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Free, publiclyaccessible full text available August 1, 2025

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coeﬃcient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier expansion in the time domain and using the recently developed polynomialexponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasilinear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the eﬃcacy of our method. The method not only delivers precise solutions but also showcases remarkable computational eﬃciency.more » « lessFree, publiclyaccessible full text available July 1, 2025

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Free, publiclyaccessible full text available September 30, 2024

Free, publiclyaccessible full text available September 30, 2024

We propose a globally convergent numerical method to compute solutions to a general class of quasilinear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasireversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented.more » « less

This work extends the applicability of our recent convexification based algorithm for constructing images of the dielectric constant of buried or occluded target. We are orientated towards the detection of explosivelike targets such as antipersonnel land mines and improvised explosive devices in the noninvasive inspections of buildings. In our previous work, the method is posed in the perspective that we use multiple source locations running along a line of source to get a 2D image of the dielectric function. Mathematically, we solve a 1D coefficient inverse problem for a hyperbolic equation for each source location. Different from any conventional Born approximationbased technique for syntheticaperture radar, this method does not need any linearization. In this paper, we attempt to verify the method using several 3D numerical tests with simulated data. We revisit the global convergence of the gradient descent method of our computational approach.more » « less

The aim of this paper is to solve an important inverse source problem which arises from the wellknown inverse scattering problem. We propose to truncate the Fourier series of the solution to the governing equation with respect to a special basis of L2. By this, we obtain a system of linear elliptic equations. Solutions to this system are the Fourier coefficients of the solution to the governing equation. After computing these Fourier coefficients, we can directly find the desired source function. Numerical examples are presented.more » « less