Cauchy Data Research Articles

Determining sources in the bioluminescence tomography problem

Abstract In this paper, we revisit the bioluminescence tomography\,(BLT) problem, where one seeks to reconstruct bioluminescence signals (an internal light source) from external measurements of the Cauchy data. As one kind of optical imaging, the BLT has many merits such as high signal-to-noise ratio, non-destructivity and cost-effectiveness etc., and has potential applications such as cancer diagnosis, drug discovery and development as well as gene therapies and so on. In the literature, BLT is extensively studied based on diffusion approximation (DA) equation, where the distribution of peak sources is to be reconstructed and no solution uniqueness is guaranteed without adequate a priori information. Motivated by the solution uniqueness issue, several theoretical results are explored. The major contributions in this work that are new to the literature are two-fold: first, we show the theoretical uniqueness of the BLT problem where the light sources are in the shape of $C^2$ domains or polyhedral- or corona-shaped; second, we support our results with plenty of problem-orientated numerical experiments.

Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain

<p>In this paper, a regularized solution to the Cauchy problem for matrix factorization of the Helmholtz equation in a three-dimensional unbounded domain is constructed explicitly based on the Carleman matrix. When solving applied problems, in addition to an approximate solution, the derivative of the approximate solution is found. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with precisely specified Cauchy data. An explicit formula for continuing the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the original Cauchy data, their continuous approximations with a specified error in the uniform metric are given. As a result, the stability of the solution to the Cauchy problem in the classical sense is estimated.</p>

An Efficient Computational Algorithm for a Class of Nonlinear Singular Perturbation Problems of Convection Diffusion Type with Cauchy Data

An Efficient Computational Algorithm for a Class of Nonlinear Singular Perturbation Problems of Convection Diffusion Type with Cauchy Data

Height-function-based 4D reference metrics for hyperboloidal evolution

Hyperboloidal slices are spacelike slices that reach future null infinity. Their asymptotic behaviour is different from Cauchy slices, which are traditionally used in numerical relativity simulations. This work uses free evolution of the formally-singular conformally compactified Einstein equations in spherical symmetry. One way to construct gauge conditions suitable for this approach relies on building the gauge source functions from a time-independent background spacetime metric. This background reference metric is set using the height function approach to provide the correct asymptotics of hyperboloidal slices of Minkowski spacetime. The present objective is to study the effect of different choices of height function on hyperboloidal evolutions via the reference metrics used in the gauge conditions. A total of 10 reference metrics for Minkowski are explored, identifying some of their desired features. They include 3 hyperboloidal layer constructions, evolved with the non-linear Einstein equations for the first time. Focus is put on long-term numerical stability of the evolutions, including small initial gauge perturbations. The results will be relevant for future (puncture-type) hyperboloidal evolutions, 3D simulations and the development of coinciding Cauchy and hyperboloidal data, among other applications.

Determination of the modified exterior Steklov eigenvalues via the reciprocity gap method

The modified exterior Steklov eigenvalues (MESEs) arise from the inverse scattering problem for inhomogeneous media with a cavity and may serve as potential target signatures in nondestructive testing. In this paper we are interested in the determination of the MESEs from the measured Cauchy data of the total field on some manifold inside the cavity due to interior point sources. To this end, the reciprocity gap (RG) method based on a linear integral equation is employed. We provide the related theory and show that the blow-up property of the approximate solution to the integral equation can be used to characterize the MESEs. Numerical examples are presented to demonstrate the viability of our method.

Shape reconstruction of a cavity with impedance boundary condition via the reciprocity gap method

We consider an interior inverse scattering problem of reconstructing the shape of a cavity with impedance boundary condition from measured Cauchy data of the total field. The incident point sources and the measurements are distributed on two different manifolds inside the cavity. We first prove that the boundary of the cavity and the surface impedance can be uniquely determined by the scattered field data on the measurement manifold. Then we develop a reciprocity gap (RG) method to reconstruct the cavity. The theoretical analysis shows the uniquely solvability and existence of the approximate solution for the linear integral equation constructed in the RG method. We also prove that the shape of the cavity can be characterized by the blow-up property of the approximate solution of the proposed integral equation. Numerical examples are presented to verify the feasibility of the RG method.

Inverse coefficient problems for one-dimensional time-fractional diffusion equations

We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.

The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensor

Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the 'U' and 'V' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds, a priori, to the spatial metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.

Lipschitz stability of an inverse conductivity problem with two Cauchy data pairs

Abstract In 1996 Seo proved that two appropriate pairs of current and voltage data measured on the surface of a planar homogeneous object are sufficient to determine a conductive polygonal inclusion with known deviating conductivity. Here we show that the corresponding linearized forward map is injective, and from this we deduce Lipschitz stability of the solution of the original nonlinear inverse problem. We also treat the case of an insulating polygonal inclusion, in which case a single pair of Cauchy data is already sufficient for the same purpose.

On the reconstruction of cardiac transmembrane potential pattern from body surface measurement

Electrocardiographic imaging (ECGI) is a diagnostic tool designed for the noninvasive detection of electrical activity in the heart. Mathematically, this imaging process can be modelled by an inverse problem for the coupled elliptic system with Cauchy data. For the bidomain model describing the electrical activity in the myocardium, we reconstruct the transmembrane potential on the heart surface from ECG recordings. The reconstruction process is split into two steps: firstly computing the electrical potential and current on heart surface from torso surface recordings and then recovering the transmembrane potential on the heart surface. The first step is essentially a Cauchy problem for elliptic equation which is well-known to be severely ill-posed. We realize this step in terms of boundary integral system by a quasi-regularization scheme to deal with the ill-posedness. The second step is implemented by an integral equation of the second kind with nontrivial null space. We remove the non-uniqueness and then provide a numerical scheme by the boundary element method to obtain the transmembrane potential on the heart surface. The regularized solution obtained from noisy measurement data is proven rigorously to be convergent to the exact solution as the measurement error in the torso surface tends to zero. For solving the regularizing system numerically, we establish quadrature formulas for boundary potentials for 3-dimensional bio-tissue in terms of the concept of solid angles. The numerical realizations for different configurations are finally presented to show the validity of the proposed scheme.

Constant Q-curvature metrics with Delaunay ends: the nondegenerate case

We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.

Geometric flavours of quantum field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution

In this series of papers we aim to provide a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this second part we use the tools of Gaussian analysis in infinite dimensional spaces introduced in the first part to describe rigorously the procedures of geometric quantization in the space of Cauchy data of a scalar theory. This leads us to discuss and establish relations between different pictures of QFT. We also apply these tools to describe the geometrization of the space of pure states of quantum field theory as a Kähler manifold. We use this to derive an evolution equation that preserves the geometric structure and avoid norm losses in the evolution. This leads us to a modification of the Schrödinger equation via a quantum connection that we discuss and exemplify in a simple case.

The nonlinear stability of (n + 1)-dimensional FLRW spacetimes

We prove nonlinear Lyapunov stability of a family of “([Formula: see text])”-dimensional cosmological models of general relativity locally isometric to the Friedman–Lemaître–Robertson–Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein–Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant [Formula: see text] case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics with constant negative spatial scalar curvature. To accomplish this result, we employ an energy method for the coupled Einstein–Euler field equations in constant mean extrinsic curvature spatial harmonic (CMCSH) gauge. In order to handle Euler’s equations, we construct energy from a current that is similar to the one derived by Christodoulou [The Formation of Shocks in 3-dimensional Fluids, EMS Monographs in Mathematics, Vol. 2 (European Mathematical Society, 2007)] (and which coincides with Christodoulou’s current on the Minkowski space) and show that this energy controls the desired norm of the fluid degrees of freedom. The use of a fluid energy current together with the CMCSH gauge condition casts the Einstein–Euler field equations into a coupled elliptic–hyperbolic system. Utilizing the estimates derived from the elliptic equations, we first show that the gravity-fluid energy functional remains uniformly bounded in the expanding direction. Using this uniform boundedness property, we later obtain sharp decay estimates if a positive cosmological constant [Formula: see text] is included, which confirms that the accelerated expansion of the physical universe that is induced by the positive cosmological constant is sufficient to control the nonlinearities in the case of small data. A few physical consequences of this stability result are discussed.

A toy model for damped water waves

We consider a toy model for a damped water waves system in a domain [Formula: see text]. The toy model is based on the paradifferential formulation of the water waves system derived in the work of Alazard–Burq–Zuily [On the water-waves equations with surface tension, Duke Math. J. 158 (2011) 413–499]. The form of damping we consider is a modified sponge layer proposed for the [Formula: see text] water waves system by Clamond et al. in [An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption, J. Comput. Phys. 205 (2005) 686–705]. We show that, in the case of small Cauchy data, solutions to the toy model exhibit a quadratic lifespan. This is done via proving energy estimates with Klainerman vector fields.

Logarithmic type stability for the simultaneous identification of Robin coefficient and heat flux in an elliptic equation

In this paper, we aim to reconstruct the unknown Robin coefficient and unknown heat flux simultaneously in an elliptic system from Cauchy data on arbitrarily small portion of accessible boundary. A novel logarithmic type stability in terms of the negative Sobolev norms is established.

Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Stable reconstruction of anisotropic scattering objects from electromagnetic Cauchy data

ABSTRACT This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance.

Energy behavior for Sobolev solutions to viscoelastic damped wave models with time‐dependent oscillating coefficient

AbstractIn this work, we study the asymptotic behavior of the structurally damped wave equations arising from the viscoelastic mechanics. We are particularly interested in the complicated interaction of the time‐dependent oscillating coefficients on the Dirichlet Laplacian operator and the structurally damped terms. On the one hand, by the application of WKB analysis, we explore the asymptotic energy estimates of the wave equations influenced by four types of oscillating mechanisms. On the other hand, in order to prove the optimality of the energy estimates for the critical cases, typical coefficients and initial Cauchy data will be constructed to show the lower bound of the energy growth rate by the application of instability arguments.

Simultaneous reconstruction of the shape and surface impedance from Cauchy data for the Helmholtz equation

In this paper, we are concerned with the coefficients identification for the Helmholtz equation. This problem consists of determining the shape and surface impedance from boundary Cauchy data. We propose two reconstruction algorithms to simultaneously recover the shape and the surface impedance of the obstacle within a body. This problem is ill-posed, thus we apply regularization techniques in order to improve the corresponding approximation. The numerical results show that the proposed algorithms are stable and effective.

Solution theory to semilinear stochastic equations of Schrödinger type on curved spaces I: operators with uniformly bounded coefficients

We study the Cauchy problem for Schrödinger type stochastic semilinear partial differential equations with uniformly bounded variable coefficients, depending on the space variables. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.