Refine
Document Type
- Article (16)
- In Proceedings (11)
- ZIB-Report (4)
- Software (1)
Language
- English (32) (remove)
Is part of the Bibliography
- no (32)
Keywords
- flight planning (3)
- free flight (3)
- optimal control (3)
- shortest path (3)
- discrete optimization (2)
- discrete-continuous algorithm (2)
- Newton's method (1)
- global optimization (1)
- higher order finite elements (1)
- mixed precision (1)
Institute
- Modeling and Simulation of Complex Processes (32) (remove)
Solving PDEs on unstructured grids is a cornerstone of engineering and scientific computing. Heterogeneous parallel platforms, including CPUs, GPUs, and FPGAs, enable energy-efficient and computationally demanding simulations.
In this article, we introduce the HPM C++-embedded DSL that bridges the abstraction gap between the mathematical formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different programming models on the other hand.
Thus, the HPM DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HPM DSL, and demonstrate its usage with three examples. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters, is presented.
A code generator and a matching back end allow the acceleration of HPM code with GPUs. Finally, the achievable performance and scalability are demonstrated for different example problems.
Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster.
Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests.
These algorithms have applications in scientific computing, geographic information systems and computer-aided design.
With floating-point arithmetic, these geometric predicates can incur round-off errors that may lead to incorrect results and inconsistencies, causing computations to fail.
This issue has been addressed using a combination of exact arithmetic for robustness and floating-point filters to mitigate the computational cost of exact computations.
The implementation of exact computations and floating-point filters can be a difficult task, and code generation tools have been proposed to address this.
We present a new C++ meta-programming framework for the generation of fast, robust predicates for arbitrary geometric predicates based on polynomial expressions.
We combine and extend different approaches to filtering, branch reduction, and overflow avoidance that have previously been proposed.
We show examples of how this approach produces correct results for data sets that could lead to incorrect predicate results with naive implementations.
Our benchmark results demonstrate that our implementation surpasses state-of-the-art implementations.
Flight planning, the computation of optimal routes in view of flight time and fuel consumption under given weather conditions, is traditionally done by finding globally shortest paths in a predefined airway network. Free flight trajectories, not restricted to a network, have the potential to reduce the costs significantly, and can be computed using locally convergent continuous optimal control methods.
Hybrid methods that start with a discrete global search and refine with a fast continuous local optimization combine the best properties of both approaches, but rely on a good switchover, which requires error estimates for discrete paths relative to continuous trajectories.
Based on vertex density and local complete connectivity, we derive localized and a priori bounds for the flight time of discrete paths relative to the optimal continuous trajectory, and illustrate their properties on a set of benchmark problems. It turns out that localization improves the error bound by four orders of magnitude, but still leaves ample opportunities for tighter bounds using a posteriori error estimators.
We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.
We present an efficient algorithm that finds a globally optimal solution to the 2D Free Flight Trajectory Optimization Problem (aka Zermelo Navigation Problem) up to arbitrary precision in finite time. The algorithm combines a discrete and a continuous optimization phase. In the discrete phase, a set of candidate paths that densely covers the trajectory space is created on a directed auxiliary graph. Then Yen’s algorithm provides a promising set of discrete candidate paths which subsequently undergo a locally convergent refinement stage. Provided that the auxiliary graph is sufficiently dense, the method finds a path that lies within the convex domain around the global minimizer. From this starting point, the second stage will converge rapidly to the optimum. The density of the auxiliary graph depends solely on the wind field, and not on the accuracy of the
solution, such that the method inherits the superior asymptotic convergence properties of the optimal control stage.
Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton's method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.
We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our DisCOptER method discrete-continuous optimization for enhanced resolution) computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are goverened by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.
Globally optimal free flight trajectory optimization can be achieved with a combination of discrete and continuous optimization. A key requirement is that Newton's method for continuous optimization converges in a sufficiently large neighborhood around a minimizer. We show in this paper that, under certain assumptions, this is the case.
Convergence Properties of Newton's Method for Globally Optimal Free Flight Trajectory Optimization
(2023)
The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.
Convergence Properties of Newton’s Method for Globally Optimal Free Flight Trajectory Optimization
(2023)
The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.
CINDy: Conditional gradient-based Identification of Non-linear Dynamics – Noise-robust recovery
(2021)
Governing equations are essential to the study of nonlinear dynamics, often enabling the prediction of previously unseen behaviors as well as the inclusion into control strategies. The discovery of governing equations from data thus has the potential to transform data-rich fields where well-established dynamical models remain unknown. This work contributes to the recent trend in data-driven sparse identification of nonlinear dynamics of finding the best sparse fit to observational data in a large library of potential nonlinear models. We propose an efficient first-order Conditional Gradient algorithm for solving the underlying optimization problem. In comparison to the most prominent alternative algorithms, the new algorithm shows significantly improved performance on several essential issues like sparsity-induction, structure-preservation, noise robustness, and sample efficiency. We demonstrate these advantages on several dynamics from the field of synchronization, particle dynamics, and enzyme chemistry.
Efficient numerical methods for simulating cardiac electrophysiology with cellular resolution
(2023)
The cardiac extracellular-membrane-intracellular (EMI) model enables the precise geometrical representation and resolution of aggregates of individual myocytes. As a result, it not only yields more accurate simulations of cardiac excitation compared to homogenized models but also presents the challenge of solving much larger problems. In this paper, we introduce recent advancements in three key areas: (i) the creation of artificial, yet realistic grids, (ii) efficient higher-order time stepping achieved by combining low-overhead spatial adaptivity on the algebraic level with progressive spectral deferred correction methods, and (iii) substructuring domain decomposition preconditioners tailored to address the complexities of heterogeneous problem structures. The efficiency gains of these proposed methods are demonstrated through numerical results on cardiac meshes of different sizes.
Aims. Detection and quantification of myocardial scars are helpful both for diagnosis of heart diseases and for building personalized simulation models. Scar tissue is generally characterized by a different conduction of electrical excitation. We aim at estimating conductivity-related parameters from endocardial mapping data, in particular the conductivity tensor. Solving this inverse problem requires computationally expensive monodomain simulations on fine discretizations. Therefore, we aim at accelerating the estimation using a multilevel method combining electrophysiology models of different complexity, namely the monodomain and the eikonal model.
Methods. Distributed parameter estimation is performed by minimizing the misfit between simulated and measured electrical activity on the endocardial surface, subject to the monodomain model and regularization, leading to a constrained optimization problem. We formulate this optimization problem, including the modeling of scar tissue and different regularizations, and design an efficient iterative solver. We consider monodomain grid hierarchies and monodomain-eikonal model hierarchies in a recursive multilevel trust-region method.
Results. From several numerical examples, both the efficiency of the method and the estimation quality, depending on the data, are investigated. The multilevel solver is significantly faster than a comparable single level solver. Endocardial mapping data of realistic density appears to be just sufficient to provide quantitatively reasonable estimates of location, size, and shape of scars close to the endocardial surface.
Conclusion. In several situations, scar reconstruction based on eikonal and monodomain models differ significantly, suggesting the use of the more accurate but more expensive monodomain model for this purpose. Still, eikonal models can be utilized to accelerate the computations considerably, enabling the use of complex electrophysiology models for estimating myocardial scars from endocardial mapping data.
The electric conductivity of cardiac tissue determines excitation propagation and is important for quantifying ischemia and scar tissue and for building personalized models.
Estimating conductivity distributions from endocardial mapping data is a challenging inverse problem due to the computational complexity of the monodomain equation, which describes the cardiac excitation.
For computing a maximum posterior estimate, we investigate different optimization approaches based on adjoint gradient computation: steepest descent, limited memory BFGS, and recursive multilevel trust region methods, which are using mesh hierarchies or heterogeneous model hierarchies. We compare overall performance, asymptotic convergence rate, and pre-asymptotic progress on selected examples in order to assess the benefit of our multifidelity acceleration.
The locality of solution features in cardiac electrophysiology simulations calls for adaptive methods. Due to the overhead incurred by established mesh refinement and coarsening, however, such approaches failed in accelerating the computations. Here we investigate a different route to spatial adaptivity that is based on nested subset selection for algebraic degrees of freedom in spectral deferred correction methods. This combination of algebraic adaptivity and iterative solvers for higher order collocation time stepping realizes a multirate integration with minimal overhead. This leads to moderate but significant speedups in both monodomain and cell-by-cell models of cardiac excitation, as demonstrated at four numerical examples.
Recently developed Concentric Tube Continuum Robots (CTCRs) are widely exploited in, for example in minimally invasive surgeries which involve navigating inside narrow body cavities close to sensitive regions. These CTCRs can be controlled by extending and rotating the tubes in order to reach a target point or perform some task. The robot must deviate as little as possible from this narrow space and avoid damaging neighbouring tissue. We consider \emph{open-loop} optimal control of CTCRs parameterized over pseudo-time, primarily aiming at minimizing the robot's working volume during its motion. External loads acting on the system like tip loads or contact with tissues are not considered here. We also discussed the inclusion of tip's orientation in the optimal framework to perform some tasks. We recall a quaternion-based formulation of the robot configuration, discuss discretization, develop optimization objectives addressing different criteria, and investigate their impact on robot path planning for several numerical examples. This optimal framework can be applied to any backbone based continuum robots.
On the Accuracy of Eikonal Approximations in Cardiac Electrophysiology in the Presence of Fibrosis
(2023)
Fibrotic tissue is one of the main risk factors for cardiac arrhythmias. It is therefore a key component in computational studies. In this work, we compare the monodomain equation to two eikonal models for cardiac electrophysiology in the presence of fibrosis. We show that discontinuities in the conductivity field, due to the presence of fibrosis, introduce a delay in the activation times. The monodomain equation and eikonal-diffusion model correctly capture these delays, contrarily to the classical eikonal equation. Importantly, a coarse space discretization of the monodomain equation amplifies these delays, even after accounting for numerical error in conduction velocity. The numerical discretization may also introduce artificial conduction blocks and hence increase propagation complexity. Therefore, some care is required when comparing eikonal models to the discretized monodomain equation.
Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the \textsc{Dune} interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables.
We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way.
A Balancing Domain Decomposition by Constraints (BDDC) preconditioner is constructed and analyzed for the solution of hybrid Discontinuous Galerkin discretizations of reaction-diffusion systems of ordinary and partial differential equations arising in cardiac cell-by-cell models. The latter are different from the classical Bidomain and Monodomain cardiac models based on homogenized descriptions of the cardiac tissue at the macroscopic level, and therefore they allow the representation of individual cardiac cells, cell aggregates, damaged tissues and nonuniform distributions of ion channels on the cell membrane. The resulting discrete cell-by-cell models have discontinuous global solutions across the cell boundaries, hence the proposed BDDC preconditioner is based on appropriate dual and primal spaces with additional constraints which transfer information between cells (subdomains) without influencing the overall discontinuity of the global solution. A scalable convergence rate bound is proved for the resulting BDDC cell-by-cell preconditioned operator, while numerical tests validate this bound and investigate its dependence on the discretization parameters.