arXiv subjects

Sebastian Acosta

Publications and source records attributed to Sebastian Acosta.

At least 19 recordsLinked to original sources

arXiv2025

Inverse photoacoustic tomography problem in media with fractional attenuation

We investigate the inverse problem of recovering an initial source for the wave equation with fractional attenuation, motivated by photoacoustic tomography (PAT). The attenuation is modeled by a Caputo fractional derivative of order $\alpha\in(0,1)$. We establish uniqueness under a geometric foliation condition via an adaptation of two types of Carleman estimates to the fractional setting, prove stability through continuity inequalities for fractional time-derivatives of wave solutions, and derive a reconstruction scheme based on a Neumann series. While our results apply directly to PAT, we expect that the analytic approach and tools employed might be of broader relevance to the analysis of PDEs with memory effects and to inverse problems for attenuated wave models.

math.AP↗

arXiv2025

Simultaneous determination of wave speed, diffusivity and nonlinearity in the Westervelt equation using complex time-periodic solutions

We consider an inverse problem governed by the Westervelt equation with linear diffusivity and quadratic-type nonlinearity. The objective of this problem is to recover all the coefficients of this nonlinear partial differential equation. We show that, by constructing complex-valued time-periodic solutions excited from the boundary time-harmonically at a sufficiently high frequency, knowledge of the first- and second-harmonic Cauchy data at the boundary is sufficient to simultaneously determine the wave speed, diffusivity and nonlinearity in the interior of the domain of interest.

math.AP↗

arXiv2024

A new interpolated pseudodifferential preconditioner for the Helmholtz equation in heterogeneous media

This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well-suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a univariate setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.

math.NA↗

arXiv2023

Pseudodifferential Models for Ultrasound Waves with Fractional Attenuation

To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudodifferential factorization of the wave operator with fractional attenuation. This factorization allows us to approximately solve the Helmholtz equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We provide explicitly the three highest order terms of the pseudodifferential expansion to incorporate the well-known square-root first order symbol for wave propagation, the zeroth order symbol for amplitude modulation due to changes in wave speed and damping, and the next symbol to model fractional attenuation. We also propose wide-angle Pade approximations for the pseudodifferential operators corresponding to these three highest order symbols. Our analysis provides insights regarding the role played by the frequency and the Pade approximations in the estimation of error bounds. We also provide a proof-of-concept numerical implementation of the proposed method and test the error estimates numerically.

math.NA↗

arXiv2023

Method of virtual sources using on-surface radiation conditions for the Helmholtz equation

We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green's function by a small distance $h>0$. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter $h$ controls the conditioning of the discrete system. We provide mathematical guidance to choose $h$, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The results are compared to analytical solutions for canonical problems. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.

math.NA↗

arXiv2022

Nonlinear ultrasound imaging modeled by a Westervelt equation

We consider the ultrasound imaging problem governed by a nonlinear wave equation of Westervelt type with variable wave speed. We show that the coefficient of nonlinearity can be recovered uniquely from knowledge of the Dirichlet-to-Neumann map. Our proof is based on a second order linearization and the use of Gaussian beam solutions to reduce the problem to the inversion of a weighted geodesic ray transform. We propose an inversion algorithm and report the results of a numerical implementation to solve the nonlinear ultrasound imaging problem in a transmission setting in the frequency domain.

math.AP↗

arXiv2021

Local on-surface radiation condition for multiple scattering of waves from convex obstacles

We propose a novel on-surface radiation condition to approximate the outgoing solution to the Helmholtz equation in the exterior of several impenetrable convex obstacles. Based on a local approximation of the Dirichlet-to-Neumann operator and a local formula for wave propagation, this new method simultaneously accounts for the outgoing behavior of the solution as well as the reflections arising from the multiple obstacles. The method involves tangential derivatives only, avoiding the use of integration over the surfaces of the obstacles. As a consequence, the method leads to sparse matrices and O(N) complexity. Numerical results are presented to illustrate the performance of the proposed method. Possible improvements and extensions are also discussed.

physics.comp-ph↗

arXiv2020

Solvability for Photoacoustic Imaging with Idealized Piezoelectric Sensors

Most reconstruction algorithms for photoacoustic imaging assume that the pressure field is measured by ultrasound sensors placed on a detection surface. However, such sensors do not measure pressure exactly due to their non-uniform directional and frequency responses, and resolution limitations. This is the case for piezoelectric sensors that are commonly employed for photoacoustic imaging. In this paper, using the method of matched asymptotic expansions and the basic constitutive relations for piezoelectricity, we propose a simple mathematical model for piezoelectric transducers. The approach simultaneously models how the pressure waves induce the piezoelectric measurements and how the presence of the sensors affects the pressure waves. Using this model, we analyze whether the data gathered by piezoelectric sensors leads to the mathematical solvability of the photoacoustic imaging problem. We conclude that this imaging problem is well-posed in certain normed spaces and under a geometric assumption. We also propose an iterative reconstruction algorithm that incorporates the model for piezoelectric measurements. A numerical implementation of the reconstruction algorithm is presented.

math.AP↗

arXiv2020

High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods

Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.

math.NA↗

arXiv2020

A weight-adjusted discontinuous Galerkin method for wave propagation in coupled elastic-acoustic media

This paper presents a high-order discontinuous Galerkin (DG) scheme for the simulation of wave propagation through coupled elastic-acoustic media. We use a first-order stress-velocity formulation, and derive a simple upwind-like numerical flux which weakly imposes continuity of the normal velocity and traction at elastic-acoustic interfaces. When combined with easily invertible weight-adjusted mass matrices, the resulting method is efficient, consistent, and energy stable on curvilinear meshes and for arbitrary heterogeneous media, including anisotropy and sub-cell (micro) heterogeneities. We numerically verify the high order accuracy and stability of the proposed method, and investigate its performance for applications in photoacoustic tomography.

math.NA↗

arXiv2019

Well-posedness for Photoacoustic Tomography with Fabry-Perot Sensors

In the mathematical analysis of photoacoustic imaging, it is usually assumed that the acoustic pressure (Dirichlet data) is measured on a detection surface. However, actual ultrasound detectors gather data of a different type. In this paper, we propose a more realistic mathematical model of ultrasound measurements acquired by the Fabry--Perot sensor. This modeling incorporates directional response of such sensors. We study the solvability of the resulting photoacoustic tomography problem, concluding that the problem is well-posed under certain assumptions. Numerical reconstructions are implemented using the Landweber iterations, after discretization of the governing equations using the finite element method.

math.AP↗

arXiv2019

Recovery of Pressure and Wave Speed for Photoacoustic Imaging under a Condition of Relative Uncertainty

In this paper, we study the photoacoustic tomography problem for which we seek to recover both the initial state of the pressure field and the wave speed of the medium from the knowledge of a single boundary measurement. The goal is to propose practical assumptions to define a set of initial conditions and wave speeds over which uniqueness for this inverse problem is guaranteed. The main result of the paper is that given two sets of wave speeds and pressure profiles, they cannot produce the same acoustic measurements if the relative difference between the wave speeds is much smaller than the relative difference between the pressure profiles. Implications for iterative joint-reconstruction algorithms are discussed.

math.AP↗

arXiv2017

Thermoacoustic tomography for an integro-differential wave equation modeling attenuation

In this article we study the inverse problem of thermoacoustic tomography (TAT) on a medium with attenuation represented by a time- convolution (or memory) term, and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters. Under the assumption of being able to measure data on the whole boundary, we prove uniqueness and stability, and propose a convergent reconstruction method for a class of smooth variable sound speeds. By a suitable modification of the time reversal technique, we obtain a Neumann series reconstruction formula.

math.AP↗

arXiv2017

High order surface radiation conditions for time-harmonic waves in exterior domains

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (J. Math. Anal. Appl. 229:184-211, 1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.

physics.comp-ph↗

arXiv2016

High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions

We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary $S$ enclosing the scatterer, the original unbounded domain $\Omega$ is decomposed into a bounded computational domain $\Omega^{-}$ and an exterior unbounded domain $\Omega^{+}$. Then, we define interface conditions at the artificial boundary $S$, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region $\Omega^{+}$. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on $\Omega^{-}$, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the order of the error induced by this ABC can easily match the order of the numerical method in $\Omega^{-}$. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.

physics.comp-ph↗

arXiv2016

Photoacoustic imaging taking into account thermodynamic attenuation

In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary. We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of the initial acoustic profile in the weak thermoelastic coupling regime. This approach is constructive, yielding a solvable equation for the unknown acoustic profile. Moreover, the solution to this reconstruction equation can be approximated numerically using the conjugate gradient method. If certain geometrical conditions for the wave speed are satisfied, this approach is well--suited for variable media and for measurements on a subset of the boundary. We also present a numerical implementation of the proposed reconstruction algorithm.

math.AP↗

arXiv2015

Multiwave imaging in an enclosure with variable wave speed

In this paper we consider the mathematical model of thermo- and photo-acoustic tomography for the recovery of the initial condition of a wave field from knowledge of its boundary values. Unlike the free-space setting, we consider the wave problem in a region enclosed by a surface where an impedance boundary condition is imposed. This condition models the presence of physical boundaries such as interfaces or acoustic mirrors which reflect some of the wave energy back into the enclosed domain. By recognizing that the inverse problem is equivalent to a statement of boundary observability, we use control operators to prove the unique and stable recovery of the initial wave profile from knowledge of boundary measurements. Since our proof is constructive, we explicitly derive a solvable equation for the unknown initial condition. This equation can be solved numerically using the conjugate gradient method. We also propose an alternative approach based on the stabilization of waves. This leads to an exponentially and uniformly convergent Neumann series reconstruction when the impedance coefficient is not identically zero. In both cases, if well-known geometrical conditions are satisfied, our approaches are naturally suited for variable wave speed and for measurements on a subset of the boundary.

math.AP↗

arXiv2015

Source estimation with incoherent waves in random waveguides

We study an inverse source problem for the acoustic wave equation in a random waveguide. The goal is to estimate the source of waves from measurements of the acoustic pressure at a remote array of sensors. The waveguide effect is due to boundaries that trap the waves and guide them in a preferred (range) direction, the waveguide axis, along which the medium is unbounded. The random waveguide is a model of perturbed ideal waveguides which have flat boundaries and are filled with known media that do not change with range. The perturbation consists of fluctuations of the boundary and of the wave speed due to numerous small inhomogeneities in the medium. The fluctuations are uncertain in applications, which is why we model them with random processes, and they cause significant cumulative scattering at long ranges from the source. The scattering effect manifests mathematically as an exponential decay of the expectation of the acoustic pressure, the coherent part of the wave. The incoherent wave is modeled by the random fluctuations of the acoustic pressure, which dominate the expectation at long ranges from the source. We use the existing theory of wave propagation in random waveguides to analyze the inverse problem of estimating the source from incoherent wave recordings at remote arrays. We show how to obtain from the incoherent measurements high fidelity estimates of the time resolved energy carried by the waveguide modes, and study the invertibility of the system of transport equations that model energy propagation in order to estimate the source.

math.AP↗