ?t ?xk k where ? (k) A0 = N (k) ajj xj + j=1 j=k (k) A1 = k?1 (k) ? 2ajl xl ? (21) l=j+1 l=k N (k) 2ajk xj + ? N ? xj ? j=1 j=k j=1 (k) N ?1 (k) 2akj xj (22) j=k+1 (k) A2 = akk (23) (k) (k) (k) (k) B0 (t) = ?A2 ?k (t) ? A1 ?k (t) ? A0 ?k (t) (k) (k) ?xk B1 (t) ? xk B2 (t) (k) B1 (t) = (24) k?1 N (i) (i) 2aik ?i (t) ? i=k+1 2aki ?i (t) k?1 k?1 (i) N (i) + i=1 2akj xj ?i (t) j=1 2ajk xj + j=k+1 j=i N N (i) (i) k?1 ?i (t) 2a x + x + i=k+1 j=k+1 2a j j j=1 jk kj i=1 j=i (25) and (k) B2 (t) = N (i) akk ?i (t) (26) i=1 i=k The initial condition, which accompanies equation (20) can be found by using equation (4) and HDMR components of f and g functions. f0 (0) + fk (xk , 0) = g0 + gk (xk ) (27) Using equation (17), the following equality can be written. (1) fk (xk , 0) = gk (xk ) (28) To obtain the solution of equation (20), a structure for the solution can be assumed. For this purpose we de?ne a lower dimensional Lie Operator, given as below. ? (k) (k) (k) Lk ? A0 + A1 xk + A2 x2k , ?xk xk ? (??, ?) (29) The solution can be written in the form. (1) fk (xk , t) ? etLk ?k (xk , t) (1) Lk fk (xk , t) = Lk etLk ?k (xk , t) = etLk Lk ?k (xk , t) (30) (31) c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 284 I?. Yaman and M. Demiralp: High Dimensional Model Representation... Using this property of Lie Operators, a partial differential equation can be obtained. ??k (xk , t) (k) (k) (k) = B0 (t) + B1 (t)etLk xk + B2 (t)etLk x2k , ?t ?k (xk , 0) = gk (xk ) (32) An Evolution Operator with a ?rst order partial derivative argument satis?es the following formula, which can be shown by using the Leibnitz rule for the derivative of a product. etLk (F1 F2 ) = etLk F1 etLk F2 (33) Using this property the effect of the Evolution Operator on xk squared function can be written as 2 e?tLk xk 2 ? e?tLk xk (34) We de?ne a function represented by Hk , which is produced by applying the Evolution Operator on xk . Hk (xk , t) ? e?tLk xk (35) Differentiation of both sides of equation (35) with respect to t gives the following differential equation is obtained. ?Hk (xk , t) (k) (k) (k) = ?A0 ? A1 Hk (xk , t) ? A2 Hk (xk , t)2 , ?t Hk (xk , 0) = xk (36) If the structure of Hk function is used in equation (32), the solution of this equation can be obtained as below. t t t (k) (k) (k) d? B0 (? ) + d? B1 (? )Hk (xk , ? ) + d? B2 (? )Hk (xk , ? )2 (37) ?k (xk , t) = gk (xk ) + 0 0 0 Using equation (30), the ?rst order univariate HDMR components are evaluated. t t (1) (k) (k) fk (xk , t) = etLk gk (xk ) + 0 d? B0 (? ) + 0 d? B1 (? )Hk (xk , ? ? t) t (k) + d? B2 (? )Hk (xk , ? ? t)2 0 (38) Applying Evolution Operator on the univariate HDMR components of inital function g, the new form of equation (38) can be written as t t (1) (k) (k) fk (xk , t) = gk (Hk (xk , ?t)) + 0 d? B0 (? ) + 0 d? B1 (? )Hk (xk , ? ? t) t (k) + d? B2 (? )Hk (xk , ? ? t)2 (39) 0 where, ?k (t) = + + ?k (t) = + + ?p (t) = + + b dxk Wk (xk ) ?x? k gk (Hk (xk , ?t)) a b t (k) d? B1 (? ) a dxk Wk (xk ) ?x? k Hk (xk , ? ? t) 0 b t (k) d? B2 (? ) a dxk Wk (xk ) ?x? k Hk (xp , ? ? t)2 0 (40) b dxk Wk (xk )xk ?x? k gk (Hk (xk , ?t)) a b (k) d? B1 (? ) a dxk Wk (xk )xk ?x? k Hk (xk , ? ? t) 0 b t (k) d? B2 (? ) a dxk Wk (xk )xk ?x? k Hk (xk , ? ? t)2 0 t (41) b dxk Wk (xk )x2k ?x? k gk (Hk (xk , ?t)) a b (k) d? B1 (? ) a dxk Wk (xk )x2k ?x? k Hk (xk , ? ? t) 0 b t (k) d? B2 (? ) a dxk Wk (xk )x2k ?x? k Hk (xk , ? ? t)2 0 t (42) c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 285 Now, we must evaluate function Hk . Equation (36) is Ricatti type differential equation and contains constant coef?cients. To solve equation (36) we make the following de?nition. (k) 2 ?k = A1 (k) (k) ? 4A2 A0 (43) 1. Case: If ?k > 0, (k) (k) 2 (k) A2 Hk (xk , t) + A1 Hk (xk , t) + A0 = (Hk (xk , t) ? h1 )(Hk (xk , t) ? h2 ) (44) Using this de?nition, we get the solution of equation (36) as below. Hk (xk , t) = h1 (xk ? h2 ) ? h2 e?t(h1 ?h2 ) (xk ? h1 ) (xk ? h2 ) ? e?t(h1 ?h2 ) (xk ? h1 ) (45) 2. Case: If ?k = 0, (k) (k) 2 (k) A2 Hk (xk , t) + A1 Hk (xk , t) + A0 = (Hk (xk , t) ? h)2 Hk (xk , t) = (xk ? h) +h 1 ? t(xk ? h) (46) (47) 3. Case: If ? < 0, It can be considered that the following constant term is de?ned. (k) A3 (k) = (k) A0 (k) 2 ? A1 (48) (k) 4A2 (k) If A3 and A2 have the same sign: ? ? (k) ?? (k) A A (k) (k) Hk (xk , t) = 3(k) tan ??t (A2 A3 ) + arctan ?xk 2(k) ?? A2 A3 (k) (49) (k) If A3 and A2 have different signs: (k) (k) (k) (k) (k) xk + A3(k) + xk ? A3(k) e2t A3 A4 A A4 A4 Hk (xk , t) = 3(k) (k) (k) (k) (k) A3 A3 A4 x + ? x ? e2t A3 A4 k (k) A4 k (50) (k) A4 where (k) (k) A4 = ?A2 (51) If Hk function is introduced into the equations (40), (41) and (42), ?k , ?k and ?k functions is obtained by solving Convolution type integral equations. Then, these functions are used to evaluate fk functions by using equations (24), (25), (26) and (39). Now, we should obtain the ?rst order constant HDMR component to ?nd the First Order Approximation. To obtain constant term we use equation (11). Since we want to obtain the First Order Approximation, only the contributions of constant and univariate components are retained. (1) N (i) N df0 (t) (i) (i) i?1 = ? (t) + 2a x + 2a x a ?i (t)+ i j j ii ji ij i=1 j=1 j=i+1 dt N N ?1 (i) (i) N j=1 a ?i (t) (52) k=j+1 2a j=1 xj jj xj + jk xk j=i j=i k=i c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 286 I?. Yaman and M. Demiralp: High Dimensional Model Representation... The constant component can be evaluated to yield, t N (i) N (1) (i) (i) i?1 ? (? ) + 2a x + 2a x f0 (t) = 0 d? a ?i (? )+ i j j ii ji ij i=1 j=1 j=i+1 N N ?1 (i) (i) N j=1 a ?i (? ) + g0 k=j+1 2a j=1 xj jj xj + jk xk j=i k=i j=i (53) Finally, the First Order HDMR Approximation of f function can be written as (1) f (x1 , и и и , xN , t) ? f0 (t) + N (1) fi (xi , t) (54) i=1 3 A Simple Illustrative Example Assume that L= N i=1 x2i ? ?xi (55) and f (x1 , и и и , xN , 0) = g(x1 , и и и , xN ) = x1 + и и и + xN (56) We attempt to solve the equation. etL g(x1 , и и и , xN ) = f (x1 , и и и , xN , t) (57) Temporal differentiation of both sides of this equation gives, ? f (x1 , и и и , xN , t) ? Lf (x1 , и и и , xN ) = 0 ?t (58) 3.1 Solving Equation via Seperation of Variables f (x1 , и и и , xN , t) = F1 (x1 , t) + и и и + FN (xN , t) (59) Substituting this form of f function, into equation (58) gives, ?F1 (x1 , t) ?F1 (x1 , t) ?F1 (xN , t) ?FN (xN , t) ? x21 ? x2N + иии + =0 ?t ?x1 ?t ?xN (60) The xi variables are independent variables, therefore we can write the equation. ?Fi (xi , t) ?Fi (xi , t) ? x2i = ki (t), ?t ?xi Fi (xi , 0) = xi (61) ki functions satisfy the equality below. N ki (t) = 0 (62) i=1 We de?ne the following function. Ti (xi , t) = Fi (xi , t) ? 0 t d? ki (? ) (63) If this form of Ti is injected into the differential equation, which is given as (61), the following equation is obtained. ?Ti (xi , t) ?Ti (xi , t) ? x2i = 0, ?t ?xi Ti (xi , 0) = xi (64) c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 287 The solution of this equation is considered to have the form, ? txi 2 ?x Ti (xi , t) = e i ui (xi ) (65) Using the initial condition which is de?ned for Ti and given in equation (64), we get the equation. ? txi 2 ?x Ti (xi , t) = e i xi (66) The following variable transformation is made to solve equation (66). ? txi 2 yi (xi ) ?y Ti (xi , t) = e i xi (67) where, x2i yi (xi ) = 1 (68) The solution of this differential equation is, yi (xi ) = ? 1 +c xi (69) This implies that xi = 1 c ? yi (70) Using equation (68) and equation (70), the differential equation, given in equation (66) becomes, ? t ?y Ti (xi , t) = e i 1 c ? yi (71) The following equality can be written by using the properties of Evolution Operators. ? et ?y q(y) = q(y + t) (72) If we use this equality and equation (70), we can write the solution of equation (71) as, Ti (xi , t) = xi 1 ? txi (73) Substituting this form of Ti into equation (63), the solution of equation (61) is evaluated. Fi (xi , t) = xi + 1 ? txi t 0 d? ki (? ), 1?i?N (74) Injecting this form of Fi functions into equation (59) the solution of the differential equation, which is given as equation (58) is obtained as below. f (x1 , и и и , xN , t) = N i=1 xi + 1 ? txi 0 t d? (k1 (? ) + и и и + kN (? )) (75) Equation (62) enables us to write the following equality. f (x1 , и и и , xN , t) = N i=1 xi 1 ? txi (76) c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 288 I?. Yaman and M. Demiralp: High Dimensional Model Representation... 3.2 Solving Equation via HDMR Using equation (1) and equation (55) we obtain the ?i functions. ?i = xi 2 , 1?i?N (77) If ?i functions have these forms, the following equalities can be written. (k) aij = ?ij ?jk , 1 ? j, k ? N (i) (78) (i) (i) We can use these equalities, in obtaining A0 , A1 and A2 which were de?ned in equations (21), (22) and (23). (i) A0 = 0 (79) (i) A1 = 0 (80) (i) A2 = 1 (i) (i) (81) (i) Similarly B0 , B1 and B2 functions can be obtained by using equations (24), (25) and (26) and above equalities. (i) B0 (t) = ??i (t) (82) (i) B1 (t) = 0 (83) (i) B2 (t) = 0 (84) Now, the HDMR components of initial function g should be obtained. g(x1 , и и и , xN ) = x1 + и и и + xN (85) Multiplying both sides of this equation with the product of the weight functions, integrating over the integral range and using the vanishing properties of components, we get the constant HDMR component of g. g0 = N (a + b) 2 (86) Similarly, the univariate components can be evaluated multiplying both sides of equation (85) with the product of weight functions, integrating over the integral range, excluding the integration over the variable xi , using the vanishing properties of components and equation (86). gi (xi ) = xi ? (a + b) 2 (87) To obtain function Hi , we should ?nd ?i , which has been given in equation (43). Using equation (79), (80) and (81), ?i can be found as ?i = 0 (88) Since ?i is equal to zero, we must use the equality of Hi function as equation (47). Equations (79), (80) and (81) enable us to say, that h = 0 in equation (47). And using this equality, Hi function can be obtained as below. Hi (xi , t) = xi 1 ? txi (89) If we use equation (87) and equation (89) together, we obtain, gi (Hi (xi , ?t)) = xi (a + b) ? 1 ? txi 2 (90) c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Appl. Num. Anal. Comp. Math. 1, No. 1 (2004) / www.interscience.wiley.com 289 Substituting equation (90) into equation (39), the ?rst order univariate HDMR components can be obtained. t xi (a + b) (1) ? fi (xi , t) = ? d? ?i (? ) (91) 1 ? txi 2 0 t N N N xi N (a + b) (1) ? fi (xi , t) = ? d? ?i (? ) (92) 1 ? txi 2 0 i=1 i=1 i=1 Similarly, using equations (53), (78) and (86) the ?rst order constant HDMR component can be obtained as follows. t N N (a + b) (1) (93) d? ?i (? ) + f0 (t) = 2 0 i=1 Finally the First Order HDMR Approximation of f function is evaluated to give (1) f (x1 , и и и , xN , t) ? f0 (t) + N i=1 (1) fi (xi , t) = N i=1 xi 1 ? txi (94) 4 Conclusion As can be seen from equation (94), with a special choice of ?i functions, we obtain the exact solution by using the First Order HDMR Approximation. Hence, this implies the ef?ciency and strong approximating capability of HDMR. In this study we have approximated a multivariate Evolution Operator with univariate Evolution Operators. Although in the present study we have retained up to ?rst order approximations only, the results give us many clues on how the higher order approximations can be evaluated. HDMR results compare perfectly with those obtained via the seperation of variables method. Acknowledgements We are grateful to Doc?. Dr. N. A. Baykara for his careful reading of the manuscript and his invaluable comments. The authors wish to acknowledge the ?nancial support from the State Planning Organization (DPT) of Turkey. The second author, Metin Demiralp, is also grateful to Turkish Academy of Sciences for its partial support. References [1] M. Demiralp and H. Rabitz, Lie Algebraic Factorization of Multivariable Evolution Operators: De?nition and the Solution of the Canonical Problem, International Journal of Engineering Science, 31, 307(1993). [2] M. Demiralp and H. Rabitz, Lie Algebraic Factorization of Multivariable Evolution Operators: Convergence Theorems for the Canonical Case, International Journal of Engineering Science, 31, 333 (1993). [3] M. Demiralp and H. Rabitz, Factorization of Certain Evolution Operators Using Lie Operator Algebra: Convergence Theorems, Journal of Mathematical Chemistry, 6, 193(1991). [4] M. Demiralp and H. Rabitz, Factorization of Certain Evolution Operators Using Lie Algebra: Formulation of the Method, Journal of Mathematical Chemistry, 6, 164(1991). [5] I. M. Sobol, Sensitivity Estimates for Nonlinear Mathematical Models, Mathematical Modelling and Computational Experiments (MMCE), 1(4), 407(1993). [6] H. Rabitz, O?. F. Al?s?, J. Shorter, and K. Shim, Ef?cient Input-Output Model Representations, Computer Physics Communications, 117, 11(1999). [7] J.A. Shorter, P.C. Ip, and H. Rabitz, An Ef?cient Chemical Kinetics Solver Using High Dimensional Model Representation, Journal of Physical Chemistry A, 103, 7192(1999). [8] H. Rabitz and O?.F. Al?s?, Managing the Tyranny of Parameters in Mathematical Modelling of Physical Systems,in Sensitivity Analysis, A. Saltelli, K. Chan, and M. Scott (eds.), 199(2000). [9] O?. F. Al?s? and H. Rabitz, Ef?cient Implementation of High Dimensional Model Representations, Journal of Mathematical Chemistry, 29, 127(2001). [10] G. Li, C. Rosenthal, and H. Rabitz, High Dimensional Model Representations, Journal of Physical Chemistry A, 105, 7765(2001). [11] G. Li, S.-W. Wang, C. Rosenthal, and H. Rabitz, High dimensional Model Representations Generated from Low Dimensional Data Samples. I. mp-Cut-HDMR, Journal of Mathematical Chemistry, 30, 1(2001). [12] H. Rabitz and O?.F. Al?s?, Additive and Multiplicate High Dimensional Representation General Foundations of High Dimensional Model Representations, Journal of Mathematical Chemistry, 25, 197(1999). c 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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