Schemes. Abstract. This work deals with the consistency of nite dierence approximations. We investigate

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1 A Note on Consistency and Adointness for Numerical Schemes Alain Sei William Symes y Abstract This ork deals ith the consistency of nite dierence approimations. We investigate the relation beteen the consistency of a numerical scheme and the consistency of its adoint. We ehibit eamples of numerical schemes hich are consistent ith a (direct) equation and hose adoint is not consistent ith the adoint equation. This undesirable feature appears in the application of the adoint state technique hich requires an adointness relation to be satised. Therefore, the numerical scheme for the adoint equation is determined by the choice of the numerical scheme on the direct equation. We conclude that in general consistency is not conserved by adointness. Key Words. Finite Dierences, Adoint Schemes, Consistency AMS(MOS) subect classications. 65M6, 65M2 Introduction The equivalence theorem (cf [, 2]) is the fundamental tool to derive convergent nite dierence approimations of linear partial dierential equations. It states that if a scheme is consistent then stability is equivalent to convergence. Consequently a lot of ork has been devoted to stability problems (cf [, 3, 4]). Indeed the problem of deriving a consistent scheme is generally easily treated by using Taylor's formula. Hoever the problem of consistency can become acute if the choice of the scheme is restricted or even imposed. Such is the case hen the adoint state technique is used. This technique (cf [5, 6, 7]) essentially used in optimiation and optimal control gives the gradient of a non linear functional as the solution of a so called adoint equation. This equation is intrinsic to the functional considered and is dened by satisfying an adointness relation. The numerical method used to compute the gradient must also satisfy this adointness condition. That is hy once a numerical scheme has been chosen on the \direct" equation (adoint ofthe adoint), the numerical scheme for the adoint equation is implicitly determined. A natural question to ask is hether the numerical approimation for the adoint equation converges if the numerical approimation for the direct equation does. In other ords, is convergence conserved by adointness? With the equivalence theorem in mind this question can be rephrased as \Are stability and consistency conserved by adointness?". To anser that question negatively e ill ehibit a fe simple eamples here consistency of the numerical scheme on the direct equation does not imply consistency for the adoint scheme of the Department of Applied Mathematics, California Institute of Technology, Pasadena, California 925 (sei@ama.caltech.edu) 92. y Department of Computational and Applied Mathematics, Rice University,P.O Bo 92, Houston, Teas 7725-

2 Alain Sei and William W. Symes adoint equation. The paper is organied as follos. In section 2 e recall briey the principle of the adoint state method. In section 3 e apply this technique to the particular problem of traveltime tomography inversion. In section 4 e give a fe eamples of non consistent approimation for the adoint equation and e present our conclusions in section 5. 2 Adoint state technique The adoint state technique is commonly used in optimiation and optimal control to compute gradients of non linear functionals. This technique consists in computing an auiliary eld (the adoint state) by solving a so called adoint equation. This adoint state enters directly in the computation of the gradient of the functional. For eample let's consider the folloing least squares problem : Find the Minimier m of the functional J(m) F (m) 2 D2 D 7 here F is a non linear operator from a certain Hilbert \model" space M into a Hilbert \data" space D: F : M! D m! F (m) d To nd the minimier m ith local optimiation techniques, e need to compute the gradient of the functional J. The derivative ofjat m in the direction m is given by: (2.) J (m):m ( F (m):m; F (m) D ) D here (:; :) D is the scalar product in D. The derivative F (m) of the functional F at m is a linear operator from M to D. Ife assume that F (m) is a one to one mapping then it admits an inverse L (F (m)) dened as follos: L : D! M d 7! L d m No let's introduce a ne eld solution of the adoint equation: L F (m) D Then equation (2.) can be ritten as follos: J (m):m ( F (m):m; F (m) D ) D ( F (m):m; L ) D ( LF (m):m; ) M ( m; ) M here (:; :) M is the scalar product in M. Therefore since the gradient G of J is dened as the element of Msuch that e haveg. J (m):m (m; G ) M 2

3 Consistency and Adointness for Numerical Schemes 3 Tomography inversion We consider as a practical case the inverse tomography problem in seismology. Given travel times data in a domain e ant to determine the sloness eld m (reciprocal of the velocity) minimiing the mist beteen computed traveltimes and traveltimes data. We ant to minimie the folloing functional: J(m) 2 (; ; m) here (; ; m) solves the eikonal equation (cf [, ]): (3.) < : d (; ) 2 d d m 2 (; ) 2 (; ) on We assume that the non linear operator F associates to each sloness m a unique travel time solution of (3.). The derivative ofjat m in the direction m is : J (m):m F (m):m (; ; m) d d d Let us set F (m):m. The traveltime perturbation is caused by the perturbation m in the sloness eld. Given m, is the solution of the folloing linear equation (cf. Appendi A): (3.2) r :r m (; ) 2 m (; ) on Thus the derivative operator F (m), dened by F (m):m solution of (3.2), is the solution operator of (3.2). Let L be the inverse operator of F (m) (e assume it eists), then L m. Therefore, L is dened by equation (3.2). We dene the adoint state as the solution of the adoint equation (3.3) L (m) d that is : > < (3.4) Then e can rite r: r m J (m):m (m) d (; ) 2 (; ) on (m) d d d Using the adointness relation (3.5) (L ) d d e have Therefore is the L 2 gradient. J (m):m (L) dd m: d d L dd 3

4 Alain Sei and William W. Symes 4 Numerical Methods The crucial point of the method is the adoint relation (3.5), hich for dierential operators reduces to integration by parts. When e compute the adoint state, relation (3.5) must also be satised by the discrete operators. To solve the adoint equation (3.3), e approimate the operator L by a discrete operator L h. The discrete equivalent of the adointness condition (3.5) is satised if this operator is the adoint of an operator L h hich must be an approimation of L. We ill sho that some simple schemes do not have this property and therefore that consistency is not conserved by the operation of taking the adoint. We illustrate this property on the eample introduced in section 2. We ant to compute the solution u of equation (3.2). To simplify the problem, e assume that (; ; m) 6 in the domain, that is, there are no turning rays. So e can divide through by and therefore u is the solution of : < (4.) : ith a(; ) (4.2) u + a(; )u f (; ) 2 u (; ) on and f m. The adoint equation of (4.) for the L 2 scalar product is : < : (a(; )) g (; ) 2 (; ) on These to equations satisfy the folloing adointness relation: (u + au ): d d ( (a) ):u d d We can transform (4.) and (4.2) into initial value problems (the variable being considered as time) by a udicious choice of f and g. With f(; ) a(; ) () equation (4.) transforms, for the function ~u u +, into the initial value problem: (4.3) < : ~u + a(; )~u (; ) 2 ~u (; ) on and ith g(; ) (a(; ) ()) equation (4.2) transforms, for the function ~ +,into the initial value problem: (4.4) < : ~ (a(; )~) (; ) 2 ~ (; ) on Both equation (4.3) and (4.4) are hyperbolic equations. For the particular choice of a(; ) e can solve equation (4.3) and (4.4) analytically. So e can compare the numerical solution ith the eact solution. In the sequel e a(; ). We start ith the simplest rst order upind scheme. 4. Upind Schemes 4.. First Order Scheme We consider the folloing domain ] ; [] ; [, the boundary being the line.we discretie (4.3) ith a rst order approimation in and an upind derivative in as follos (cf 4

5 Consistency and Adointness for Numerical Schemes [] pp 2): > < (4.5) u n+ u n +(a + ) n u n u n +(a ) n u n + u n u here a + ma( ; ), a min( ; ). So (a+ ) n for and (a ) n for, here corresponds to. Equation (4.3) being hyperbolic e epect conditional stability. This scheme is stable if and are chosen such that : ma a (4.6) as is easily seen by a plane ave (or Von Neumann) analysis (cf [3]). The truncation error of this scheme is O( +) for any point of the (; ) grid. The adoint scheme of (4.5) derived in appendi B,is given by: (4.7) n n (a + ) n + n + (a + ) n n (a ) n n (a ) n n n n J n2::n N ::J This scheme is stable under condition (4.6). Let's eamine the truncation in. Wehaveat that point (of inde ), (a + ) n (a ) n. Therefore e can rite: (a + ) n + n + (a + ) n n (a ) n n (a ) n n (a+ ) n + n + (a ) n n 2:(a) (;)+O() The adoint scheme is not consistent ith the adoint equation. We illustrate this phenomenon in the case here ] ; []; 3[ and () 2. The eact solution of equation (4.4) derived in appendi C is given by (; ) ( ) We plot belo the eact and the numerical solution. The inconsistency of the numerical scheme in isobvious. We can notice that convergence is assured everyhere but at the point. This is in agreement ith the equivalence theorem (cf [, 2]) hich implies that if a scheme is stable and is not convergent then it cannot be consistent. 5

6 Alain Sei and William W. Symes Figure 4.: The eact solution (solid line) and the numerical solution (dashed line) of the adoint equation for dierent depth. The depth 3 is the initial data curve. The scheme is not consistent in. The inconsistency of the adoint scheme seems to be a direct consequence of the upind character of the direct scheme of equation (4.5). The adoint scheme approimates the folloing continuous equation: (a + (; )) (a (; )) But since a + and a are dened using the functions Ma and Min hich are not dierentiable in, e have inconsistency. This analysis is supported by using another upind scheme. 6

7 Consistency and Adointness for Numerical Schemes 4..2 Second Order Scheme We choose to use a second order one sided approimation in space. Since the -derivative is not a problem e keep a rst order approimation in. So, e consider the folloing scheme: (4.) here D2 u n+ u n +(a + ) n D2 un +(a ) n D2+ un u (resp D2 + ) is the left (resp. right) second order approimation given by: D2 u n un 2 4:u n +3:un 2 D2 + un un +2 4:u n + +3:u n 2 This scheme is consistent ith equation (3.4) and the truncation error is O( + 2 )atevery point (; ) of the grid. The adoint scheme is given by: (4.9) n n D2 + ((a+ ) n n ) D2 ((a )n n ) n n J n2::n N since the adoint ofd2 is D2 + and the adoint ofd2 + is D2. In ehave again (a + ) n (a ) n, and so: D2 + ((a + ) n n )+D2 ((a ) n n ) (a+ ) n +2 n +2 4:(a + ) n + n + +3:(a + ) n n 2 (a ) n 2 n 2 4:(a ) n n +3:(a + ) n n 2 (a+ ) n +2 n +2 4:(a + ) n + n + 2 (a ) n 2 n 2 4:(a ) n n 2 2(a) (;) 4(a) (;)+O() 2(a) (;)+O() Therefore this scheme is not consistent ith equation (4.4) in. Furthermore it is not consistent at.for eample at, corresponding to +ehave(a ) n (a ) n and by denition of a e have(a ) n (a )n + therefore: D2 + ((a + ) n n )+D2 ((a ) n n ) (a+ ) n +2 n +2 4:(a + ) n + n + +3:(a + ) n n 2 (a ) n 2 n 2 4:(a ) n n +3:(a + ) n n 2 (a+ ) n +2 n +2 (a ) n n 2 4:(a + ) n + n + 2 3:(a + ) n n 4:(a + ) n + 2(a) (; ) n + 3:(a + ) n n + O() 2 as a result, the scheme is not consistent at(and also by symmetry). 7

8 Alain Sei and William W. Symes Figure 4.2: The eact solution (solid line) and the numerical solution (dashed line) of the adoint equation for dierent depth. The depth 3 is the initial data curve. The scheme is not consistent in. The inconsistency of the adoint scheme seems to be caused by the upind character of the numerical method chosen. But as e shall see in the net section, problems even occur ith centered schemes. We illustrate this point on the La-Wendro scheme (cf [] pp ). 4.2 Centered Scheme The La-Wendro scheme is a centered, dissipative approimation of (4.3) second order in and. To get a second order approimation in e simply use a centered nite dierence approimation. To get a second order approimation in e use the modied equation approach (cf [9, 3]). We have u (au ) a u au a u + a(au ) a u + a 2 u + aa u

9 Consistency and Adointness for Numerical Schemes We use this epression to derive a second order accurate scheme in the interior of the domain as follos: (4.) D + un here e have used : 2 D + un un+ (a n ) 2 u n + a n Do an Do un D + an Do un + a n Do un u n un + 2u n + un On the boundary e use a rst order upind scheme D + un J +(a+ ) n u n J u n J J D + un +(a ) n u n 2 u n D o un un + u n 2 This scheme is at any point (; ) second order and second order in. Its adoint is given by taking the adoint of each operator in (4.). Using the folloing relations: (D + ) (D ) D u n un u n ( ) ( ) (D o ) (D o ) the adoint scheme in the interior of the domain is given by: (4.) D n 2 ((a n )2 n ) Do (an Do an n )+Do (D+ an n ) +D o (an n ) On the boundary e use the adoint of the rst order upind scheme : D u n J D u n (a + ) n J un J (a + ) n J un J (a ) n 2 un 2 (a ) n un This scheme, unlike the rst or second order upind scheme, is consistent ith (3.4). But it is not second order as one ould epect. The approimation of the derivative is given by : D n 2 hich should be a second order approimation of ((a n )2 n ) Do (an Do an n )+Do (D+ an n ).ByTaylor's epansion e have: Therefore, D n 2 + O( 2 ) Q ((a n ) 2 n ) D o (a n Do an n )+D o (D + an n ) 9

10 Alain Sei and William W. Symes should be an approimation of. This quantity satises : But using equation (4.4) e have Q (a 2 ) (aa ) +(a ) +O( 2 ) (a) (a) (a + a ) (a ) +(a(a) ) We can see at once that Q is not an approimation of since for instance the term ith a has the rong sign. Therefore since this term is the correction term multiplied by the adoint scheme is consistent ith the adoint equation. Hoever it cannot be second order as the scheme (4.). 5 Conclusions We have shon in this paper that the consistency of a numerical scheme ith a continuous equation does not imply the consistency of the adoint scheme ith the adoint equation. This property should be epected since L h consistent ith L means that the truncation error goes to ero ith the mesh sie. That is if P h is the proection from V ot V h, L 2L(V; V ) and L h 2L(V h ;V h )e have P h L L h P h L(V;V)! hen h! The consistency of the adoint scheme L h means therefore that P h L L h P h L(V;V)! hen h! So ecept hen L and L h are self-adoint (cf [2]) the consistency of L h does not imply the consistency of L h.furthermore hen both the chosen scheme and its adoint are consistent, they do not have necessarily have the same order of accuracy. Therefore the adoint state technique needs to be applied carefully at the discrete level. In particular the numerical method chosen to compute the adoint state should have a consistent adoint. Acknolegement: This ork as partially supported by the National Science Foundation, the Oce of Naval Research, the Air Force Oce of Scientic Research, the Teas Geophysical Parallel Computation Proect, the Schlumberger Foundation, IBM, and The Rice Inversion Proect. TRIP Sponsors for 995 are Advance Geophysical, Amerada Hess, Amoco Production Co., Conoco Inc., Cray Research Inc., Discovery Bay, Eon Production Research Co., Interactive Netork Technologies, Mobil Research and Development Corp. A Derivation of the perturbed equation We consider a sloness perturbation m. The travel time (m+m) associated to the perturbation satises : < r(m + m) 2 (m + m) 2 in : (m + m) on We ant to nd the equation satised by (m):m. Since (r(m + m)) 2 (r((m)+ (m):m + o(m 2 ))) 2 (r(m)) 2 +2:r(m):r (m):m + o(m 2 ))

11 Consistency and Adointness for Numerical Schemes e can rite 2:r(m): (m):m + o(m 2 )) (r(m + m)) 2 (r(m)) 2 2m:m + o(m 2 ) Dropping the term of order greater or equal to to (because e are looking for the rst derivative) and dividing by m e nd that is the solution of : (A.) r :r(; ) m(; ) (; ) 2 m (; ) (; ) 2 B Adoint Upind Scheme We are looking for the adoint equation of equation (4.5) for the discrete L 2 scalar product. We use the notation (:; :) h for that scalar product. Let us note P the adoint operator of P, dened by the discrete equation (4.5). We have (P ; u) h (Pu;) h J N 2 n n Let us treat the rst integral I J N 2 n J 2 J 2 J N 2 n n u n+ u n+ N n N n2 n u n u n n un+ n u n n The second integral can be ritten as I 2 J N 2 n N n N n (a + ) n N n N n u n u n n un n un!! 2 +(a ) n u n + u n! J u n + N u N since u n (a + ) n u n u J J 2 (a + ) n n un (a + ) n + n + un J (a + ) n A n un J 2 (a + ) A n n un

12 Alain Sei and William W. Symes J N 2 n Let us treat the last integral I 3 J N 2 n N n N n J N 2 n (a + ) n + n + (a + ) n n n (a Finally e can rite J N 2 n J 2 n N + n J J 3 )n u n + u n (a ) n n un + (a ) n n un J 2 N u n + (a + ) n J n J un J since (a + ) n 2 n (a + ) n A n un J 2 (a ) n A n un (a ) n n (a ) n n N u n (a ) n n un 2 since (a )n J n n n u n+ (a ) n n un 2 (a + ) n + n + (a + ) n n u n J Choosing such that : < : (a + ) n N u N u n u n N n +(a n + (a )n n (a ) n n ) n u n + (a + ) n J n J un J n n J n2::n N 2::J u n!! u n e nd that the adointscheme of (4.5) is given by (4.7). Using the right handside of equation (4.5) and equation (4.7) e can rite (B.) N J n 2 u n Resn N J n 2 n s n (s) n ( ) n C Eact solution of equations (4.3) and (4.4) In the specic case here a(; ) equations (4.3) and (4.4) are solvable analytically. We use the method of characteristics (cf [3]). We indicate the solution of the adoint equation (4.4), the method for equation (4.3) being similar. First e rerite (4.4) ( ) 2

13 Consistency and Adointness for Numerical Schemes as follos: The characteristics of equation (4.4) are the curves (t);(t);(t) solution of the ODE system: d dt ) (t) t d dt ) (t) (t) d dt ) (t) (t) Since for t ehave() and (),ehave ( ; ) ( ) ( ). Therefore (; ) ( ). References [] P.D La and R.D Richmyer, Survey of the Stability of Linear Finite Dierence Equations. Communications on Pure and Applied Mathematics, Vol I, 956. [2] L.V Kantorovich, Functional Analysis and Applied Mathematics, Armed Services Technical Information Agency, 952. [3] R.D Richmyer and K.W Morton, Dierence Methods for Initial Value Problems. Interscience Publishers, 967. [4] R. Courant, K. Friedrichs and H. Ley, On the Partial Dierence Equations of Mathematical Physics. Mathematische Annalen Vol, 32-74, 92. English Translation in IBM Journal March 967. [5] J.L. Lions, Contr^ole optimal de systemes gouvernes par des equations au derivees partielles. Dunod-Gauthier Villard, Paris, 96 (English Translation : Springer-Verlag, Ne-York,92) [6] G. Chavent, Identication of functional parameters in partial dierential equations, Identi- cation of parameter distributed systems: R.E. Goodson, and Polis, Ed., Ne York, ASME, 974. [7] A. Bamberger, G. Chavent, Ch. Hemon and P. Lailly, Inversion of normal incidence seismograms, Geophysics, Vol 47, pp , 92. [] R.J. Leveque, Numerical Methods for conservation las Birkhauser Verlag, Basel, 992. [9] P.D La and B. Wendroff, Dierence Schemes for Hyperbolic Equations ith High Order of Accuracy. Communications on Pure and Applied Mathematics, Vol VII, pp 3-39, 964. [] M. Born and E. Wolf, Principles of Optics; electromagnetic theory of propagation, interference and diraction of light, Macmillan Co, Ne York, 964. [] R.K. Luneburg, Mathematical Theory of Optics, 964, University of California Press. [2] F. Ries and B. S.-Nagy, Functional Analysis, Ungar Publishing Co, Ne York,

14 Alain Sei and William W. Symes [3] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol 2, Interscience Publishers, Ne York,

CHAPTER 3 THE COMMON FACTOR MODEL IN THE POPULATION. From Exploratory Factor Analysis Ledyard R Tucker and Robert C. MacCallum

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