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Geometric Numerical Methods: From Random Fields to Shape Matching

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Titel: Geometric Numerical Methods: From Random Fields to Shape Matching
Autoren: Jansson, Erik, 1996
Schlagwörter: hydrodynamics, Stochastic partial differential equations, Lie–Poisson systems, shape analysis, geometric numerical integration, optimal transport, surface finite element methods, Gaussian random fields
Beschreibung: Geometry is central to many applied problems, though its influence varies. Some problems are inherently geometric, requiring numerical methods that preserve the underlying structure to remain accurate. Others are well understood in Euclidean space but demand different techniques when extended to curved settings. This thesis addresses such geometric challenges through studying numerical methods for two main types of problems: matching problems and stochastic (partial) differential equations. It is based on seven papers: the first three focus on SPDEs and SDEs, while the remaining consider matching problems and related differential equations. The first develops a numerical method for fractional SPDEs on the sphere, combining a recursive splitting scheme with surface finite elements. The second studies a Chebyshev–Galerkin approach for simulating non-stationary Gaussian random fields on hypersurfaces. The third introduces a geometric integrator for stochastic Lie–Poisson systems, derived via a reduction of the implicit midpoint method for canonical Hamiltonian systems. The fourth explores sub-Riemannian shape matching, where shapes are matched using constrained motions, and shows how this problem can be interpreted as a neural network. The fifth studies the convergence of a gradient flow for the Gaussian Monge problem. The sixth adapts geometric shape matching to recover protein conformations from single-particle Cryo-EM data by using rigid deformations of chains of particles. The seventh investigates the numerical signature of blow-up in hydrodynamic equations, showing that numerical solutions can be used to detect the onset in a class of hydrodynamic equations.
Dateibeschreibung: electronic
Zugangs-URL: https://research.chalmers.se/publication/545974
https://research.chalmers.se/publication/545974/file/545974_Fulltext.pdf
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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Geometric Numerical Methods: From Random Fields to Shape Matching
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Jansson%2C+Erik%22">Jansson, Erik</searchLink>, 1996
– Name: Subject
  Label: Subject Terms
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22hydrodynamics%22">hydrodynamics</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+partial+differential+equations%22">Stochastic partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Lie–Poisson+systems%22">Lie–Poisson systems</searchLink><br /><searchLink fieldCode="DE" term="%22shape+analysis%22">shape analysis</searchLink><br /><searchLink fieldCode="DE" term="%22geometric+numerical+integration%22">geometric numerical integration</searchLink><br /><searchLink fieldCode="DE" term="%22optimal+transport%22">optimal transport</searchLink><br /><searchLink fieldCode="DE" term="%22surface+finite+element+methods%22">surface finite element methods</searchLink><br /><searchLink fieldCode="DE" term="%22Gaussian+random+fields%22">Gaussian random fields</searchLink>
– Name: Abstract
  Label: Description
  Group: Ab
  Data: Geometry is central to many applied problems, though its influence varies. Some problems are inherently geometric, requiring numerical methods that preserve the underlying structure to remain accurate. Others are well understood in Euclidean space but demand different techniques when extended to curved settings. This thesis addresses such geometric challenges through studying numerical methods for two main types of problems: matching problems and stochastic (partial) differential equations. It is based on seven papers: the first three focus on SPDEs and SDEs, while the remaining consider matching problems and related differential equations. The first develops a numerical method for fractional SPDEs on the sphere, combining a recursive splitting scheme with surface finite elements. The second studies a Chebyshev–Galerkin approach for simulating non-stationary Gaussian random fields on hypersurfaces. The third introduces a geometric integrator for stochastic Lie–Poisson systems, derived via a reduction of the implicit midpoint method for canonical Hamiltonian systems. The fourth explores sub-Riemannian shape matching, where shapes are matched using constrained motions, and shows how this problem can be interpreted as a neural network. The fifth studies the convergence of a gradient flow for the Gaussian Monge problem. The sixth adapts geometric shape matching to recover protein conformations from single-particle Cryo-EM data by using rigid deformations of chains of particles. The seventh investigates the numerical signature of blow-up in hydrodynamic equations, showing that numerical solutions can be used to detect the onset in a class of hydrodynamic equations.
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      – Text: English
    Subjects:
      – SubjectFull: hydrodynamics
        Type: general
      – SubjectFull: Stochastic partial differential equations
        Type: general
      – SubjectFull: Lie–Poisson systems
        Type: general
      – SubjectFull: shape analysis
        Type: general
      – SubjectFull: geometric numerical integration
        Type: general
      – SubjectFull: optimal transport
        Type: general
      – SubjectFull: surface finite element methods
        Type: general
      – SubjectFull: Gaussian random fields
        Type: general
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      – TitleFull: Geometric Numerical Methods: From Random Fields to Shape Matching
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          Name:
            NameFull: Jansson, Erik
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
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