Numerical Solution of Elliptic Problems

Birkhoff, Garrett; Lynch, Robert E.

doi:10.1137/1.9781611970869

Cited by 101 publications

(52 citation statements)

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“…(6) and inhomogeneous mixed boundary conditions. Finite element discretization of the boundary value problem produces a linear system of equations, which has a unique solution [20]. Although the rank of this system of linear equations is large (the number of equations is equal to the number of pixels in the reconstruction area), the left-hand side coefficient matrix is very sparse and a solution can be obtained extremely quickly.…”

Section: Phase-retrieval Boundary Value Problem and A Priori Informationmentioning

confidence: 99%

Boundary value problem for phase retrieval from unidirectional X-ray differential phase images

Gasilov¹,

Mittone²,

Horng³

et al. 2015

Opt. Express

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Abstract:The phase retrieval problem can be reduced to the second order partial differential equation. In order to retrieve the absolute values of the X-ray phase and to minimize the reconstruction artifacts we defined the mixed inhomogeneous boundary condition using available a priori information about the sample. Finite element technique was used to solve the boundary value problem. The approach is validated on numerical and experimental phantoms. In order to demonstrate a possible application of the method, we have processed an entire tomographic set of differential phase images and estimated the magnitude of the refractive index decrement for some tissues inside complex biomedical samples. References and links1. R. Fitzgerald, "Phase-sensitive X-ray imaging," Phys. Today 53, 23-27 (2000). 2. A. Bravin, P. Coan, and P. Suortti, "X-ray phase-contrast imaging: from pre-clinical applications towards clinics," Phys. Med. Biol 58, R1-R35 (2013). 3. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335-337 (1996). 4. V. N. Ingal and E. A. Beliaevskaya, "X-ray plane-wave topography observation of the phase contrast from a non-crystalline object," J. Phys. D 28, 2314-2318 (1995). 5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, "Quantitative X-ray phase imaging with a grating interferometer," Opt. Lett. 108, 158102 (2012). 11. J. Sperl, D. Bequ, G. Kudielka, K. Mahdi, P. Edic, and C. Cozzini, "A Fourier-domain algorithm for total-variation regularized phase retrieval in differential X-ray phase contrast imaging," Opt. Express 22, 450-462 (2014 F. Pfeiffer, and J. Herzen, "Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum," Opt. Express 23, 523-535 (2015).

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Section: Phase-retrieval Boundary Value Problem and A Priori Informationmentioning

confidence: 99%

Boundary value problem for phase retrieval from unidirectional X-ray differential phase images

Gasilov¹,

Mittone²,

Horng³

et al. 2015

Opt. Express

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“…conjugate-gradient (CG) type -iterative schemes. Like the classical stationary Richardson and Frankel (alias second-order Richardson) schemes (Birkhoff and Lynch, 1984, chpt. 5) they effectively generate an approximate solution of the governing elliptic problem, say (15), recurrently by linearly combining the initial guess φ ′ 0 with the subsequent iterates of the initial residual error, r 0 = L(φ ′ 0 ) − R, under the linear operator L, such that…”

Section: Highlights Of the Elliptic Solvermentioning

confidence: 99%

“…Smolarkiewicz and Margolin, 2000, and references therein) may be more appealing to a physicist. Just like the first-and second-order Richardson methods may be viewed as discrete integrations of the heat and damped oscillation equations (Birkhoff and Lynch, 1984, sections 4.9 and 4.15, respectively), the preconditioned GCR(k) algorithm can be derived 6 by augmenting BVPs of the preceding section with a kth-order damped oscillation equation…”

Section: Highlights Of the Elliptic Solvermentioning

confidence: 99%

A nonhydrostatic unstructured-mesh soundproof model for simulation of internal gravity waves

Smolarkiewicz

Szmelter

2011

Acta Geophys.

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Citation: SMOLARKIEWICZ, P.K. and SZMELTER, J., 2011. A nonhydrostatic unstructured-mesh soundproof model for simulation of internal gravity waves. Acta Geophysica, 59 (6), pp. 1109 -1134 Additional Information:• This article was published in the journal, Acta Geophysica [Versita / AbstractA semi-implicit edge-based unstructured-mesh model is developed that integrates nonhydrostatic soundproof equations, inclusive of anelastic and pseudo-incompressible systems of partial differential equations. The model builds on nonoscillatory forwardin-time MPDATA approach using finite-volume discretization and unstructured meshes with arbitrarily shaped cells. Implicit treatment of gravity waves benefits both accuracy and stability of the model. The unstructured-mesh solutions are compared to equivalent structured-grid results in simulations of an intricate multiscale internal wave phenomenon of a non-Boussinesq amplification and breaking of deep stratospheric gravity waves. The departures of the anelastic and pseudo-incompressible results are quantified in reference to a recent asymptotic theory

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“…To construct schemes with an improved convergence defect, it is tempting to use a technique based on locally condensing rectangular meshes. So, in the case of regular boundary value problems whose solutions have singularities, an improvement of the accuracy of a discrete solution can be achieved by means of a priori and/or a posteriori local refinement of a rectangular mesh in those subdomains where the errors of the discrete solution are larger (see, e.g., [1,3,13]). …”

Section: Is Said To Be Unimprovable With Respect To the Values Ofmentioning

confidence: 99%

“…The use of a technique developed to improve the accuracy of the solutions in the case of regular boundary value problems (e.g., the technique of a priori/a posteriori adaptive meshes; see [1,3,6] and bibliography therein), turns out to be ineffective in the case of singularly perturbed problems (see [10] and the statement of Theorem 4.1 in Section 4). Therefore, the quest for conditions necessary (and for specific discrete methods also sufficient) for the ε-uniform convergence of numerical methods, for problems with moving transition layers is relevant.…”

Section: Introductionmentioning

confidence: 99%

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

Shishkin

Shishkina

Hemker

2004

Comput. Methods Appl. Math.

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-We study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm, even under the condition≈ ε, where ε is the perturbation parameter and N and N 0 denote the number of mesh points with respect to x and t. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer, there are no schemes that converge uniformly in ε even under the very restrictive condition≈ ε. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges 'almost ε-uniformly ', viz., under

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Numerical Solution of Elliptic Problems

Cited by 101 publications

References 0 publications

Boundary value problem for phase retrieval from unidirectional X-ray differential phase images

Boundary value problem for phase retrieval from unidirectional X-ray differential phase images

A nonhydrostatic unstructured-mesh soundproof model for simulation of internal gravity waves

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

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