arXiv subjects

Yavar Kian

Publications and source records attributed to Yavar Kian.

At least 19 recordsLinked to original sources

arXiv2026

Inverse coefficient problem for the wave equation with nonlocal attenuation

In this article, we investigate the inverse problem of determining time-dependent first- and zeroth-order coefficients for a wave equation with nonlocal-in-time attenuation on a Riemannian manifold, using boundary measurements. This problem is motivated by applications in imaging and viscoelasticity, where such equations arise naturally, and we aim to characterize the properties of the underlying medium by recovering these coefficients. Our main objective is to exploit the memory properties of solutions induced by nonlocal attenuation, establishing the recovery of a general class of lower-order coefficients from boundary measurements collected over an arbitrarily small time interval. As a byproduct of our analysis, we also prove recovery results for time-dependent coefficients using measurements supported on disjoint time intervals. While such data restrictions are unavailable, and generally impossible, for the classical wave equation, we demonstrate that they become feasible under standard assumptions due to the presence of nonlocal-in-time attenuation. Our analysis combines techniques from differential geometry, the theory of partial differential equations with nonlocal terms, complex analysis, and the theory of nonlocal operators.

math.AP↗

arXiv2026

Numerical Analysis of Space-Time Dependent Source Identification in Subdiffusion Equations

In this work, we propose an easy-to-implement fixed-point algorithm for reconstructing a space-time dependent source in a subdiffusion model from lateral boundary measurements. The numerical scheme combines a Galerkin finite element method for spatial discretization with a finite difference method for temporal discretization. We establish the linear convergence of the fixed-point iteration and derive an error bound that depends explicitly on the discretization parameters and the noise level. The error analysis relies on stability properties of the continuous inverse problem and technical estimates for the associated direct problem with limited-regularity data. Numerical experiments are presented to support and complement the theoretical analysis.

math.NA↗

arXiv2026

Stability and Reconstruction of a Nonlinearity in a Parabolic Equation from Partial Boundary Data

In this work, we investigate the inverse problem of determining a semilinear term in a nonlinear parabolic equation from one single boundary flux measurement taken on an arbitrary subset of the boundary. More precisely, we address both uniqueness and stability issues of the inverse problem and establish new Hölder-type stability estimates. The Hölder exponent depends explicitly on the measurement configuration as well as on regularity properties of the semilinear term. The analysis relies on a novel approach based on the derivation of a suitable integral identity involving solutions of the associated adjoint equation. This allows reformulating the inverse problem as an inverse source problem with a sign-changing source term. The main results are obtained by combining fundamental properties of parabolic equations, including maximum principle and appropriate energy estimates. Finally, we complement the theoretical analysis with an iterative reconstruction algorithm inspired by inverse source problems, and illustrate its accuracy on several numerical experiments.

math.AP↗

arXiv2026

Recovery of a Measure-valued Source in the Heat Equation from Sparse Boundary Measurements

This article is devoted to the inverse source problem of uniquely determining a measure-valued source from sparse boundary measurements. The measurements considered consist of flux observations over a time interval at two distinct points on the boundary of the domain. The main objective of this work is to extend the existing literature on inverse source problems from sparse boundary measurements, which has so far been limited to point sources or L2 sources, to the identification of a general class of Radon measures. Our approach combines several analytical tools, including regularity properties, boundary representations, and the time analyticity of solutions to the diffusion equation with singular sources. Our theoretical analysis is complemented by a numerical study of the problem. In particular, we investigate the reconstruction of point sources and of a source supported on a curve, and present numerical experiments illustrating the recovery of such sources from sparse boundary flux measurements.

math.AP↗

arXiv2026

Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.

math.AP↗

arXiv2026

Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements

In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in $\mathbb{R}^d$ ($d\geq2$), and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth and bounded in $\mathbb{R}^2$, under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.

math.AP↗

arXiv2026

Unique Determination of Variable Order in Subdiffusion from a Single Measurement

We study the inverse problem of recovering a spatially dependent variable order in a time-fractional diffusion model from the boundary flux measurement generated by a single boundary excitation. It arises in the identification of heterogeneous media in anomalous diffusion processes. In this work, we establish several new uniqueness results for the inverse problem in the case of piecewise constant variable orders, without any monotonicity condition. The analysis follows a new approach that combines properties of harmonic functions, a linearization technique in the Laplace domain, and tools from complex, asymptotic, and geometrical analysis. In addition, we weaken the regularity assumptions on the problem data and extend the analysis of previous contributions to higher-dimensional settings.

math.AP↗

arXiv2026

Inverse problem for the geometric Navier-Stokes equations

We consider the inverse problem of determining a compact Riemannian manifold with boundary from fixed time observations of the solution, restricted to a small subset in space, for the Navier-Stokes system with a local source on the manifold. Our approach is based on a reduction to an inverse problem for an auxiliary hyperbolic Stokes system, via linearization and spectral techniques. We solve the resulting inverse problem by a new generalization of the Boundary Control method.

math.AP↗

arXiv2026

Recovery of nonlinear material parameters in a quasilinear Lamé system

We investigate the inverse problem of determining nonlinear elastic material parameters from boundary stress measurements corresponding to prescribed boundary displacements. The material law is described by a nonlinear, space-independent elastic tensor depending on both the displacement and the strain, and gives rise to a general class of quasilinear Lamé systems. We prove the unique and stable recovery of a wide class of space-independent nonlinear elastic tensors, including the identification of two nonlinear isotropic Lamé moduli as well as certain anisotropic tensors. The boundary measurements are assumed to be available at a finite number of boundary points and, in the isotropic case, at a single point. Moreover, the measurements are generated by boundary displacements belonging to an explicit class of affine functions. The analysis is based on structural properties of nonlinear Lamé systems, including asymptotic expansions of the boundary stress and tensorial calculus.

math.AP↗

arXiv2026

Recovery of Sturm-Liouville operators from partial boundary spectral data and applications

We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues together with the normal derivatives of the corresponding eigenfunctions at both endpoints. We present two novel applications of our spectral result in inverse coefficient determination problems for evolutionary PDEs that include passive wave-based imaging of a medium and active imaging for the time-dependent Schrödinger equation with unknown internal sources. Our results yield finite time measurement bounds for such inverse coefficient determination problems. A central innovation is the use of Kahane's interpolation theorem to analyze endpoint time traces of solutions, enabling the recovery without requiring analyticity assumptions or infinite-time data, as in previous approaches. Finally, in the appendix, we present a spectral interpolation theorem for one-dimensional Schrödinger operators, which may be of independent interest.

math.AP↗

arXiv2025

Unique and Stable Recovery of Space-Variable Order in Multidimensional Subdiffusion

In this work we investigate the unique identifiability and stable recovery of a spatially dependent variable-order in the subdiffusion model from the boundary flux measurement. We establish several new unique identifiability results from the observation at one point on the boundary without / with the knowledge of medium properties, and a conditional Lipschitz stability estimate when the observation is available on the whole boundary. The analysis crucially employs resolvent estimates in the $L^r(Ω)$ ($r>d$) spaces, solution representation in the Laplace domain and novel asymptotic expansions of the Laplace transform of the boundary flux at $p= 0$ and $p=1$.

math.AP↗

arXiv2025

Uniqueness and stability in determining the wave equation from a single passive boundary measurement

This article addresses the inverse problem of simultaneously recovering both the wave speed coefficient and an unknown initial condition (acting as the source) for the multidimensional wave equation from a single passive boundary measurement. Specifically, we establish uniqueness and Hölder stability estimates for determining these parameters in the wave equation on $\mathbb{R}^3$, where only a single boundary measurement of the solution--generated by the unknown source--is available. Our work connects to thermoacoustic and photoacoustic tomography (TAT/PAT) for the physically relevant case of piecewise constant sound speeds. We significantly relax the stringent conditions previously required for resolving this problem, extending results to general classes of piecewise constant sound speeds over inclusions with unknown locations. Moreover, we do not require decay properties in time of solutions to the wave equation, which enables our study to accommodate a much broader class of unknown sources. The approach combines low frequency-domain solution representations with distinctive properties of elliptic and hyperbolic equations.

math.AP↗

arXiv2025

Simultaneous Identification of Coefficients and Source in a Subdiffusion Equation from One Passive Measurement

This article is devoted to the detection of parameters in anomalous diffusion from a single passive measurement. More precisely, we consider the simultaneous identification of coefficients as well as a time-dependent source term appearing in a time-fractional diffusion equation from a single boundary or internal passive measurement. We obtain several uniqueness results in dimension one as well as a multidimensional extension under some symmetry assumptions. Our analysis relies on spectral representation of solutions, complex and harmonic analysis combined with some known inverse spectral results for Sturm-Liouville operators. The theoretical results are complemented by a corresponding reconstruction algorithm and numerical simulations.

math.AP↗

arXiv2025

Stable Determination and Reconstruction of a Quasilinear Term in an Elliptic Equation

In this work, we investigate the inverse problem of determining a quasilinear term appearing in a nonlinear elliptic equation from the measurement of the conormal derivative on the boundary. This problem arises in several practical applications, e.g., heat conduction. We derive novel Hölder stability estimates for both multi- and one-dimensional cases: in the multi-dimensional case, the stability estimates are stated with one single boundary measurement, whereas in the one-dimensional case, due to dimensionality limitation, the stability results are stated for the Dirichlet boundary condition varying in a space of dimension one. We derive these estimates using different properties of solution representations. We complement the theoretical results with numerical reconstructions of the quasilinear term, which illustrate the stable recovery of the quasilinear term in the presence of data noise.

math.AP↗

arXiv2024

Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.

math.AP↗

arXiv2024

Simultaneous stable determination of quasilinear terms for parabolic equations

In this work, we consider the inverse problem of simultaneously recovering two classes of quasilinear terms appearing in a parabolic equation from boundary measurements. It is motivated by several industrial and scientific applications, including the problems of heat conduction and population dynamics, and we study the issue of stability. More precisely, we derive simultaneous Lipschitz and Hölder stability estimates for two separate classes of quasilinear terms. The analysis combines different arguments including the linearization technique with a novel construction of singular solutions and properties of solutions of parabolic equations with nonsmooth boundary conditions. These stability results may be useful for deriving the convergence rate of numerical reconstruction schemes.

math.AP↗

arXiv2024

Identification of a Spatially-Dependent Variable Order in One-Dimensional Subdiffusion

In this work we investigate an inverse problem of identifying a spatially variable order in the one-dimensional subdiffusion model from the boundary flux measurement. The model involves a generalized Caputo derivative in time, and arises in the mathematical modeling of anomalous diffusion in heterogeneous media. We prove the unique recovery of a monotone piecewise constant variable order and its range for known and unknown media, respectively. The analysis is based on a delicate asymptotic expansion of the Laplace transform of the data as $p\to0$, which is of independent interest.

math.AP↗

arXiv2024

Determination and reconstruction of a semilinear term from point measurements

In this article we study the inverse problem of determining a semilinear term appearing in an elliptic equation from boundary measurements. Our main objective is to develop flexible and general theoretical results that can be used for developing numerical reconstruction algorithm for this inverse problem. For this purpose, we develop a new method, based on different properties of solutions of elliptic equations, for treating the determination of the semilinear term as a source term from a point measurement of the solutions. This approach not only allows us to make important relaxations on the data used so far for solving this class of inverse problems, including general Dirichlet excitation lying in a space of dimension one and measurements located at one point on the boundary of the domain, but it also allows us to derive a novel algorithm for the reconstruction of the semilinear term. The effectiveness of our algorithm is corroborated by extensive numerical experiments. Notably, as demonstrated by the theoretical analysis, we are able to effectively reconstruct the unknown nonlinear source term by utilizing solely the information provided by the measurement data at a single point.

math.NA↗