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1. Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

   https://doi.org/10.1093/gji/ggab151  

   Iversen, Einar; Ursin, Bjørn; Saksala, Teemu; Ilmavirta, Joonas; de Hoop, Maarten V (May 2021, Geophysical Journal International)

   null (Ed.)

   SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model. 

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2. The C∞ -isomorphism property for a class of singularly-weighted x-ray transforms

   https://doi.org/10.1088/1361-6420/aca8cb  

   Mishra, Rohit Kumar; Monard, François; Zou, Yuzhou (December 2022, Inverse Problems)

   Abstract We study a one-parameter family of self-adjoint normal operators for the x-ray transform on the closed Euclidean disk D , obtained by considering specific singularly weighted L 2 topologies. We first recover the well-known singular value decompositions in terms of orthogonal disk (or generalized Zernike) polynomials, then prove that each such realization is an isomorphism of C ∞ ( D ) . As corollaries: we give some range characterizations; we show how such choices of normal operators can be expressed as functions of two distinguished differential operators. We also show that the isomorphism property also holds on a class of constant-curvature, circularly symmetric simple surfaces. These results allow to design functional contexts where normal operators built out of the x-ray transform are provably invertible, in Fréchet and Hilbert spaces encoding specific boundary behavior. 

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3. Fast algorithms for Jacobi expansions via nonoscillatory phase functions

   https://doi.org/10.1093/imanum/drz016  

   Bremer, James; Yang, Haizhao (April 2019, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm–Liouville eigentransforms and calculating Gauss–Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters $$\alpha $$ and $$\beta $$ in Jacobi’s differential equation are of magnitude less than $1/2$, is based on the well-known fact that in this regime Jacobi’s differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available. 

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4. Periodic Staircase Matrices and Generalized Cluster Structures

   https://doi.org/10.1093/imrn/rnaa148  

   Gekhtman, M.; Shapiro, M.; Vainshtein, A. (July 2022, International mathematics research notices)

   As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot– Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators. 

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5. Fast algorithms using orthogonal polynomials

   https://doi.org/10.1017/S0962492920000045  

   Olver, Sheehan; Slevinsky, Richard Mikaël; Townsend, Alex (May 2020, Acta Numerica)

   We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications. 

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