| 51. |
- Bogfjellmo, Geir, 1987, et al.
(författare)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Tidskriftsartikel (refereegranskat)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
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| 52. |
- Elfverson, Daniel, et al.
(författare)
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Multiscale methods for problems with complex geometry
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 321, s. 103-123
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Tidskriftsartikel (refereegranskat)abstract
- We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.
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| 53. |
- Frahm, J., et al.
(författare)
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An extension problem related to the fractional Branson-Gover operators
- 2020
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 278:5
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Tidskriftsartikel (refereegranskat)abstract
- The Branson-Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson-Gover operators. For Euclidean spaces we show that the fractional Branson-Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli-Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of their properties. (C) 2019 Elsevier Inc. All rights reserved.
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| 54. |
- Jensen, M., et al.
(författare)
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Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions
- 2022
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Ingår i: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 42:1, s. 199-228
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Tidskriftsartikel (refereegranskat)abstract
- We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional, the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the difference in regularity within the domain.
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| 55. |
- Khesin, Boris, et al.
(författare)
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Newton's Equation on Diffeomorphisms and Densities
- 2017
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Ingår i: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). - Cham : Springer International Publishing. - 1611-3349 .- 0302-9743. - 9783319684451 ; 10589, s. 873-873
-
Konferensbidrag (refereegranskat)abstract
- We develop a geometric framework for Newton-type equations on the infinite-dimensional configuration space of probability densities. It can be viewed as a second order analogue of the "Otto calculus" framework for gradient flow equations. Namely, for an n-dimensional manifold M we derive Newton's equations on the group of diffeomorphisms Diff(M) and the space of smooth probability densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Poisson reduction of Newton's equation on Diff(M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton-Jacobi equation of fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrodinger-type and Newton's equations is a symplectomorphism between the corresponding phase spaces T* Dens(M) and PL2 (M, C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kahler map provided that the space of densities is equipped with the (prolonged) Fisher-Rao information metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher-Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton-Jacobi, and linear and nonlinear Schrodinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and mu-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.
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| 56. |
- Maurelli, M., et al.
(författare)
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Incompressible Euler equations with stochastic forcing: A geometric approach
- 2023
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Ingår i: Stochastic Processes and Their Applications. - : Elsevier BV. - 0304-4149. ; 159, s. 101-148
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a stochastic version of Euler equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden (1970). For the Euler equations on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic non-linear partial differential equations.(c) 2023 Elsevier B.V. All rights reserved.
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| 57. |
- Modin, Klas, 1979, et al.
(författare)
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Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey
- 2021
-
Ingår i: Arnold Mathematical Journal. - : Springer Science and Business Media LLC. - 2199-6792 .- 2199-6806. ; 7:3, s. 357-385
-
Tidskriftsartikel (refereegranskat)abstract
- Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for N= 2 , 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.
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| 58. |
- Nyagahakwa, Venuste, et al.
(författare)
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Sets with the Baire property in topologies formed from a given topology and ideals of sets
- 2017
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Ingår i: Questions and answers in General Topology. - Osaka : Symposium of General Topology / Iso Shinpojumu. - 0918-4732. ; 35:1, s. 59-76
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Tidskriftsartikel (refereegranskat)abstract
- Let X be a set, τ1, τ2 topologies on X and Bp(X, τi) the family of all subsets of X possessing the Baire property in (X, τi), i = 1, 2. In this paper we study conditions on τ1 and τ2 that imply a relationship (for example, inclusion or equality) between the families Bp(X, τ1) and Bp(X, τ2). We are mostly interested in the case where the topology τ2 is formed with the help of a local function defined by the topology τ1 and an ideal of sets I on X. We also consider several applications of the local function defined by the usual topology on the reals and the ideal of all meager sets there, for proving some known facts.
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| 59. |
- Rowlett, Julie, 1978, et al.
(författare)
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How to hear the corners of a drum
- 2019
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Ingår i: Matrix Annals. - Cham : Springer International Publishing. ; 2017, s. 243-278, s. 243-278
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Konferensbidrag (refereegranskat)abstract
- We prove that the existence of corners in a class of planar domain, which includes all simply connected polygonal domains and all smoothly bounded domains, is a spectral invariant of the Laplacian with both Neumann and Robin boundary conditions. The main ingredient in the proof is a locality principle in the spirit of Kac’s “principle of not feeling the boundary,” but which holds uniformly up to the boundary. Albeit previously known for Dirichlet boundary condition, this appears to be new for Robin and Neumann boundary conditions, in the geometric generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle are insufficient to obtain the analogous result. However, we describe how one may be able to harness powerful microlocal methods and combine these with the locality principles demonstrated here to show that corners are a spectral invariant; this is current work-in-progress (Nursultanov et al., Preprint).
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| 60. |
- Samuelsson Kalm, Håkan, 1976
(författare)
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The dbar-equation, duality, and holomorphic forms on a reduced complex space
- 2021
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 31:2, s. 1786-1820
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Tidskriftsartikel (refereegranskat)abstract
- We solve the $\bar\partial$-equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the partial derivative-equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.
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