Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

Ingerman, David; Druskin, Vladimir; Knizhnerman, Leonid

doi:10.1002/1097-0312(200008)53:8<1039::aid-cpa4>3.0.co;2-i

Cited by 69 publications

(33 citation statements)

References 15 publications

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“…Exponential clustering of poles at singularities has been part of the landscape of rational approximation for half a century, though not many works in the area focus on this effect. (A fascinating previous example is [ 20 ].) Our motivation is that this clustering is what makes rational approximations so powerful, and understanding it enables one to improve existing numerical algorithms and develop new ones.…”

Section: Discussionmentioning

confidence: 99%

“…2 and 4 , other functions with endpoint singularities give similar results. For a related discussion of clustered poles and root exponential convergence see [ 20 ], where rational approximations associated with continued fractions are connected with optimal finite difference grids.…”

Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning

confidence: 99%

“…Evidently in a wide range of rational approximations, both best and near-best, the distances of poles to singularities is well approximated by the formula for some constants and , that is, for some and . It is interesting that a hint of tapering can also be seen in the optimal finite differencing grids plotted in Figure 3.3 of [ 20 ].…”

Section: Tapered Exponential Clusteringmentioning

confidence: 99%

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Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

2021

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Section: Discussionmentioning

confidence: 99%

Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning

confidence: 99%

Section: Tapered Exponential Clusteringmentioning

confidence: 99%

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Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

2021

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“…They are called optimal because they give spectral accuracy of the DtN map with finite volumes on coarse grids. Optimal grids were introduced and analyzed in [3,23,24,29], for forward problems. The first inversion method on optimal grids was proposed in [7], for Sturm-Liouville inverse spectral problems in one dimension.…”

Section: Introductionmentioning

confidence: 99%

Circular resistor networks for electrical impedance tomography with partial boundary measurements

Borcea¹,

Druskin²,

Mamonov³

2010

Inverse Problems

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Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11,49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting trade-off between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary.

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“…9 Let us note that the net condensation flux J cd obtained in this way will be much less than the individual condensation and evaporation rates in Eq. ͑5͒, we can now find the time-dependent boundary condition at x = 0 for Eq.…”

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confidence: 97%

Dynamic condensation blocking in cryogenic refueling

Осипов

Muratov

2008

Applied Physics Letters

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Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

Cited by 69 publications

References 15 publications

Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

Circular resistor networks for electrical impedance tomography with partial boundary measurements

Dynamic condensation blocking in cryogenic refueling

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