The solution of some direct and inverse problems for acoustic equation by statistical simulation method
by
Irina I. Belinskaya
Institute of Computational Mathematics and Mathematical Geophysics, Novosibirsk,Russia

The main method to solve the inverse problem for the acoustics equation and oscillation equation ( definition of the equation's coefficients)

c-2\frac\partial2U\partialt2 = \DeltaU - Ñlnq(x) ÑU,        x in R2×R+, t in R+,

which was used in the paper, is the combination of proectional method, dynamical variant of Gelfand-Levitan method and Monte Carlo method. The problem is to determine p(x) out of F(t) from the equation

\frac\partial\partialt2 u = \frac\partial\partialx2 u - p'(x)p-1(x)ux, x, t in R+

(1)

if we know that

uk | t=0 \equiv 0, ux | x=0 = \delta(t)p(0)

(2)

u | x=0 = F(t)

(3)

where F(t) - media response, p(x), u_x, u(x, t), F(t)-square matrices of the dimention n. Let p(x) be smooth sufficiently with x > 0, that is p_ij(x) in C²(R₊). Notice that system (1) is N-approximation of acoustic equation, mentioned above.

Note that the solution of the inverse problem for acoustic equation on the base of the reducing it to the integral equation in the form

W(x, t) + ó õ x -x  W(x, s)F'(t-s)ds = -\frac12 [F'(t+x)+F'(t-x)],     |t| < x

(4)

has the peculiarity: actually we need to find not the whole solution but it's local value in fixed point t=x. This task can be solved by Monte Carlo methods comparativelly more effective than the other methods considered. In this connection there is unique dependence between W(x, t) and matrix-coefficient p(x) of the system (1)-(3). The problem is ill-posed in classical sense. If we reduce the problem (1)-(3) to the equivalent integral equation (4), we obtain the correctness of setting and stability of solution.

The matrix estimate of Monte Carlo method for W(x, x) is constructed on the base of relations :

W(x, x) = M\xix,     \xix = H+ N å n=1   Qn K(x, xn)

Q0 = E,     Qn+1 = Qn \fracK(xn, xn+1)p(xn, xn+1),

E-unit matrix dimension n×n.

We solve the inverse problem in two steps : a) solving the direct problem : the obtaining of additional information u | _x=0 = F(t) for solving the inverse problem. b) solving the inverse problem. On the step a) we also can use Monte Carlo method, solving the Cauchy problem for the acoustic equation. Here we reduce the initial Cauchy problem to the Volterra eqution of the second kind:

U(x, t) = U0(x, t) - \frac14\pi ó õ t 0  \rho é ë ó õ ó õ \omega   q(x+\alpha\rho) U(x+\alpha\rho, t-\rho) d\omega ù û d\rho

Then we construct the local Monte Carlo estimate

I\delta = (U, \delta(x-x0)) = (U0, U*) = U(x0)

with finite variance. In the paper were constructed and approbated the Monte Carlo algorithms for finding the estimation of the Cauchy problem's solution for telegraph equation in the space of Poison's pathes. Also was constructed the solution of the inverse problem for telegraph equation and geoelectric problem in quasistationar approach, using Monte Carlo method.

Date received: April 17, 1999