Analysis of Wave Propagation in 1D Inhomogeneous Media

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1 Analysis of Wave Propagation in D Inhomogeneous Media Patrick Guidotti, Knut Solna, and James V. Lambers Author Query AQ Au: Please provide keywords. AQ Au: Primary and Secondary classifications are required to drive our online search engine and to help readers locate your article more easily. So please provide the appropriate primary and secondary classifications. AQ Au: Please provide received and accepted dates. AQ Au: Remark inserted here, is it correct? AQ Au: Please update. AQ Au: Please provide complete information. AQ Au: Is it book or journal? Please clarify. AQ Au: These references [,, ] are not cited in text. Please cite inside the text or delete from reference lists.

2 Numerical Functional Analysis and Optimization, ():, Copyright Taylor & Francis Group, LLC ISSN: - print/- online DOI:./ ANALYSIS OF WAVE PROPAGATION IN D INHOMOGENEOUS MEDIA Patrick Guidotti and Knut Solna Department of Mathematics, University of California at Irvine, Irvine, California, USA James V. Lambers Department of Petroleum Engineering, Stanford University, Stanford, California, USA In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the highand low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation. Keywords. AMS Subject Classification. AQ Received ; Accepted. Address correspondence to James V. Lambers, Department of Petroleum Engineering, Stanford University, Stanford, CA -, USA; lambers@standford.edu AQ AQ

3 Guidotti et al.. INTRODUCTION In this paper, we consider the one-dimensional inhomogeneous wave equation { tt u c (x) xx u = in (, ), (.) u + x u = on,, with = and c L (, ) and strictly positive. Our results remain valid for any but, for the sake of brevity, we shall present them for the case = only. After some remarks on the structure of the solution operator and on the implications for its numerical approximability, we turn to the main focus of the paper: Spectral asymptotics and numerical resolution of (.). As for the asymptotic spectral properties of the generator, the general result of [, Theorem..] would readily imply that k ( / ) dx k, k large c(x) for the eigenvalues of the generator = c (x) xx. In this particular case, however, led by the physical meaning of the coefficient c, we are able to obtain information about the asymptotic behavior of the nodal points of high-frequency eigenfunctions and, from that, infer about their asymptotic shape. The analysis is based on a shooting method for the computation of the eigenvalue/eigenfunction pairs and the self-similar nature of the problem in combination with the use of a canonical transformation. In particular, we observe that a sort of self-homogenization occurs at high frequencies (cf. Section.). It turns out that the same ideas can be profitably employed to obtain homogenization results for rapidly varying coefficents. These are similar to the result derived in, for instance, [] for the self-adjoint case using a variational approach. Here, however, our focus is an asymptotic approximation for the spectrum, and we present the approach in Section.. Then, in Section, we integrate some of the ideas developed into a numerical approach to high order/large time step resolution of (.). This approach employs Krylov subspace spectral methods, first introduced in []. These methods are Galerkin methods in which each component of the solution in the chosen basis of trial functions is computed using an approximation of the propagator belonging to a low-dimensional Krylov subspace of the operator. Each approximation is based on the use of Gaussian quadrature to evaluate Riemann Stieltjes integrals over the spectral domain as described in []. Because the Krylov subspace approximation of is constructed using Gaussian rules that are tailored

4 Wave Propagation in D Inhomogeneous Media to each component, all components can be resolved more accurately than with traditional spectral methods. Based on the encouraging results for the one-dimensional case, we intend to pursue the possibility of adapting the techniques used in this paper to perform similar analysis in the higher dimensional case.. ANALYTIC STRUCTURE OF THE PROPAGATOR In this section, we derive a spectral representation formula for the solution of the inhomogeneous wave equation in a bounded onedimensional interval as given in (.). In order to do so, we need to analyze the spectral properties of non self-adjoint boundary value problem (, B) given by = c (x) xx, (.) B = j, j =,. (.) where j denotes the trace operator at j =,. This is done in Section.. The analytic structure of the solution makes the relation between the conservation and reversibility properties of the equation apparent (Section.). In particular, they can be concisely formulated in terms of a functional relation satisfied by the propagator (evolution operator)... Properties of the Generator We start by collecting some information about the spectral properties of the generator, that is, of the boundary value problem (.) (.). We therefore study c (x) xx u = u, (.) u(j) =, j =,. (.) Lemma.. All eigenvalues of (.) (.) are strictly positive real and simple. The eigenfunction corresponding to the first (smallest) eigenvalue can be chosen to be positive. Proof. Assume that is an eigenvalue of (.) (.) and u an associated eigenfunction, then u(x) c (x) dx = x u(x) dx which implies the positivity of the eigenvalue. Moreover, an eigenvalue of (.) (.) is given when the boundary conditions are linearly dependent, and therefore the kernel has always at most dimension one, which gives

5 Guidotti et al. simplicity of the eigenvalues. Finally, because the operator has empty kernel and compact resolvent, the spectrum is a pure point spectrum, which concludes the proof. Borrowing from the self-adjoint terminology, we call the first eigenvalue the principal eigenvalue. Next, we show that it is a strictly monotone function of the size of the domain. Lemma.. Let x (, ) and (x ) be the principal eigenvalue for the Dirichlet problem for c (x) xx on [, x ]. Then Proof. (x )>(x ), < x < x. Normalizing eigenfunctions by the requirement x () = we can look for them by considering the initial value problem xx u = u, x [, ], c (x) u() =, x u() =. (.) For =, no nontrivial solution can be found, but, by increasing its magnitude, the value of the solution at x = can be reduced until it becomes zero for the first time. This gives and for [, ]. It is therefore also obvious that needs to be further increased to obtain a zero at x <, which, in its turn, determines and for the interval [, x ]. It turns out that we can determine all other eigenvalues and order them according to their size or equivalently according to the number of their zeros. Lemma.. For every n there is exactly one simple eigenvalue n > for the Dirichlet problem for c (x) xx on [, ] such that the associated eigenfunction n has exactly n + zeros (counting the boundary points). Proof. By using exactly the same arguments as in the proof of Lemma., one can obtain all eigenfunctions as solutions the initial value problem (.) by gradually increasing in order to produce, one by one, new zeros in the interval [, ]. They therefore can be numbered by using their zeros.

6 Wave Propagation in D Inhomogeneous Media Next we introduce a functional setting that allows us to obtain a spectral representation of the operator. Let L (, ) be the Lebesgue space of square integrable functions. Denote by A the L (, )-realization of with domain of definition given by dom(a) = H (, ) H (, ), the space of H functions that vanish on the boundary. Because A has compact resolvent, it allows for a spectral calculus. Lemma.. The operator A can be represented by A = n n, n, (.) n= where ( n ) n and ( n ) n are the eigenfunctions of A and A to the eigenvalue n, respectively. Here A is given by the L -realization of xx (c (x) ) with Dirichlet boundary conditions. Proof. Because all eigenvalues are simple and = is not one of them, the operator A does not contain any nontrivial Jordan blocks nor does it contains a quasi-nilpotent operator. It follows that the operator A allows for the claimed spectral representation. See [] for more details. One needs only to observe that the spectral projection E n is given by n,, where n can be defined through n span k : n = k and n, n = and can be easily verified to be an eigenfunction of A to the eigenvalue n. Remark.. In general, the functions ( n ) n are not an orthogonal system. They are, however, asymptotically orthogonal for smooth c and almost orthogonal for small perturbations of a constant c as we shall see in the next sections. It should be observed that the operator A becomes self-adjoint with respect to the weighted scalar product (u v) = u(x)v(x)/c (x)dx. This provides a different point of view but produces the same spectral resolution of A. See also Remark.... Structure of the Solution The spectral representation of the generator A allows us to obtain a representation of the solution operator (propagator) in terms of the AQ

7 Guidotti et al. sine and cosine families generated by A by a simple functional calculus. Introduce R (t) = A / sin(t A) := R (t) = cos(t A) := n= sin(t n ) n n, n, (.) cos(t n ) n, n, (.) n= where taking the square root of the operator poses no problem even though the operator is not self-adjoint. Then the propagator of (.) can be written as [ ] R (t) R (t) P (t) =. (.) AR (t) R (t) Remark.. The fact that the wave equation is reversible can be seen through the identities R (t) + AR (t) = id L (,), R (t)r (t) = R (t)r (t), t (.) which imply P (t)p ( t) = P ( t)p (t) = [ idl (,) id L (,) ] (.) We observe that our ultimate goal is an efficient numerical scheme for the solution of (.). We are in particular interested in nondissipative and nondispersive schemes. The functional relations (.) make the constraints apparent that such a scheme should satisfy. Next we introduce an appropriate energy norm A and show that it is conserved along solutions of (.). This is done by means of the basis development in terms of the eigenfunctions ( n ) n. Let u L (, ), then we can write u = n }{{, u n. } u n n= Then, taking (u, v) H (, ) L (, ), we define (u, v) A = ( n un + n) v whenever the right-hand side is finite. n= (.)

8 Wave Propagation in D Inhomogeneous Media Remark.. It is not a priori clear that (.) does define a norm that is equivalent to the standard norm of H (, ) L (, ). This follows from the fact that, in the described setting, n = n/c (x) and the fact that the speed of propagation is bounded above and below. The relation between the eigenfunctions is a manifestation of Remark.. This also means that ( n ) n is a frame and (n ) n its dual frame. For a definition and characterization of frames, we refer []. Lemma.. For any solution of (.) with =, one has (u(t), u(t)) A = (u(), u()) A, t. (.) Proof. Denote the initial value (u(), u()) by (u, u ). Then, by developing in the basis of eigenfunctions, we can write the solution as ( [ (u(t), u(t)) = cos ( t ) n u n + sin ( t ] ) n u n n, n n= n= A simple computation then shows that n= [ n sin ( t ) n u n + cos ( t ) ) n u n ] n. (.) [ { n cos ( t ) n u n + sin ( t ] ) n u n n [ + n sin ( t ) n u n + cos ( t ) ] } n u n =.. High-Frequency Spectral Asymptotics n (u n ) + ( u n ). In this section, we show that a sort of self-homogenization occurs at high frequency, which makes the asymptotic behavior of the eigenvalues and eigenfunctions of A very simple. We begin with the following lemma concerning small perturbations of the constant coefficient case. Lemma.. Assume that c C ([, ]) is almost constant, that is, that c. Then the spectrum of A is a small perturbation of that of the operator A given by dom A = H (, ) H (, ), Au = c xx u, u dom(a) (.) for c = ( dx). c(x) n= AQ

9 Guidotti et al. Proof. Introducing the change of variables given by y = (x) = c x d (.) c() which leaves the interval invariant; the operator A in the new variables takes on the form c (x) xx = c yy cc ( (y)) y. The result then follows from the continuous dependence of the operator on its coefficient functions. Remark.. It should be observed that the coefficient c can be thought of as the speed of propagation through the medium in the interval [, ]. Then the integral dx can be interpreted as the time it takes to go c(x) from one end to other of the medium. Thus the averaged coefficient actually measures the effective size of the interval. It turns out that this kind of averaging is always taking place regardless of the size and shape of the coefficient c, at least at high frequencies. The next Proposition makes this precise and also gives an approximation for the high-frequency eigenfunctions. Proposition.. A is given by For large n, the asymptotic behavior of the eigenvlaues of n ( (n) d). (.) c() Moreover, the eigenfunctions n have the following asymptotic shape ( n (x) sin x x ) j, x [x j, x j ] (.) x j x j where = x < x < < x n = have to be chosen such that xj x j c() d = n d, j =,, n. (.) c() Proof. We know from Lemmata.. that the nth eigenfunction n has n + zeros in [, ]. Denote them by = x < x < < x n =.

10 Wave Propagation in D Inhomogeneous Media If n is the associated eigenvalue, then it is also the principal eigenvalue j n of the problems { c (x) xx u = u, in [x j, x j ] u(x j ) = u(x j ) =. for j =,, n. So, in particular one has j n = n, j =,, n. By blowing up the intervals [x j, x j ] to the fixed interval [, ] by means of we obtain x = x j + y(x j x j ), y [, ] c (y) (x j x j ) yyũ = ũ, in [, ] u() = u() =. where now c(y) = c(x j + y(x j x j )) is a slowly varying coefficient provided n is large. Lemma. therefore gives ( j n (x j x j ) x j x j and subsequently that xj ) x j c() d = xj x j c() d = n c() d ( xj x j c() d). because we know already that n = =n n = n. We conclude that the subintervals are uniquely determined. Lemma. also entails that the eigenfunctions on the subintervals all have approximately the form ( j n (x) = sin x x ) j, x [x j, x j ]. x j x j Remark.. It is interesting to observe that to first order the asymptotic behavior of the eigenvalues only contains average information concerning the coefficient, whereas the asymptotic behavior of eigenfunctions reflects local averages taken at the scale determined by the number of its zeros.

11 Guidotti et al... Low Frequency Spectral Asymptotics In this section, we consider the asymptotic behavior of the low-frequency part of the spectrum. We describe it in the regime where the length x of the medium is large. That is, we consider the problem { c (x) xx u = u, x [, x ], (.) u() =, u(x ) = in the limit x. Observe that we allow for large () fluctuations in the local speed c. Ifweset = /x and make the change of variables y = x, this problem becomes ( ) y c yy u = u = u, u () =, u () =. y [, ], (.) This is a homogenization scaling. The self-adjoint case when is periodic is discussed in [], for instance, and the case when is random and varies on a microscale is discussed in []. Wave propagation in the quasistatic limit correpsonding to a scaling of the above type is discussed in, for instance, [, ] where the group velocity in this limit is derived from the homogenized equations. Here we consider the leading part of the spectrum of the non-self-adjoint problem with rapidly varying coefficients. It can be characterized by the following proposition. Proposition.. Let x and ( n (x ), n (x; x ) be the nth pair of eigenvalue and function of the Dirichlet problem (.). For f C assume that y ( ) s y c f (s)ds = c f (s)ds( + ( p )) (.) with Then c as x. = lim c (s/)ds, < c < c(x) < c <, p >. n (x ) (n) c /x (.) n (x; x ) sin(nx/x ) (.) x

12 Wave Propagation in D Inhomogeneous Media Proof. As in the proof of Proposition., we use a shooting argument to solve the eigenvalue problem. It involves normalizing the eigenfunction by requiring y () = and writing (.) as: ( ) y yy = c, y [, ], (.) () =, y () =. Again, for = no nontrivial solution can be found. By increasing, the value of can be reduced until it becomes zero for the first time. This value gives the first eigenvalue and the corresponding leading eigenfunction. In order to describe these for x large we introduce v = (v, v ) T = (,,y )T and obtain the initial value problem [ ] v y = c v, v() = (, ) T. Then, we construct an approximating sequence v n by letting and [ ] v y = c v, [ ] v n y = c v n, n, (.) with v n () = (, ). The increments v n = v n v (n ) solve the same equation (.) and we find v n y (y) ( + ( c/c) ) v (n ) (s)ds. Observe next that v [ ] (y) = y (c c )v (s)ds < p c y, and v n (y) (c y) n p, n, n!

13 Guidotti et al. where here and below c j are constants independent of. Thus, v n form a Cauchy sequence and sup v (y) p e c. y (,) We next establish that v is close to the principal eigenfunction associated with the constant speed c. This follows because explicitly v (y) = sin( /c y), /c moreover, v () p exp(c ) and y = is the first zero of, which is a positive function, thus > such that : (c ) p c sup sin(y) (y) p c y (,) Finally, upon a normalization and a change of argument, we arrive at (.) for n =. Next, we consider the case with general n that follows by induction. Let n be the nth eigenfunction associated with (.), which can be constructed as above via a shooting procedure where is successively increased. The eigenvalues are again identified with those values for that give a new zero in the interval [, ], because the additional zero only can enter at y = because yy = only for =. Now assume that (.) hold for the first n eigenfunctions. Then, > so that for y n () > /, it follows that c (n) > such that n+ n c (n). By an argument as above, c (n) such that sup n+ (y) sin( n+/c y) y (,) n+ /c p e c (n) and we find c (n) > such that sin ( n+/c y) has exactly n + zeros in [, + p c (n)]. Now, by proceeding as above, (.) follows for n + and therefore for general n. Remark.. The condition (.) is satisfied with p = if for instance c (x) = c ( + (x))

14 Wave Propagation in D Inhomogeneous Media with (x) being a bounded function. More generally for we find y with c ( s c (x) = c ( + (x)) ) ( y ( f (s)ds = c f (s)ds + Y Y (x) = x ( ) y y f (y) Y (s)ds. Thus, if Y (x) = (x p ) with p >, then (.) is satisfied.. KRYLOV SUBSPACE SPECTRAL METHODS ( ) )) s f (s)ds In this section, we apply Krylov subspace spectral methods developed in [] to the problem (.) with = and the initial conditions u(x,) = f (x), u t (x,) = g (x), x (, ). (.).. Symmetrization We first apply two transformations to the differential operator = c (x) xx defined in (.). As in previous discussion, we focus on the operator A that is the L (, )-realization of defined on dom(a) = H (, ) H (, ). First, we apply the change of variables (.) to obtain where, we recall, and A = c xx cc ( (x)) x, (.) (x) = c ( ) c = c() d (.) x Next, we define the transformation V by d, x (, ). (.) c() Vf (x) = (x)f (x), (.)

15 Guidotti et al. where which yields [ c x ] (x) = exp c ( ())d, (.) A s = V AVf = V c [Vf ] xx c(c )[Vf ] x = c [f + f + f ] c(c )[f + f ] [ ( ) ] [( ) ( = c f + c c(c ) f + c + [(( ) c = c f c + + c(c ) ) ] c(c ) f ) c + ( c c ) ] f (.) It is easy to see that these transformations have the property that they symmetrize the operator, and they also respect the boundary conditions; that is, if f dom(a) = H (, ) H (, ), then V (f ) dom(a), and conversely... Krylov Subspace Spectral Methods for IBVP Once we have preconditioned the operator, to obtain a new selfadjoint operator, we can use Krylov subspace methods developed in [] to compute an approximate solution. These methods are Galerkin methods that use an approximation for each coefficient of the solution in the chosen basis that is, in some sense, optimal.... Reduction to Quadratic Forms Using a standard Galerkin approach, we begin with an orthonormal set of N trial functions (x) = sin(x), < N, (.) that satisfy the boundary conditions. We seek an approximate solution ũ(x, t) = N ũ (t) (x), (.) = that lies in the space spanned by the trial functions, where each coefficient ũ, =,, N, is an approximation of the quantity u (t) =, u(, t). (.)

16 Wave Propagation in D Inhomogeneous Media Because the exact solution u(x, t) is given by u(x, t) = R (t)f (x) + R (t)g (x), (.) where R (t) and R (t) are defined in (.), (.), we can obtain ũ by approximating each of the quadratic forms where is a nonzero constant, because u (t) = c + (t) c (t) c + (t) = + f, R (t)[ + f ] (.) c (t) = f, R (t)[ f ] (.) s + (t) = + g, R (t)[ + g ] (.) s (t) = g, R (t)[ g ], (.) + s+ (t) s (t). (.) Similarly, we can obtain the coefficients ṽ of an approximation of u t (x, t) by approximating the quadratic forms c + (t) = + f, AR (t)[ + f ] (.) c (t) = f, AR (t)[ f ] (.) s + (t) = + g, R (t)[ + g ] (.) s (t) = g, R (t)[ g ]. (.) As noted in [], this approximation to u t (x, t) does not introduce any error due to differentiation of our approximation of u(x, t) with respect to t the latter approximation can be differentiated analytically. It follows from the preceding discussion that we can compute an approximate solution ũ(x, t) at a given time T using the following algorithm. Algorithm. (Krylov Subspace Spectral Method for IBVP). Given functions c(x), f (x), and g (x) defined on the interval (, ), a final time T, and an orthonormal set of functions,, N that satisfy the boundary conditions, the following algorithm computes a function ũ(x, t) of the form (.) that approximately solves the problem (.), (.) from t = to t = T. t = Choose a nonzero constant while t < T do Select a time step t f (x) =ũ(x, t) g (x) =ũ t (x, t)

17 Guidotti et al. for = to N do Compute the quantities c + (t), c (t), s+ (t), s (t), c + (t), c (t), s + (t), and s (t) ũ (t) = (c + (t) c (t)) + (s+ (t) s (t)) ṽ (t) = (c + (t) c (t) ) + (s+ (t) s (t) ) end ũ(x, t + t) = N = (x)ũ (t) ũ t (x, t + t) = N = (x)ṽ (t) t = t + t end... Computation of the Quadratic Forms We now discuss the approximation of quantities of the form I [f ]= v N, f (A)v N (.) where A is the L (, )-realization of a self-adjoint differential operator defined on dom(a) = H (, ) H (, ), f is a given analytic function, and v N (x) is a function of the form v N (x) = N u (x). (.) Given this representation of v N, we can approximate this quantity by = I N [f ]=v T N f (A N )v N (.) where v N =[v N (x ) v N (x N )] T, and A N is an N N symmetric matrix that approximates the operator A on the space spanned by,, N. For example, we may choose [A N ] ij = N k (x i ) k, A l l (x j ), (.) k,l= or use a finite-difference approximation that takes the boundary conditions into account. In particular, if we use a three-point stencil, then A N is a tridiagonal matrix. We can compute this quadratic form using techniques described in []. Let A N have eigenvalues a = N = b, (.)

18 Wave Propagation in D Inhomogeneous Media with corresponding eigenvectors q,, q N. Then I N [f ]=v T N f (A N )v N (.) N = f ( j ) q T j v N (.) = j= b where () is the piecewise constant measure <a n q T j v N i < i+ () = j=i n q T j v N b j= a f ()d() (.) (.) We can approximate the value of this Riemann Stieltjes integral using Gaussian quadrature. Applying the symmetric Lanczos algorithm to A N with initial vector v N, we can construct a sequence of polynomials p,, p K that are orthogonal with respect to the measure (). These polynomials satisfy a three-term recurrence relation j+ p j+ () = ( j+ )p j () j p j (), (.) p (), p () =, v N (.) which can be written in matrix-vector notation as p K () = J K p K () + K p K ()e K (.) where p () p K () =., J K = (.) p K () K K It follows that the eigenvalues of J K are the zeros of p K (), which are the nodes for Gaussian quadrature. It can be shown (see []) that the K

19 Guidotti et al. corresponding weights are equal to the squares of the first components of the normalized eigenvectors of J K.... Accuracy of the Approximate Solution We now state and prove a result concerning the accuracy of each component of the approximate solution. We first use the following result from []. Lemma.. LetAbeanN N symmetric positive definite matrix. Let u and v be fixed vectors, and define u = u + v. For j a positive integer, let g j () be defined by g j () = et T j e u, (.) where T is the K K Jacobi matrix produced by the K iterations of the symmetric Lanczos algorithm applied to A with starting vector u. Then, for some satisfying <<, g j () g j ( ) j K = u T A j v + k=k [ j K + k=k e T e T [ T k X T X T A k ] re T K T j k e u T u [ ] r T k X T X T Ak e T K T j k e u T ] u = (.) Proof. See []. Corollary.. for j < K. Under the assumptions of the lemma, g j () g j ( ) = u T A j v, (.) We can now describe the local truncation error of each component of the computed solution. Theorem.. Assume that c(x),f(x), and g (x) belong to span,, N, and let u(x, t) be the exact solution of (.), (.) at (x, t), and let ũ(x, t) be the approximate solution computed by Algorithm.. Then, u(, t) ũ(, t) = O(t K ) (.) where K is the number of quadrature nodes used in Algorithm..

20 Wave Propagation in D Inhomogeneous Media Proof. Let g () be the function from Lemma. with A = A NK, where N K = K N, u = and v = f, where f = f (x ) f (x NK )] T. Furthermore, denote the entries of T by () () () () () T = (.) K () K () K () K () K () Finally, let () = u and K () = r, and let c = [c + (t) c (t)] =, R (t)f. (.) Then, by Lemma. and Corollary.,, R (t)f c { t j = ( ) j, A j f g } j() g j ( ) (j)! = = = j= { t j j K ( ) j, A j f c T (j)! Aj N K f + j= t K (K )! et d d K t d (K )! et d k=k e T re T K T j k e } + O(t K ) d [ ] T k d X T X T Ak = N K [ ] T K X T X T AK = N K re T K T K e + O(t K ) [ K T j e K r T j= AK j N K ] = re T K T K e + O(t K ) K t d [ ] = (K )! et T K e K r T re T K d T K e + O(t K ) = = t K d [ ] r e T (K )! d T K e K + O(t = K ) = t K d ( () K () ) (K )! d + O(t K ) = = O(t K ). (.) A similar result holds for s =, R (t)g.

21 Guidotti et al. Note that the proof assumes that Algorithm. uses a discretization of A on an N K -point grid, where N K = K N. This grid refinement is used to avoid loss of information that would be incurred on an N -point grid when multiplying gridfunctions. In practice, this refinement is seen to be unnecessary when the coefficients are reasonably smooth. When it is needed to ensure sufficient accuracy, its effect on the efficiency of Algorithm. is minimized by the fact that K is typically chosen to be small (say, K = ork = ). Implementation details discussed in [] also mitigate this concern.... Non-Orthogonal Basis Functions Ideally, we would like our trial functions to be approximate eigenfunctions of the symmetrized operator obtained previously. Although the eigenfunctions of this operator are orthogonal with respect to the inner product,, we cannot assume that any basis of approximate eigenfunctions is necessarily orthogonal as well. Suppose that we refine our initial (orthogonal) basis of approximate eigenfunctions of the form (.) to obtain a new basis (x) N = so that each function (x) is a sparse combination of functions of the form (.); that is, = C (.) where the matrices and are defined by ij = j (x i ), ij = j (x i ), x i = ix, < i < N, (.) and the matrix C is sparse. Then, if we define the vector u(t) to be the values of our approximate solution ũ(x, t) at time t and the gridpoints x i, i =,, N, then we can efficiently obtain u(t + t) by computing where the vector ũ(t + t) is defined by u(t + t) = C(C T C) ũ(t + t) (.) [ũ(t + t)] =, R (t)ũ(, t) + R (t)ũ t (, t), <<N. (.) Generalizing, if we obtain the matrix representing the values of approximate eigenfunctions by a sequence of transformations C,, C k where each C j, j =,, k, has O() bandwidth, and the integer k is small, then we can still compute the solution in O(N ) time per time step... Numerical Experiments To test our algorithm, we solve the problem (.), (.) from t = to t =.

22 Wave Propagation in D Inhomogeneous Media... Construction of Test Cases In many of the following experiments, it is necessary to construct functions of a given smoothness. To that end, we rely on the following result (see []): Theorem.. Let f (x) be a -periodic function and assume that its pth derivative is a piecewise C function. Then, ˆf () constant/( p+ +). (.) Based on this result, the construction of a C p+ function f (x) proceeds as follows:. For each =,, N /, choose the discrete Fourier coefficient ˆf () by setting ˆf () = (u + iv)/ p+ +, where u and v are random numbers uniformly distributed on the interval (, ).. For each =,, N /, set ˆf ( ) = ˆf ().. Set ˆf () equal to any real number.. Set f (x) = ˆf <N / ()e ix. In the following test cases, coefficients and initial data are constructed so that their third derivatives are piecewise C, unless otherwise noted. We will now introduce some functions that will be used in the experiments described in this section. As these functions and operators are randomly generated, we will denote by R, R, the sequence of random numbers obtained using MATLAB s random number generator rand after setting the generator to its initial state. These numbers are uniformly distributed on the interval (, ). We will make frequent use of a two-parameter family of functions defined on the interval [, ]. First, we define { f j,k (x) = Re ˆf j ()( + ) k+ e }, ix j, k =,,, (.) where <N /, = ˆf j () = R jn +(+N /) + ir jn +(+N /). (.) The parameter j indicates how many functions have been generated in this fashion since setting MATLAB s random number generator to its initial state, and the parameter k indicates how smooth the function is.

23 Guidotti et al. In many cases, it is necessary to ensure that a function is positive or negative, so we define the translation operators E + and E by E + f (x) = f (x) min f (x) +, x [,] (.) E f (x) = f (x) max f (x). x [,] (.) It is also necessary to ensure that a periodic function vanishes on the boundary, so we define the translation operator E by E f (x) = f (x) f (). (.)... Discretization and Error Estimation The problem is solved using the following methods: A finite difference scheme presented by Kreiss et al. (see []). The Krylov subspace spectral method with K = Gaussian quadrature nodes and the basis (.). The Krylov subspace spectral method with K = Gaussian quadrature nodes and a basis obtained by applying two iterations of inverse iteration to each function in the basis (.). In all cases, the operator = c(x) xx is preconditioned using the transformations described in Section to obtain a self-adjoint operator. Then, the L (, )-realization of defined on H (, ) H (, ) is discretized using a matrix of the form (.) that operates on the space of gridfunctions defined on a grid consisting of N equally spaced points x j = jx, where x = /(N + ), for various values of N. The approximate solution is then computed using time steps t k = k, k =,,, so that we can analyze the temporal convergence behavior. Let u (k) (x, t), k =,,, be the approximate solution computed using time step t k. For k =,,, the relative error E k in u (k) (x, t) at t = is estimated as follows: We use the same method to solve the backward problem for (.) with end conditions u(x,) = u (k) (x, t), u t (x,) = u (k) t (x, t), x (, ). (.) Let v (k) (x, t) be the approximation solution of the inverse problem, for k =,,. Then we approximate the relative difference between v (k) (x,) and u (k ) (x,) = f (x) in the L -norm; that is, E k u(k) (,) v (k) (,) u (k) (,), (.)

24 where Wave Propagation in D Inhomogeneous Media [u (k) ] j = u (k) (x j,), [v (k) ] j = v (k) (x j,), x j = jx. (.)... Results We first solve the problem (.), (.) with smooth data c(x) = f, (x), f (x) = f, (x), g (x) = f, (x). (.) The functions c(x), f (x), and g (x) are plotted in Figs., respectively. The temporal convergence is illustrated in Fig., where N = gridpoints are used in all cases. The finite difference method of Kreiss et al. converges quadratically, whereas approximately th-order convergence is attained using the Krylov subspace spectral method. Note that the use of inverse iteration does not improve the convergence rate, but it does yield a more accurate approximation for larger time steps. In Fig., all three methods are used to solve (.), (.) with time steps t k = k and mesh sizes x k = (k+), for k =,,. The finitedifference method converges quadratically, while the Krylov subspace spectral method without inverse iteration exhibits quintic convergence. Using inverse iteration, the convergence is only superquadratic, but this is FIGURE Smooth wave speed c(x) = f, (x). F-F F F

25 Guidotti et al. FIGURE Smooth initial data u(x,) = f (x) = f, (x). FIGURE Smooth initial data u t (x,) = g (x) = f, (x).

26 Wave Propagation in D Inhomogeneous Media FIGURE Estimates of relative error in approximate solutions of the problem (.), (.) with data (.) computed using finite differencing and Krylov subspace spectral methods, with time steps t k = k, k =,,. FIGURE Estimates of relative error in approximate solutions of the problem (.), (.) with data (.) computed using finite differencing and Krylov subspace spectral methods, with time steps t k = k and mesh sizes x k = (k+), for k =,,.

27 Guidotti et al. due to the fact that the accuracy is so high at t = that the machine precision prevents attaining a faster convergence rate for smaller time steps. Both experiments are repeated with data that is not as smooth. Specifically, we use c(x) = f, (x), f (x) = f, (x), g (x) = f, (x). (.) The functions c(x), f (x), and g (x) are plotted in Figs., respectively. The results corresponding to Figs. and are illustrated in Figs. and, respectively. As expected, the accuracy and the convergence rate are impaired to some extent. This can be alleviated by refining the spatial grid during the Lanczos iteration, as described in [], in order to obtain more accurate inner products of functions; we do not do this here... Gaussian Quadrature in the Spectral Domain Consider the computation of the quadratic form, f (Ã) where (x) is defined in (.) and à is defined in (.). Figures and illustrate the relationship between the eigenvalues of à and the Gaussian quadrature nodes obtained by the symmetric Lanczos algorithm that is employed by Krylov subspace spectral methods. In Fig., the speed c(x) is defined to be c, (x), which is shown in Fig.. Because the speed is FIGURE Non-smooth wave speed c(x) = f, (x). F-F F-F F

28 Wave Propagation in D Inhomogeneous Media FIGURE Nonsmooth initial data u(x,) = f (x) = f, (x). FIGURE Nonsmooth initial data u t (x,) = g (x) = f, (x).

29 Guidotti et al. FIGURE Estimates of relative error in approximate solutions of the problem (.), (.) with data (.) computed using finite differencing and Krylov subspace spectral methods, with time steps t k = k, k =,,. FIGURE Estimates of relative error in approximate solutions of the problem (.), (.) with data (.) computed using finite differencing and Krylov subspace spectral methods, with time steps t k = k and mesh sizes x k = (k+), for k =,,.

30 Wave Propagation in D Inhomogeneous Media FIGURE Approximate eigenvalues of the operator = c, (x) xx, and Gaussian quadratures nodes of a -point rule used to approximate, f (Ã) where à is defined in (.), plotted against the wave number. For each eigenvalue, the wave number is determined by the dominant frequency of the corresponding approximate eigenfunction. smooth, (x) is an approximate eigenfunction of Ã, and it follows that the nodes are clustered around the corresponding approximate eigenvalue. In Fig., the speed is c(x) = + cos(x). Because of this oscillatory F perturbation, the eigenvalues do not define a smooth curve, as seen in the top plot. Note that the sharp oscillations in the curve traced by the eigenvalues correspond to sharp changes in the placement of the two quadrature nodes.. CONCLUSIONS We have considered wave propagation in one dimension in the case of heterogeneous and complicated coefficients. Our point of view has been to consider the analytical structure of the solution operator in order to derive asymptotic properties for the spectrum. In particular, we considered nonself-adjoint problems with small fluctuations in the coefficients and problems where the fluctuations are large and rapid, respectively. We then derived techniques for efficient and accurate numerical wave propagation that are based on using low-dimensional Krylov subspace approximations of the solution operator to obtain components of the solution in a basis of trial functions in a Galerkin-type scheme. We demonstrated that this

31 Guidotti et al. FIGURE Approximate eigenvalues of the operator = ( + cos(x) xx, and Gaussian quadratures nodes of a -point rule used to approximate, f (Ã) where à is defined in (.), plotted against the wave number. For each eigenvalue, the wave number is determined by the dominant frequency of the corresponding approximate eigenfunction. approach gives a high-order approximation that converges faster than competing methods in the problems that we have considered. Developing theory and numerical procedures for wave propagation in rough and multiscale media is important in a number of applications, such as analysis and design of algorithms for solving inverse problems related to propagation in the ocean, the atmosphere, or in the heterogeneous earth, for instance. Such applications require us, however, to consider propagation in several spatial dimensions, and our main aim is to generalize and integrate further our approach to deal with multiple spatial dimensions. ACKNOWLEDGMENT P.G. was partially supported by NSF CMS. R.S. was supported by NSF OMS and OMS. REFERENCES. A. Bensoussan, J. L. Lions, and G. Papanicolaou (). Asymptotic Analysis of Periodic Structures. North Holland, Amsterdam.. P. G. Casazza (). The art of frame theory. Taiwanese J. Math. ():.. N. Dunford and J. T. Schwarz (). Linear Operators, Part III. Wiley, New York.

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