Neumann Data Research Articles

Simultaneous identification of three parameters in a time-fractional diffusion-wave equation by a part of boundary Cauchy data

Abstract This paper is devoted to determine the fractional order, the initial flux speed and the boundary Neumann data simultaneously in a one-dimensional time-fractional diffusion-wave equation from part boundary Cauchy observation data. We prove the uniqueness result for this inverse problem by using a new result for the Mittag-Leffler function and Laplace transform combining with analytic continuation. Then we use the iterative regularizing ensemble Kalman method in Bayesian framework to solve the inverse problem numerically. And four numerical examples are provided to show the effectiveness and stability of the proposed algorithm.

A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data

Abstract We propose a new numerical method for the solution of the problem of the reconstruction of the initial condition of a quasilinear parabolic equation from the measurements of both Dirichlet and Neumann data on the boundary of a bounded domain. Although this problem is highly nonlinear, we do not require an initial guess of the true solution. The key in our method is the derivation of a boundary value problem for a system of coupled quasilinear elliptic equations whose solution is the vector function of the spatially dependent Fourier coefficients of the solution to the governing parabolic equation. We solve this problem by an iterative method. The global convergence of the system is rigorously established using a Carleman estimate. Numerical examples are presented.

Boundary-Element Methods for Field Reconstruction in Accelerator Magnets

Magnetic fields in the aperture of particle accelerator magnets can be represented by boundary potentials, exploiting Kirchhoff's integral equation. Depending on the formulation, magnetic measurement data can be represented by the discrete approximations of Dirichlet or Neumann data at the domain boundary. The missing Cauchy data, which are related to the tangential-field components, can then be computed by the boundary-element method (BEM) in a numerical post-processing step. Evaluating the integral equation for field reconstruction inside the domain of interest will reduce measurement uncertainties and approximation errors due to the smoothing property of Green's kernel. Applications to the reconstruction of 2-D fields (integrated quantities from stretched-wire measurements) and 3-D fields (local quantities from measurements with moving induction-coil magnetometers) are presented.

Estimates for the L q -mixed problem in C 1,1-domains

ABSTRACT We consider the -mixed problem in domains in with -boundary. We assume that the boundary between the sets where we specify Neumann and Dirichlet data is Lipschitz. With these assumptions, we show that we may solve the -mixed problem for q in the range .

The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data

Let \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document} be a bounded smooth domain, we study the following anisotropic elliptic problem \begin{document}$ \begin{cases} -\nabla(a(x)\nabla \upsilon)+a(x)\upsilon = 0& \text{in}\, \, \, \, \, \Omega, \dfrac{\partial \upsilon}{\partial\nu} = e^\upsilon-s\phi_1-h(x) & \text{on}\, \, \partial\Omega, \end{cases} $\end{document} where \begin{document}$ \nu $\end{document} denotes the outer unit normal vector to \begin{document}$ \partial\Omega $\end{document} , \begin{document}$ h\in C^{0, \alpha}( \partial\Omega) $\end{document} , \begin{document}$ s>0 $\end{document} is a large parameter, \begin{document}$ a(x) $\end{document} is a positive smooth function and \begin{document}$ \phi_1 $\end{document} is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large \begin{document}$ s $\end{document} , which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of \begin{document}$ a(x)\phi_1 $\end{document} as \begin{document}$ s\rightarrow+\infty $\end{document} .

Isogeometric boundary element methods and patch tests for linear elastic problems: Formulation, numerical integration, and applications

An Isogeometric Boundary Element Method for solving three-dimensional boundary-value problems of classical linear elasticity theory is proposed. The method is developed as a generalization of the author’s earlier work on Laplace’s equation to Navier’s equations. As a result the proposed method features (i) proper basis functions for approximating Dirichlet and Neumann data, (ii) high-order collocation schemes for weakly singular, singular, and hyper-singular integral operators, (iii) state-of-the-art numerical integration schemes capable of handling geometries with disparate dimensions, (iv) well-conditioned linear algebraic systems, with the condition number independent of the mesh size. Boundary Element Patch Tests, as extensions of concepts widely used for finite element methods, are also introduced. It is shown how these tests can be used to assess the veracity of boundary element formulations and numerical integration schemes, implementations, and geometric precision of Computer Aided Design models. The method is applied to two challenging case studies, representative of industrial applications.

Correction: Zhang, H.; Zhang, X. Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum. Mathematics 2019, 7, 667

The authors wish to make the following corrections to this paper [...]

Existence Results of Two Mixed Boundary Value Elliptic PDEs in $$\mathbb {R}^N$$

We study the existence of a solution to the mixed boundary value problem for Helmholtz and Poisson type equations in a bounded Lipschitz domain $$\Omega \subset \mathbb {R}^N$$ and in $$\mathbb {R}^N{\setminus }\Omega $$ for $$N\ge 3$$. The boundary $$\partial \Omega $$ of $$\Omega $$ is the decomposition of $$\Gamma _1,\Gamma _2\subset \partial \Omega $$ such that $$\partial \Omega =\Gamma =\overline{\Gamma }_1\cup \Gamma _2=\Gamma _1\cup \overline{\Gamma }_2$$ and $$\Gamma _1\cap \Gamma _2=\emptyset $$. We have shown that if the Neumann data $$f_2\in H^{-\frac{1}{2}}(\Gamma _2)$$ and the Dirichlet data $$f_1\in H^{\frac{1}{2}}(\Gamma _1)$$ then the Helmholtz problem with mixed boundary data admits a unique solution. We have also shown the existence of a weak solution to a mixed boundary value problem governed by the Poisson equation with a measure data and the Dirichlet, Neumann data belongs to $$f_1\in H^{\frac{1}{2}}(\Gamma _1)$$, $$f_2\in H^{-\frac{1}{2}}(\Gamma _2)$$, respectively.

Generalized Tikhonov Method and Convergence Estimate for the Cauchy Problem of Modified Helmholtz Equation with Nonhomogeneous Dirichlet and Neumann Datum

We investigate a Cauchy problem of the modified Helmholtz equation with nonhomogeneous Dirichlet and Neumann datum, this problem is ill-posed and some regularization techniques are required to stabilize numerical computation. We established the result of conditional stability under an a priori assumption for an exact solution. A generalized Tikhonov method is proposed to solve this problem, we select the regularization parameter by a priori and a posteriori rules and derive the convergence results of sharp type for this method. The corresponding numerical experiments are implemented to verify that our regularization method is practicable and satisfied.

Prescribing a heat flux coming from a wave equation

Abstract What happens when one prescribes a heat flux which is proportional to the Neumann data of a solution of the wave equation in the whole space on the surface of a heat conductive body? It is shown that there is a difference in the asymptotic behavior of the indicator function in the most recent version of the time domain enclosure method, which aims at extracting information about an unknown cavity embedded in the body.

An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

We consider the wave equation, [Formula: see text], in fixed flat Friedmann–Lemaître–Robertson–Walker and Kasner spacetimes with topology [Formula: see text]. We obtain generic blow up results for solutions to the wave equation toward the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to the solutions that blow up in an open set of the Big Bang hypersurface [Formula: see text]. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate [Formula: see text]-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the [Formula: see text] norms of the solutions blow up toward the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates, respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

Variable exponent Calderón's problem in one dimension

We consider one-dimensional Calder\'on's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.

From reflections to a uniform elliptic growth

We use reflections involving analytic Dirichlet and Neumann data on a real-analytic curve in order to find a representation of solutions to Cauchy problems for harmonic functions in the plane. We apply this representation for finding solutions to Hele-Shaw problems. We also generalize the results by deriving the corresponding formulae for the Helmholtz equation and applying them to a uniform elliptic growth.

Solving the 3D Cauchy problems of nonlinear elliptic equations by the superposition of a family of 3D homogenization functions

We solve the 3D Cauchy problem of a nonlinear elliptic equation in a cuboid, using the derived family of 3D homogenization functions of different orders. When the solution is expressed by the weight superposition of a family of 3D homogenization functions, the unknown boundary data and the solution can be recovered quickly by solving a small scale linear system. It deserves to note that the superposition of homogenization functions method (SHFM) does not need to solve nonlinear equations and regularization, and is quite accurate to find the solution in the whole domain with the errors smaller than the level of noise being imposed on the overspecified Neumann data. Another advantage of the SHFM is that it can solve the Cauchy problem in a large size of the cuboid.

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

In this article, we investigate the problem of identifying the source term f in the elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data (j, g) with noise level In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn–Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to a sought source function. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution, we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

Resolution improving filter for time-reversal (TR) with a switching TR mirror in a halfspace.

This paper addresses the issue of using a switching time-reversal (TR) mirror for wave energy focusing to subsurface targets. The motivation stems from applications in geophysics, hydro-geology, environmental engineering, and even in therapeutic medicine. Using TR concepts, wave-focusing is straightforward and efficient, but only under ideal conditions that are, typically, unattainable in practice. The unboundedness of the subsurface that hosts the target, the TR mirror's limited aperture, and, worse, the practical need for a switching TR mirror, where recorded Dirichlet data are time-reversed as Neumann data (switching mirror), all contribute to the deterioration of the focusing resolution at the target. Herein, the development of a data filter is discussed, which is shown to be capable of overcoming the switching mirror's shortcoming, leading to improved focusing resolution. The filter's effect is demonstrated with numerical examples.

Homogenization of Linear Parabolic Equations with a Certain Resonant Matching Between Rapid Spatial and Temporal Oscillations in Periodically Perforated Domains

In this article, we study homogenization of a parabolic linear problem governed by a coefficient matrix with rapid spatial and temporal oscillations in periodically perforated domains with homogeneous Neumann data on the boundary of the holes. We prove results adapted to the problem for characterization of multiscale limits for gradients and very weak multiscale convergence.

Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

Solving the inverse conductivity problems of nonlinear elliptic equations by the superposition of homogenization functions method

The inverse conductivity problem of a nonlinear elliptic equation is solved by using two sets of single-parameter homogenization functions as the bases for solution and conductivity function. When the solution is obtained by solving a linear system to satisfy the over-specified Neumann boundary condition on a partial boundary, the unknown conductivity function can be recovered by solving another linear system generated from the governing equation by collocation technique. The maximum absolute error of the recovered conductivity is smaller than the noise being imposed on the Neumann data. The superposition of homogenization functions method (SHFM) is quite accurate to find the whole solution and the conductivity function, and the required extra data are parsimonious.