arXiv subjects

Guido Kanschat

Publications and source records attributed to Guido Kanschat.

At least 19 recordsLinked to original sources

arXiv2025

A multigrid method for CutFEM and its implementation on GPU

We present a multigrid method for an unfitted finite element discretization of the Dirichlet boundary value problem. The discretization employs Nitsche's method to implement the boundary condition and additional face based ghost penalties for stabilization. We apply standard intergrid operators, relying on the fact that the relevant domain of computation does not grow under mesh refinement. The smoother is a parallel implementation of the multiplicative vertex-patch smoother with inconsistent treatment of ghost penalties. Our computational results show that we obtain a fast converging method. Furthermore, runtime comparison to fitted methods show that the losses are moderate although many optimizations for Cartesian vertex patches cannot be applied on cut patches.

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arXiv2024

Tensor-product vertex patch smoothers for biharmonic problems

We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations. However, in three-dimensional cases, the additive smoother outperforms its multiplicative counterpart due to the latter's lower potential for parallelism. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 percent.

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arXiv2024

Subsweep: Extensions to the Sweep method for radiative transfer

We introduce the radiative transfer postprocessing code Subsweep. The code is based on the method of transport sweeps, in which the exact solution to the scattering-less radiative transfer equation is computed in a single pass through the entire computational grid. The radiative transfer module is coupled to radiation chemistry, and chemical compositions as well as temperatures of the cells are evolved according to photon fluxes computed during radiative transfer. Subsweep extends the method of transport sweeps by incorporating sub-timesteps in a hierarchy of partial sweeps of the grid. This alleviates the need for a low, global timestep and as a result Subsweep is able to drastically reduce the amount of computation required for accurate integration of the coupled radiation chemistry equations. We succesfully apply the code to a number of physical tests such as the expansion of HII regions, the formation of shadows behind dense objects, and its behavior in the presence of periodic boundary conditions.

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arXiv2024

Multigrid methods for the Stokes problem on GPU systems

This paper presents a matrix-free multigrid method for solving the Stokes problem, discretized using $H^{\text{div}}$-conforming discontinuous Galerkin methods. We employ a Schur complement method combined with the fast diagonalization method for the efficient evaluation of the local solver within the multiplicative Schwarz smoother. This approach operates directly on both the velocity and pressure spaces, eliminating the need for a global Schur complement approximation. By leveraging the tensor product structure of Raviart-Thomas elements and an optimized, conflict-free shared memory access pattern, the matrix-free operator evaluation demonstrates excellent performance numbers, reaching over one billion degrees of freedom per second on a single NVIDIA A100 GPU. Numerical results indicate efficiency comparable to that of the three-dimensional Poisson problem.

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arXiv2024

Smoothers with localized residual computations for geometric multigrid methods

We improve the performance of multigrid solvers on many-core architectures with cache hierarchies by reorganizing operations in the smoothing step to minimize memory transfers. We focus on patch smoothers, which offer robust convergence rates with respect to the finite element degree for various equations, in the setting of multiplicative subspace correction for numerical efficiency. By combining the computation of local residuals with local solvers, we increase the locality of the problem and thus reduce data transfers. The thread-parallel implementation of this algorithm is based on coloring, which contradicts cache efficiency. We improve data locality by rearranging the loop into batches so that more data can be reused. The organization of consecutive batches prioritizes data locality.

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arXiv2024

An implementation of tensor product patch smoothers on GPU

We present a GPU implementation of vertex-patch smoothers for higher order finite element methods in two and three dimensions. Analysis shows that they are not memory bound with respect to GPU DRAM, but with respect to on-chip scratchpad memory. Multigrid operations are optimized through localization and reorganized local operations in on-chip memory, achieving minimal global data transfer and a conflict free memory access pattern. Performance tests demonstrate that the optimized kernel is at least 2 times faster than the straightforward implementation for the Poisson problem, across various polynomial degrees in 2D and 3D, achieving up to 36% of the peak performance in both single and double precision on Nvidia A100 GPU.

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arXiv2024

Multilevel Interior Penalty Methods on GPUs

We present a matrix-free multigrid method for high-order discontinuous Galerkin (DG) finite element methods with GPU acceleration. A performance analysis is conducted, comparing various data and compute layouts. Smoother implementations are optimized through localization and fast diagonalization techniques. Leveraging conflict-free access patterns in shared memory, arithmetic throughput of up to 39% of the peak performance on Nvidia A100 GPUs are achieved. Experimental results affirm the effectiveness of mixed-precision approaches and MPI parallelization in accelerating algorithms. Furthermore, an assessment of solver efficiency and robustness is provided across both two and three dimensions, with applications to Poisson problems.

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arXiv2024

Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions

This paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting $H^{div}$-conforming approximations of the solid displacement and fluid flux fields. For the resulting saddle-point problem, we construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms. Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method.

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arXiv2024

Homogeneous multigrid for hybrid discretizations: application to HHO methods

We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.

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arXiv2023

Ensemble Kalman Inversion for Image Guided Guide Wire Navigation in Vascular Systems

This paper addresses the challenging task of guide wire navigation in cardiovascular interventions, focusing on the parameter estimation of a guide wire system using Ensemble Kalman Inversion (EKI) with a subsampling technique. The EKI uses an ensemble of particles to estimate the unknown quantities. However since the data misfit has to be computed for each particle in each iteration, the EKI may become computationally infeasible in the case of high-dimensional data, e.g. high-resolution images. This issue can been addressed by randomised algorithms that utilize only a random subset of the data in each iteration. We introduce and analyse a subsampling technique for the EKI, which is based on a continuous-time representation of stochastic gradient methods and apply it to on the parameter estimation of our guide wire system. Numerical experiments with real data from a simplified test setting demonstrate the potential of the method.

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arXiv2023

Discrete tensor product BGG sequences: splines and finite elements

In this paper, we provide a systematic discretization of the Bernstein-Gelfand-Gelfand (BGG) diagrams and complexes over cubical meshes of arbitrary dimension via the use of tensor-product structures of one-dimensional piecewise-polynomial spaces, such as spline and finite element spaces. We demonstrate the construction of the Hessian, the elasticity, and the divdiv complexes as examples for our construction.

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arXiv2023

Homogeneous multigrid method for HDG applied to the Stokes equation

We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart--Thomas, or a hybrid Brezzi--Douglas--Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Numerical experiments underline our analytical findings.

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arXiv2022

The Sweep Method for radiative Transfer in Arepo

We introduce the radiative transfer code Sweep for the cosmological simulation suite Arepo. Sweep is a discrete ordinates method in which the radiative transfer equation is solved under the infinite speed of light, steady state assumption by a transport sweep across the entire computational grid. Since Arepo is based on an adaptive, unstructured grid, the dependency graph induced by the sweep dependencies of the grid cells is non-trivial. In order to solve the topological sorting problem in a distributed manner, we employ a task-based-parallelism approach. The main advantage of the sweep method is that the computational cost scales only with the size of the grid, and is independent of the number of sources or the distribution of sources in the computational domain, which is an advantage for radiative transfer in cosmological simulations, where there are large numbers of sparsely distributed sources. We successfully apply the code to a number of physical tests such as the expansion of HII regions, the formation of shadows behind dense objects, the scattering of light, as well as its behavior in the presence of periodic boundary conditions. In addition, we measure its computational performance with a focus on highly parallel, large-scale simulations.

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arXiv2022

A Tensor-Product Finite Element Cochain Complex with Arbitrary Continuity

We develop tensor product finite element cochain complexes of arbitrary smoothness on Cartesian meshes of arbitrary dimension. The first step is the construction of a one-dimensional $C^m$-conforming finite element cochain complex based on a modified Hermite interpolation operator, which is proved to commute with the exterior derivative by means of a general commutation lemma. Adhering to a strict tensor product construction we then derive finite element complexes in higher dimensions.

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arXiv2021

Partial differential equations on hypergraphs and networks of surfaces: derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).

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arXiv2021

Analysis of injection operators in multigrid solvers for hybridized discontinuous Galerkin methods

Uniform convergence of the geometric multigrid V-cycle is proven for HDG methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers which are bounded, convergent, and consistent. Elliptic regularity is used in the proofs. The new assumptions admit injection operators local to a single coarse grid cell. Examples for admissible injection operators are given. The analysis applies to the hybridized local discontinuous Galerkin method, hybridized Raviart-Thomas, and hybridized Brezzi-Douglas-Marini mixed element methods. Numerical experiments are provided to confirm the theoretical results.

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arXiv2021

Homogeneous multigrid for embedded discontinuous Galerkin methods

We introduce a homogeneous multigrid method in the sense that it uses the same embedded discontinuous Galerkin (EDG) discretization scheme for Poisson's equation on all levels. In particular, we use the injection operator developed in [LuRK2020] for HDG and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

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arXiv2020

HMG -- Homogeneous multigrid for HDG

We introduce a homogeneous multigrid method in the sense that it uses the same HDG discretization scheme for Poisson's equation on all levels. In particular, we construct a stable injection operator and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

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