Allan Peter Engsig-Karup

E-mail: apek@dtu.dk

• 02614 at DTU: High-Performance Computing (Past teaching: GPU Computing)
• 02623 at DTU: The Finite Element Methods for Partial Differential Equations
• 02671 at DTU: Data-Driven Methods for Computational Science and Engineering (given in 2023, 2024 and 2025)
• 02687 at DTU: Scientific Computing for Ordinary and Partial Differential Equations
• 02689 at DTU: Advanced Numerical Methods for Differential Equations
• 02456 at DTU (projects): Deep Learning
• 01666 at DTU (projects): Project work - Bachelor of Mathematics and Technology
• 02122 at DTU (projects): Software Technology Project
• 02901 at DTU (projects): Advanced Topics in Machine Learning - Physics-Informed Machine Learning
• Final program of the summer school (Aug 19-23, 2024)

• Mathematics of Data Science Seminar Series

• Jens Visbech, PhD student/Inria Bordeaux/DTU Construct
• DTU Compute: Jens Visbech receives Elite Research Travel Grants 2025
• ForskerForum: Fra VVS-lærling til bølgeforsker (Danish article)
• Max Bitsch, Industrial PhD student, DTU Compute/DHI Group
• Freja Petersen, Industrial PhD student, DTU Compute/DHI Group
• Niels Skovgaard Jensen, Industrial PhD student, DTU Compute/Ticra

• Yuanxin Xia, PhD student, DTU Electro/DTU Compute
• Anton Sørensen, PhD student, DTU Sustain/DTU Compute
• Matteo Calafa, PhD student, DTU Electro/DTU Compute

We present a new efficient hybrid parameter estimation method based on the idea, that if nonlinear dynamic models are stated in terms of a system of equations that is linear in terms of the parameters, then regularized ordinary least squares can be used to estimate these parameters from time series data. We introduce the term "Physics-Informed Regression" (PIR) to describe the proposed data-driven hybrid technique as a way to bridge theory and data by use of ordinary least squares to efficiently perform parameter estimation of the model coefficients of different parameter-linear models; providing examples of models based on nonlinear ordinary equations (ODE) and partial differential equations (PDE). The focus is on parameter estimation on a selection of ODE and PDE models, each illustrating performance in different model characteristics. For two relevant epidemic models of different complexity and number of parameters, PIR is tested and compared against the related technique, physics-informed neural networks (PINN), both on synthetic data generated from known target parameters and on real public Danish time series data collected during the COVID-19 pandemic in Denmark. Both methods were able to estimate the target parameters, while PIR showed to perform noticeably better, especially on a compartment model with higher complexity. Given the difference in computational speed, it is concluded that the PIR method is superior to PINN for the models considered. It is also demonstrated how PIR can be applied to estimate the time-varying parameters of a compartment model that is fitted using real Danish data from the COVID-19 pandemic obtained during a period from 2020 to 2021. The study shows how data-driven and physics-informed techniques may support reliable and fast -- possibly real-time -- parameter estimation in parameter-linear nonlinear dynamic models.

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We present a σ-transformed fully nonlinear potential flow (FNPF) model implemented in the open-source spectral/hp element framework Nektar++. The solver splits the problem into (i) free-surface Zakharov equations and (ii) a σ-transformed Laplace continuity equation, coupled at run-time via the CWIPI code coupler. A linear/nonlinear operator splitting avoids re-factorizing the time-dependent Laplace matrix each step. Convergence, scaling, and three benchmarks (semicircular channel, Whalin shoal, and crescent waves) demonstrate accuracy and practicality.

Physics‑informed holomorphic neural networks (PIHNNs) have recently emerged as efficient surrogate models for solving differential problems. By embedding the underlying problem structure into the network, PIHNNs require training only to satisfy boundary conditions, often resulting in significantly improved accuracy and computational efficiency compared to traditional physics-informed neural networks (PINNs). In this work, we improve and extend the application of PIHNNs to two-dimensional problems. First, we introduce a novel holomorphic network architecture based on the Kolmogorov‑Arnold representation (PIHKAN), which achieves higher accuracy with reduced model complexity. Second, we develop mathematical extensions that broaden the applicability of PIHNNs to a wider class of elliptic partial differential equations, including the Helmholtz equation. Finally, we propose a new method based on Laurent series theory that enables the application of holomorphic networks to multiply‑connected plane domains, thereby removing the previous limitation to simply‑connected geometries.

This study numerically assesses the acoustic impact of different building façade design strategies for two sound source locations: one corresponding to road traffic noise and another to rooftop-mounted building services. Using a 3D nodal discontinuous Galerkin finite element method, the simulations capture wave phenomena at low frequencies more accurately than traditional methods. Results show reflections in street canyons can increase rooftop-source noise by up to 12 dB. Balconies exhibited screening or amplifying effects between −3.5 dB and +6.3 dB depending on source placement. Rooftop noise was heavily influenced by façade tilting (−8.4 dB to +4.6 dB), while balconies had minor effects (~±1 dB). The numerical results were validated experimentally with a scale model test. :contentReference[oaicite:1]{index=1}

We present a new parallel spectral element solver, FNPF-SEM, for simulating linear and fully nonlinear potential flow-based water waves and their interaction with offshore structures. The tool is designed as a general-purpose wave model for offshore engineering applications. Built within the open-source framework Firedrake, the new FNPF-SEM model is designed as a computational tool capable of capturing both linear and nonlinear wave phenomena with high accuracy and efficiency, with support for high-order (spectral) finite elements. Additionally, Firedrake provides native support for MPI-based parallelism, allowing for efficient multi-CPU distributed computations needed for large-scale simulations. We demonstrate the capabilities of the high-order spectral element model through h- and p-convergence studies, and weak and strong scaling tests. Validation is performed against analytical solutions and experimental data for several benchmark cases, including nonlinear high-order harmonic generation and linear and nonlinear wave interactions with a cylinder and a breakwater. The new FNPF-SEM model offers a numerical framework for simulating wave propagation and wave-structure interactions, with the following key features: i) the ability to represent complex geometries through flexible, unstructured finite element meshes; ii) reduced numerical diffusion and dispersion by using high-order polynomial expansions; and iii) scalability to full- and large-scale simulations over long time periods through a parallel implementation.

We present a new high-order accurate computational fluid dynamics model based on the incompressible Navier–Stokes equations with a free surface for the accurate simulation of non-linear and dispersive water waves in the time domain. The spatial discretization is based on Chebyshev polynomials in the vertical direction and a Fourier basis in the horizontal direction, allowing for the use of the fast Chebyshev and Fourier transforms for the efficient computation of spatial derivatives. The temporal discretization is done through a generalized low-storage explicit fourth-order Runge–Kutta, and for the scheme to conserve mass and achieve high-order accuracy, a velocity-pressure coupling needs to be satisfied at all Runge–Kutta stages. This results in the emergence of a Poisson pressure problem that constitutes a geometric conservation law for mass conservation. The occurring Poisson problem is proposed to be solved efficiently via an accelerated iterative solver based on a geometric -multigrid scheme, which takes advantage of the high-order polynomial basis in the spatial discretization and hence distinguishes itself from conventional low-order numerical schemes. We present numerical experiments for validation of the scheme in the context of numerical wave tanks demonstrating that the -multigrid accelerated numerical scheme can effectively solve the Poisson problem that constitute the computational bottleneck, that the model can achieve the desired spectral convergence, and is capable of simulating wave-propagation over non-flat bottoms with excellent agreement in comparison to experimental results.

Finding the focus of bacterial infections can be challenging, especially for hospitalised patients. Conventional microbiological diagnostic methods are either time consuming, expensive, or difficult to interpret due to contamination. The aim of this study was to apply machine learning (ML) to reliably predict the focus of bacterial infections. This study utilised a dataset including samples from 10,153 patients, collected from November 1, 2019 to June 3, 2023 at Rigshospitalet, Denmark. The dataset contains microbiological findings, biochemical data, and vital parameters. The dataset was analysed using ML. The ML-outputs were calibrated using Venn-ABERS calibration and model uncertainty was addressed using conformal risk control. The best performing model was the XGBoost model achieving an AUC of 0.93 ± 0.051 (mean ± SD.). Combining the model with methods from the conformal prediction framework achieves predictive capabilities that surpasses similar studies, while also accounting for model uncertainty by providing statistically robust uncertainty estimates.

This paper describes a new high-order composite numerical model for simulating moored floating offshore bodies. We focus on a floating offshore wind turbine and its static equilibrium and free decay. The composite scheme models linear to weakly nonlinear motions in the time domain by solving the Cummins equations. Mooring forces are acquired from a discontinuous Galerkin finite element solver. Linear hydrodynamic coefficients are computed by solving a pseudo-impulsive radiation problem in three dimensions using a spectral element method. Numerical simulations of a moored model-scale floating offshore wind turbine were performed and compared with experimental measurements for validation, ultimately showing a fair agreement.

To improve both scale and fidelity of numerical water wave simulations to study the evolution of wave fields within offshore engineering, it is of key practical interest to achieve high numerical efficiency. We propose a p-multigrid accelerated time-domain scheme for efficient and -scalable solution of a hybrid-spectral model for the simulation of highly nonlinear and highly dispersive water waves, the accurate calculation of wave kinematics and taking into account varying water depth. To achieve low numerical dispersive and dissipate errors, a high-order hybrid-spectral collocation scheme is implemented to solve the fully nonlinear potential flow (FNPF) equations, and utilizing a p-multigrid iterative solver scheme for solving the Laplace problem. Hereby, the numerical scheme combines the high accuracy of spectral methods, high efficiency of the fast Fourier transform (FFT) algorithm, and multigrid methods. Numerical analysis is performed to evaluate the performance and confirm the spectral accuracy of the scheme. Numerical experiments are considered for steady nonlinear wave propagation. A Fourier-continuation technique is used to extend the scheme from a periodic domain setup to perform benchmarks in a finite domain setup for numerical wave tanks utilizing conventional techniques for wave generation and absorption, and with results in excellent agreement with experimental measurements.

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov-Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

We present a general multi-fidelity (MF) framework which is applied through utilizing flexible-order explicit finite difference numerical schemes using convolutional neural networks (CNNs) by combining low-order simulation data with higher order simulation data obtained from numerical simulations based on partial differential equations (PDEs). This allows for improving the performance of low-order numerical simulation through learning from the data how to correct the numerical schemes to achieve improved accuracy. Through the lens of numerical analysis we evaluate the accuracy, efficiency and generalizability of constructed data-driven MF-models. To illustrate the concept, the construction of the MF models uses CNNs and is evaluated for numerical schemes designed for solving linear PDEs; the heat, the linear advection equation and linearized 1D shallow water equations. The numerical schemes allow for a high level of explainability of data-driven correction terms obtained via CNNs through numerical analysis of truncation errors. It is demonstrated that data-driven MF models is a means to improve the accuracy of LF models through operator correction.

This paper presents a novel, efficient, high-order accurate, and stable spectral element-based model for computing the complete three-dimensional linear radiation and diffraction problem for floating offshore structures. We present a solution to a pseudo-impulsive formulation in the time domain, where the frequency-dependent quantities, such as added mass, radiation damping, and wave excitation force for arbitrary heading angle, , are evaluated using Fourier transforms from the tailored time-domain responses. The spatial domain is tessellated by an unstructured high-order hybrid configured mesh and with solutions represented by piece-wise polynomial basis functions in the spectral element space. Fourth-order accurate time integration is employed, making the entire numerical scheme a high-order scheme. The model can use symmetry boundaries in the spatial representation to reduce the computational burden. The key piece of the numerical model – the discrete Laplace solver – is verified through - and -convergence studies. Moreover, to highlight the capabilities of the proposed model, we present proof-of-concept examples of simple floating bodies (a hemisphere and a box). Also, an oscillating water column is considered, including generalized modes representing the piston motion and wave sloshing effects inside the wave energy converter chamber. In this case, the spectral element model trivially computes the infinite-frequency added mass, which is a singular problem for conventional boundary element-type solvers. The proposed model serves its practical purpose within the field of offshore engineering for simulations of floating offshore wind turbines, wave energy converters, and much more.

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough to fulfil the Sommerfeld radiation condition. By strategically truncating the computational domain with a efficient boundary treatment, the computational cost is lowered. One commonly used boundary treatment is the perfectly matched layer (PML) that dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable and an optimization procedure of the damping functions is proposed to enhance the performance of the formulation.

We propose a new high-order finite difference numerical model for the simulation of nonlinear water waves and wave-structure interaction with fixed structures using the Navier-Stokes equations. The complete formulation is described in three spatial dimensions (3D) and preliminary validation results are here presented for two spatial dimensions (2D). A spatially fixed computational domain is defined through introducing a sigma-coordinate that transform the Navier–Stokes equations along the vertical dimension from the sea bed to the still water level. Numerical experiments highlights both the correctness of the solution, the high-order convergence property that is attractive for efficient solutions as well as the classical benchmark problem due to Beji & Battjes (1994) where experimental measurements are available and hence serve as validation of the new high-order numerical scheme.

We address the challenge of acoustic simulations in three-dimensional (3D) virtual rooms with parametric source positions, which have applications in virtual/augmented reality, game audio, and spatial computing. The wave equation can fully describe wave phenomena such as diffraction and interference. However, conventional numerical discretization methods are computationally expensive when simulating hundreds of source and receiver positions, making simulations with parametric source positions impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with parametric source positions, achieving millisecond-scale computations.

We present a massively parallel and scalable nodal discontinuous Galerkin finite element method (DGFEM) solver for the time-domain linearized acoustic wave equations. The solver is implemented using the libParanumal finite element framework with extensions to handle curvilinear geometries and frequency dependent boundary conditions of relevance in practical room acoustics. The implementation is benchmarked on heterogeneous multi-device many-core computing architectures, and high performance and scalability are demonstrated for a problem that is considered expensive to solve in practical applications. In a benchmark study, scaling tests show that multi-GPU support gives the ability to simulate large rooms, over a broad frequency range, with realistic boundary conditions, both in terms of computing time and memory requirements. Furthermore, numerical simulations on two non-trivial geometries are presented, a star-shaped room with a dome and an auditorium. Overall, this shows the viability of using a multi-device accelerated DGFEM solver to enable realistic large-scale wave-based room acoustics simulations.

We propose and demonstrate a new approach for fast and accurate surrogate modelling of urban drainage system hydraulics based on physics-guided machine learning. The surrogates are trained against a limited set of simulation results from a hydrodynamic (HiFi) model. Our approach reduces simulation times by one to two orders of magnitude compared to a HiFi model. It is thus slower than e.g. conceptual hydrological models, but it enables simulations of water levels, flows and surcharges in all nodes and links of a drainage network and thus largely preserves the level of detail provided by HiFi models.

The use of model-based numerical simulations of wave propagation in rooms for engineering applications requires that acoustic conditions for multiple parameters are evaluated iteratively, which is computationally expensive. We present a reduced basis method (RBM) to achieve a computational cost reduction relative to a traditional full-order model (FOM) for wave-based room acoustic simulations with parametrized boundaries. The FOM solver is based on the spectral-element method; however, other numerical methods could be applied. The RBM reduces the computational burden by solving the problem in a low-dimensional subspace for parametrized frequency-independent and frequency-dependent boundary conditions. The problem is formulated in the Laplace domain, which ensures the stability of the reduced-order model (ROM). We study the potential of the proposed RBM in terms of computational efficiency, accuracy, and storage requirements, and we show that the RBM leads to 100-fold speedups for a two-dimensional case and 1000-fold speedups for a three-dimensional case with an upper frequency of 2 and 1 kHz, respectively. While the FOM simulations needed to construct the ROM are expensive, we demonstrate that the ROM has the potential of being 3 orders of magnitude faster than the FOM when four different boundary conditions are simulated per room surface.

Realistic sound is essential in virtual environments, such as computer games and mixed reality. Efficient and accurate numerical methods for pre-calculating acoustics have been developed over the last decade; however, pre-calculating acoustics makes handling dynamic scenes with moving sources challenging, requiring intractable memory storage. A physics-informed neural network (PINN) method in one dimension is presented, which learns a compact and efficient surrogate model with parameterized moving Gaussian sources and impedance boundaries and satisfies a system of coupled equations. The model shows relative mean errors below 2%/0.2 dB and proposes a first step in developing PINNs for realistic three-dimensional scenes.

In marine offshore engineering, cost-efficient simulation of unsteady water waves and their nonlinear interaction with bodies are important to address a broad range of engineering applications at increasing fidelity and scale. We consider a fully nonlinear potential flow (FNPF) model discretized using a Galerkin spectral element method to serve as a basis for handling both wave propagation and wave-body interaction with high computational efficiency within a single modeling approach. We design and propose an efficient -scalable computational procedure based on geometric p-multigrid for solving the Laplace problem in the numerical scheme. The fluid volume and the geometric features of complex bodies is represented accurately using high-order polynomial basis functions and unstructured meshes with curvilinear prism elements. The new p-multigrid spectral element model can take advantage of the high-order polynomial basis and thereby avoid generating a hierarchy of geometric meshes with changing number of elements as required in geometric h-multigrid approaches.

During the COVID-19 pandemic, Denmark has pursued a mass testing strategy culminating in the testing of 12.167 individuals per 100,000 inhabitants per day during the spring of 2021. The strategy included free access to COVID-19 testing, and since 2021, compulsory documentation for negative tests or vaccination has been required for access to workplace, educational institutions, restaurants, and many other places. Testing and subsequent isolation if testing was positive were voluntary. The present study provides an analysis of whether testing frequency in Denmark showed any correlation to hospitalizations throughout the relevant stages of the pandemic. Mass testing was found not to correlate significantly with the number of hospitalizations during the pandemic. Interestingly, during the highest level of testing in spring 2021 the fraction of positive tests increased slightly; thus, the Danish mass testing strategy, at its best, failed to reduce the prevalence of COVID-19. Furthermore, the relationship between positives in antigen testing and in rt-PCR testing indicated that many patients were not tested early in their infection when the risk of transmission was at the highest. In conclusion, the Danish mass testing strategy for COVID-19 does not appear to have a detectable correlation to the number of hospitalizations due to COVID-19.

The focus is on the parallel scalability of a distributed multigrid framework, known as the DTU Compute GPUlab Library, for execution on graphics processing unit (GPU)-accelerated supercomputers. We demonstrate near-ideal weak scalability for a high-order fully nonlinear potential flow (FNPF) time domain model on the Oak Ridge Titan supercomputer, which is equipped with a large number of many-core CPU-GPU nodes. The high-order finite difference scheme for the solver is implemented to expose data locality and scalability, and the linear Laplace solver is based on an iterative multilevel preconditioned defect correction method designed for high-throughput processing and massive parallelism.

This paper presents a wave-based numerical scheme based on a spectral element method, coupled with an implicit-explicit Runge-Kutta time stepping method, for simulating room acoustics in the time domain. The scheme has certain features which make it highly attractive for room acoustic simulations, namely a) its low dispersion and dissipation properties due to a high-order spatio-temporal discretization, b) a high degree of geometric flexibility, where adaptive, unstructured meshes with curvilinear mesh elements are supported and c) its suitability for parallel implementation on modern many-core computer hardware. A method for modelling locally reacting, frequency dependent impedance boundary conditions within the scheme is developed, in which the boundary impedance is mapped to a multipole rational function and formulated in differential form. Various numerical experiments are presented, which reveal the accuracy and cost-efficiency of the proposed numerical scheme.

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

We present an arbitrary-order spectral element method for general-purpose simulation of non-overturning water waves, described by fully nonlinear potential theory. The method can be viewed as a high-order extension of the classical finite element method proposed by Cai et al. (1998) , although the numerical implementation differs greatly. Features of the proposed spectral element method include: nodal Lagrange basis functions, a general quadrature-free approach and gradient recovery using global projections. The quartic nonlinear terms present in the Zakharov form of the free surface conditions can cause severe aliasing problems and consequently numerical instability for marginally resolved or very steep waves. We show how the scheme can be stabilised through a combination of over-integration of the Galerkin projections and a mild spectral filtering on a per element basis. This effectively removes any aliasing driven instabilities while retaining the high-order accuracy of the numerical scheme. The additional computational cost of the over-integration is found insignificant compared to the cost of solving the Laplace problem.

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The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT decomposition and analyze its properties. We obtain results on the convergence of the decomposition, revealing links between the regularity of the function, the dimension of the input space, and the TT ranks. We also show that the regularity of the target function is preserved by the univariate functions (i.e., the “cores'') comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting spectral tensor-train decomposition combines the favorable dimension-scaling of the TT decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm TT-DMRG-cross to obtain the TT decomposition of tensors resulting from suitable discretizations of the target function. We assess the performance of the method on a range of numerical examples: a modified set of Genz functions with dimension up to 100, and functions with mixed Fourier modes or with local features. We observe significant improvements in performance over an anisotropic adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online (http://pypi.python.org/pypi/TensorToolbox/).

A feasibility study is presented on the effectiveness of applying nonlinear multigrid methods for efficient reservoir simulation of subsurface flow in porous media. A conventional strategy modeled after global linearization by means of Newton’s method is compared with an alternative strategy modeled after local linearization, leading to a nonlinear multigrid method in the form of the full-approximation scheme (FAS). It is demonstrated through numerical experiments that, without loss of robustness, the FAS method can outperform the conventional techniques in terms of algorithmic and numerical efficiency for a black-oil model. Furthermore, the use of the FAS method enables a significant reduction in memory usage compared with conventional techniques, which suggests new possibilities for improved large-scale reservoir simulation and numerical efficiency. Last, nonlinear multilevel preconditioning in the form of a hybrid-FAS/Newton strategy is demonstrated to increase robustness and efficiency.

A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and nonintrusive methods based on generalized polynomial chaos (PC). These methods allow us to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution at different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. Finally, we present a synthetic experiment studying the variance-based sensitivity of the wave load on an offshore structure to a number of input uncertainties. In the numerical examples presented the PC methods exhibit fast convergence, suggesting that the problem is amenable to analysis using such methods.

In this chapter, we use our library for heterogeneous and massively parallel GPU implementations. The library is written in Compute Unified Device Architecture (CUDA) C/C++ and a fully nonlinear and dispersive free surface water wave model is implemented. We describe how flexible-order finite difference (stencil) approximations to the partial differential equations of the model can be prototyped using library components provided in an in-house library. In this library hardware-specific implementation details are hidden.

Massively parallel processors, such as graphical processing units (GPUs), have in recent years proven to be effective for a vast amount of scientific applications. Today, most desktop computers are equipped with one or more powerful GPUs, offering heterogeneous high-performance computing to a broad range of scientific researchers and software developers. Though GPUs are now programmable and can be highly effective computing units, they still pose challenges for software developers to fully utilize their efficiency. Sequential legacy codes are not always easily parallelized, and the time spent on conversion might not pay off in the end. This is particular true for heterogeneous computers, where the architectural differences between the main and coprocessor can be so significant that they require completely different optimization strategies. The cache hierarchy management of CPUs and GPUs are an evident example hereof. In the past, industrial companies were able to boost application performance solely by upgrading their hardware systems, with an overt balance between investment and performance speedup. Today, the picture is different; not only do they have to invest in new hardware, but they also must account for the adaption and training of their software developers. What traditionally used to be a hardware problem, addressed by the chip manufacturers, has now become a software problem for application developers.

This paper investigates the numerical efficiency of defect correction methods preconditioned by iterative solvers for solving fully nonlinear water wave problems. The focus lies on improving the performance of solvers for the Laplace problem in a σ-transformed domain, commonly encountered in potential flow simulations of ocean waves. By analyzing various strategies within high-order numerical frameworks, the study provides insights into convergence behavior and computational efficiency relevant to marine and offshore engineering applications.

We implement and evaluate a massively parallel and scalable algorithm based on a multigrid preconditioned Defect Correction method for the simulation of fully nonlinear free surface flows. A dedicated numerical model based on the proposed algorithm is executed in parallel by utilizing affordable modern special purpose graphics processing unit (GPU). We describe and demonstrate how this approach makes it possible to do fast desktop computations for large nonlinear wave problems in numerical wave tanks (NWTs) with close to 50/100 million total grid points in double/single precision with 4 GB global device memory available. A new code base has been developed in C++ and compute unified device architecture C and is found to improve the runtime more than an order in magnitude in double precision arithmetic for the same accuracy over an existing CPU (single thread) Fortran 90 code when executed on a single modern GPU.

The flexible-order, finite difference based fully nonlinear potential flow model described in Bingham and Zhang (2007) is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss–Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature. The open source software developed in Fortran 90 by Allan Peter Engsig-Karup and examples presented in this work are available online (https://github.com/apengsigkarup/OceanWave3D-Fortran90).