We consider the inverse problem of identifying the location and shape of a finitely supported acoustic source function, separable with respect to space and frequency, from measurements of the acoustic field on a closed surface for many frequencies. A simple uniqueness proof and an error estimate for the unknown source function are presented. From the uniqueness proof an efficient numerical algorithm for the solution is developed. The algorithm is tested using numerically generated data in dimensions 2 and 3.

Boundary element methods (BEM) based near-field acoustic holography (NAH) has been used successfully in order to reconstruct the normal velocity on an arbitrarily shaped structure surface from measurements of the pressure field on a nearby conformal surface. An alternative approach for this reconstruction on a general structure utilizes the equivalent sources method (ESM). In ESM the acoustic field is represented by a set of point sources located over a surface that is close to the structure surface. This approach is attractive mainly for its simplicity of implementation and speed. In this work ESM as an approximation of BEM based NAH is studied and the necessary conditions for the successful application of this approach in NAH is discussed. A cylindrical fuselage surface excited by a point force as an example to validate the results is used.

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the three-dimensional case. We show that any regular solution of this equation admits a unique representation by a single-layer potential, so that the problem is equivalent to the solution of a linear integral equation of the first kind. We study uniqueness of reconstruction and obtain a sharp stability estimate and convergence rates for some regularization algorithms when the domain is a sphere. We have developed a boundary element code to solve the integral equation. We report numerical results with this code applied to three geometries: a sphere, a cylinder with spherical endcaps and a cylinder with a floor modelling the interior of an aircraft cabin. The exact test solution is given by a point source exterior to the surfaces with about 1% random noise added. Regularization methods using the truncated singular value decomposition with generalized cross validation and the conjugate gradient (cg) method with a stopping rule due to Hanke and Raus are compared. An interesting feature of the three-dimensional problem is the relative insensitivity of the optimal regularization parameter (number of iterations) for the cg method to the wavenumber and the multiplicity of the singular values of the integral operator.

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