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Item type:Publication, Access status: with files , Improving efficiency of parallel across the method spectral deferred corrections(Society for Industrial & Applied Mathematics (SIAM), 2025-02-11); ; Parallel across the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDCs) with a diagonal sweeper, closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency, and stability depend critically on the coefficients of the used SDC preconditioner. Previous approaches used numerical optimization to find good diagonal coefficients. Instead, we propose an approach that allows one to find optimal diagonal coefficients analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDCs, we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients, Picard iteration, and a fourth-order explicit as well as a fourth-order implicit diagonally implicit Runge-Kutta method.Publicationtype: Journal Article21 27 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Systematic construction of continuous-time neural networks for linear dynamical systems(Society for Industrial & Applied Mathematics (SIAM), 2025); ; ; Discovering a suitable neural network architecture for modeling complex dynamical systems poses a formidable challenge, often involving extensive trial and error and navigation through a high-dimensional hyperparameter space. In this paper, we discuss a systematic approach to constructing neural architectures for modeling a subclass of dynamical systems, namely, linear time-invariant (LTI) systems. We use a variant of continuous-time neural networks in which the output of each neuron evolves continuously as a solution of a first-order or second-order ordinary differential equation. Instead of deriving the network architecture and parameters from data, we propose a gradient-free algorithm to compute sparse architecture and network parameters directly from the given LTI system, leveraging its properties. We bring forth a novel neural architecture paradigm featuring horizontal hidden layers and provide insights into why employing conventional neural architectures with vertical hidden layers may not be favorable. We also provide an upper bound on the numerical errors of our neural networks. Finally, we demonstrate the high accuracy of our constructed networks on three numerical examples.Publicationtype: Journal Article51 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Super-localized orthogonal decomposition for high-frequency Helmholtz problems(Society for Industrial & Applied Mathematics (SIAM), 2024-07-18); ; We propose a novel variant of the Localized Orthogonal Decomposition (LOD) method for time-harmonic scattering problems of Helmholtz type with high wavenumber κ . On a coarse mesh of width H, the proposed method identifies local finite element source terms that yield rapidly decaying responses under the solution operator. They can be constructed to high accuracy from independent local snapshot solutions on patches of width ℓH and are used as problem-adapted basis functions in the method. In contrast to the classical LOD and other state-of-the-art multiscale methods, two- and three-dimensional numerical computations show that the localization error decays super-exponentially as the oversampling parameter ℓ is increased. This suggests that optimal convergence is observed under the substantially relaxed oversampling condition ℓ ≳ (log κ H )(d 1)/d with d denoting the spatial dimension. Numerical experiments demonstrate the significantly improved offline and online performance of the method also in the case of heterogeneous media and perfectly matched layers.Publicationtype: Journal Article18 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Spectral deferred correction methods for second-order problems(Society for Industrial & Applied Mathematics (SIAM), 2024-06-01); ; ; ; Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit, or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nystr\"om method.Publicationtype: Journal Article49 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A unified analysis framework for iterative parallel-in-time algorithms(2023-10); ; ; Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations and the convergence estimates that are available for some of them are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly and compare them directly numerically. Our framework also allows us to find new methods.Publicationtype: Journal ArticleScopus© Citations 2 234 207 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , NFFT.jl: Generic and fast Julia implementation of the nonequidistant fast fourier transform*(Society for Industrial & Applied Mathematics (SIAM), 2023); ; The nonequidistant fast Fourier transform (NFFT) is an extension of the famous fast Fourier transform (FFT) that can be applied to nonequidistantly sampled data in time/space or frequency domain. It is an approximative algorithm that allows one to control the approximation error in such a way that machine precision is reached while keeping the algorithmic complexity in the same order as a regular FFT. The NFFT plays a major role in many signal processing applications and has been intensively studied from a theoretical and computational perspective. The fastest CPU implementations of the NFFT are implemented in the low-level programming languages C and C++ and require a compromise between code generalizability, code readability, and code efficiency. The programming language Julia promises new opportunities in optimizing these three conflicting goals. In this work we show that Julia indeed allows one to develop an NFFT implementation which is completely generic and dimension-agnostic and requires about two to three times less code than the other famous libraries NFFT3 and FINUFFT while still being one of the fastest NFFT implementations developed to date.Publicationtype: Journal Article7 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A scalable algorithm for shape optimization with geometric constraints in Banach spacesThis work develops an algorithm for PDE-constrained shape optimization based on Lipschitz transformations. Building on previous work in this field, the p-Laplace operator is utilized to approximate a descent method for Lipschitz shapes. In particular, it is shown how geometric constraints are algorithmically incorporated avoiding penalty terms by assigning them to the subproblem of finding a suitable descent direction. A special focus is placed on the scalability of the proposed methods for large scale parallel computers via the application of multigrid solvers. The preservation of mesh quality under large deformations, where shape singularities have to be smoothed or generated within the optimization process, is also discussed. It is shown that the interaction of hierarchically refined grids and shape optimization can be realized by the choice of appropriate descent directions. The performance of the proposed methods is demonstrated for energy dissipation minimization in fluid dynamics applications.Publicationtype: Journal ArticleScopus© Citations 1 100 144 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , On the approximate solution and modeling of the kernel of nonlinear breakage population balance equation(2020-12); ; ; The study of collision-induced nonlinear breakage phenomenon is mostly unexplored but is important in the area of particulate processes. In this work, the volume and time dependent collisional breakage kernel function is modeled based on the population balance modeling approach. To solve the nonlinear breakage population balance equation, the weighted finite volume scheme for linear breakage process from Kumar, Saha, and Tsotsas [SIAM J. Numer. Anal., 53 (2015), pp. 1672-1689] is extended for the case of collision-induced breakage process. The weighted finite volume scheme is developed in such a way that it conserves the total mass of the system while preserving the total number of particles in the system. Moreover, an event-driven constant number Monte Carlo simulation algorithm is presented, and the simulation results are used as an alternative to experimental results. The volume dependency of the collisional breakage kernel is incorporated successfully in the Monte Carlo simulation for the first time while selecting particles for collision events. Some essential properties of any particulate process, such as the total number of particles and the size distribution of particles, are validated successfully for several breakage distribution functions using the Monte Carlo results. This offers new insights into the estimation and interpretation of collision-induced nonlinear breakage kinetics.Publicationtype: Journal Article118 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Iterative solution of saddle-point systems from radial basis function (RBF) interpolation(2019); Scattered data interpolation using conditionally positive definite radial basis functions typically leads to large, dense, and indefinite systems of saddle-point type. Due to ill-conditioning, the iterative solution of these systems requires an effective preconditioner. Using the technique of H -matrices, we propose, analyze, and compare two preconditioning approaches: transformation of the indefinite into a positive definite system using either Lagrangian augmentation or the nullspace method combined with subsequent H -Cholesky preconditioning. Numerical tests support the theoretical condition number estimates and illustrate the performance of the proposed preconditioners which are suitable for problems with up to N ≈ 40000 centers in two or three spatial dimensions.Publicationtype: Journal Article186 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Hierarchical Matrix Approximation for Kernel-Based Scattered Data Interpolation(2017-10-03); ; Scattered data interpolation by radial kernel functions leads to linear equation systems with large, fully populated, ill-conditioned interpolation matrices. A successful iterative solution of such a system requires an efficient matrix-vector multiplication as well as an efficient preconditioner. While multipole approaches provide a fast matrix-vector multiplication, they avoid the explicit setup of the system matrix which hinders the construction of preconditioners, such as approximate inverses or factorizations which typically require the explicit system matrix for their construction. In this paper, we propose an approach that allows both an efficient matrix-vector multiplication as well as an explicit matrix representation which can then be used to construct a preconditioner. In particular, the interpolation matrix will be represented in hierarchical matrix format, and several approaches for the blockwise low-rank approximation are proposed and compared, of both analytical nature (separable expansions) and algebraic nature (adaptive cross approximation). The validity of using an approximate system matrix in the iterative solution of the interpolation equations is demonstrated through a range of numerical experiments.Publicationtype: Journal Article129