nan Rakesh

Inversion of spherical means and the wave equation in even dimensions

We establish inversion formulas of the so-called filtered back-projection type to recover a function supported in the ball in even dimensions from its spherical means over spheres centered on the boundary of the ball. We also find several formulas to recover initial data of the form

The spherical mean value operator with centers on a sphere

(f,0)

The range of the spherical mean value operator for functions supported in a ball

(or

Reconstruction for an inverse problem for the wave equation with constant velocity

(0,g)

Spherical means with centers on a hyperplane in even dimensions

) for the free space wave equation in even dimensions from the trace of the solution on the boundary of the ball, provided that the initial data has support in the ball.

Uniqueness for the inverse backscattering problem for angularly controlled potentials

Let B represent the ball of radius ρ in Rn and S its boundary; consider the map , where represents the mean value of f on a sphere of radius r centered at p. We summarize and discuss the results concerning the injectivity of , the characterization of the range of , and the inversion of . There is a close connection between mean values over spheres and solutions of initial value problems for the wave equation. We also summarize the results for the corresponding wave equation problem.

An inverse problem for a layered medium with a point source

Suppose n > 1 is an odd integer, f is a smooth function supported in a ball B with boundary S, and u is the solution of the initial value problem u tt - Δ x u = 0, (x, t) ∈ R n x [0, ∞); u(x,t = 0) = 0, u t (x, t = 0) = f(x), x ∈ R n . We characterize the range of the map f → u| S×[0,∞) and give a stable scheme for the inversion of this map. This also characterizes the range of the map sending f to its mean values over spheres centred on S.

A one-dimensional inverse problem for a hyperbolic system with complex coefficients

Let Omega contained in/implied by Rn, n>1 be a bounded domain with smooth boundary. Consider utt- Delta xu+q(x)u=0 in Omega *(0,T), u(x,0)=0,ut(x,0)=0 if x in Omega and u(x,t)=f(x,t) on delta Omega *(0,T). Define the Dirichlet to Neumann map Lambda q:H1( delta Omega *(0,T)) to L2( delta Omega *(0,T)), f(x,t) to delta u/ delta n where n is the unit outward normal to delta Omega . If T>diam( Omega ), the author expresses q(x) in terms of Lambda q, and shows that if q(x) is piecewise constant then q(x) is uniquely determined by Lambda q(fi), i=1 . . . N for some integer N.

An inverse problem for the wave equation in the half plane

Given a real-valued function on R-n we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with great simplicity and economy. We apply their method to derive an inversion formula for the even n case. A feature of our inversion formula, for the even n case, is that it does not require the Fourier transform of the mean values or the use of the Hilbert transform, unlike the previously known inversion formulas for the even n case. Along the way, we extend the isometry identity of Bukhgeim and Kardakov for odd n, for solutions of the wave equation, to the even n case.

Stability for an inverse problem for a two-speed hyperbolic PDE in one space dimension

We consider the problem of recovering a smooth, compactly supported potential on from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled angular derivatives, then the two potentials are identical. In particular, if two potentials differ by a finite linear combination of spherical harmonics with radial coefficients and have the same backscattering data then the two potentials are identical.