The workshop’s scope is given by challenges and recent advances in numerical methods addressing computational tasks in numerical differential geometry and beyond. The participants will elaborate on complementary perspectives and methods as well as on connections to applications in bio-physics, systems biology, and computational physics.

Workshop attendance is only possible upon invitation by the organizer.

There are 5 talks planned from morning to early afternoon. Please see the schedule below. All talks are available via zoom: link

Program

TUESDAY, September 12, 2023
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[ 09:15 – 09:30 CET ]

Welcome and Introduction by Michael Hecht, Young Investigator Group Leader, CASUS/HZDR

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[ 9:30 – 10:00 CET ]

Grady Wright, Professor, Dept. Mathematics, Boise State University, Boise, Idaho, USA

Title: Recent advances in kernel methods for problems posed on surfaces

Abstract: Kernel approximation methods, such as radial basis functions, are advantageous for a wide range of applications that involve analyzing/synthesizing scattered data, or numerically solving partial differential equations (PDEs) on geometrically difficult domains.  Although initially considered for Euclidean domains, there has been much recent work on kernel methods for various problems posed on surfaces embedded in Euclidean space.  We survey some of these recent advances with a specific focus on localized kernel methods for solving surface PDEs, meshfree multilevel solvers for linear systems that arise from discretizing elliptic surface PDEs, and implicit surface (level-set) reconstruction techniques from point cloud data.

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[ 10:00 – 10:30 CET ]

Jan Glaubitz, PhD; Postdoctoral Associate, Department of Aeronautics and Astronautics Massachusetts Institute of Technology Cambridge, MA, USA

Title: Stability of radial basis function methods for hyperbolic conservation laws

Abstract: Radial basis function (RBF) methods have emerged as a potent tool in numerical analysis, consistently showcasing commendable performance across diverse simulations. However, their stability for hyperbolic conservation laws remains sparsely studied. The incorporation of boundary conditions often exacerbates stability concerns. This talk delves into strategies to achieve provable stability for RBF methods. We introduce a novel approach, leveraging recent advancements for summation-by-parts operators—traditionally utilized in finite difference and finite element domains—to shed new light on the stability of RBF methods.

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[ 11:00 – 11:30 CET ]

Jonah Reeger, PhD, Assistant Professor, Air Force Institute of Technology Wright-Patterson Air Force Base, USA

Title: Adaptivity in local kernel based methods for approximating the action of linear operators

Abstract: Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations is developed. This estimate is useful for determining locations in the solution domain where increased node density (equivalently, reduction in the spacing between nodes) can decrease the error in the solution. An adaptive procedure for adding nodes to the domain for both the approximation of derivatives and the approximate evaluation of definite integrals is described. This method efficiently computes the error estimate at a set of prescribed points and adds new nodes for approximation where the error is too large. Computational experiments demonstrate close agreement between the error estimate and actual absolute error in the approximation. Such methods are necessary or desirable when approximating solutions to PDE (or in the case of quadrature/cubature), where the initial data and subsequent solution (or integrand) exhibit localized features that require significant refinement to resolve and where uniform increases in the density of nodes accross the entire computational domain is not possible or too burdensome.

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[ 11:30 – 12:00 CET ]

Daniel Fortunato, PhD, Associate Research Scientist, Center for Computational Mathematics, Flatiron Institute, NYC, USA

Title: A high-order fast direct solver for surface PDEs

Abstract: We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré–Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in O(N log N) operations for a mesh with N degrees of freedom. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 17 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction–diffusion systems.

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[ 14:00 – 14:30 CET ]

Sachin K. Thekke Veetti, PhD, Postdoctoral researcher, Center for Systems Biology Dresden (CSBD)
Michael Hecht, PhD, Young Investigator Group Leader, CASUS/HZDR

Title: Global polynomial level sets for numerical differential geometry of smooth closed surfaces

Abstract: We present a computational scheme that derives a global polynomial level set param- etrization for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth surfaces by affine algebraic varieties. From such a global polynomial level set parametrization, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even fourth-order terms such as the Laplacian of mean curvature are approximated with high precision. The accuracy performance results in a gain of computational efficiency, significantly reducing the number of surface points required compared to classic alternatives that rely on surface meshes or embedding grids. We mathematically derive and empirically demonstrate the strengths and the limitations of the present approach, suggesting it to be applicable to a large number of computational tasks in numerical differential geometry.

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 Organizer

●     Dr. Michael Hecht, Young Investigator Group Leader, CASUS/HZDR