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Container Adaptors
(1999)
The C++ standard template library has many useful containers for data. The standard library includes two adpators, queue, and stack. The authors have extended this model along the lines of relational database semantics. Sometimes the analogy is striking, and we will point it out occasionally. An adaptor allows the standard algorithms to be used on a subset or modification of the data without having to copy the data elements into a new container. The authors provide many useful adaptors which can be used together to produce interesting views of data in a container.
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.
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 highly localized dynamics of cardiac electrophysiology models call for adaptive simulation methods. Unfortunately, the overhead incurred by classical mesh adaptivity turns out to outweigh the performance improvements achieved by reducing the problem size. Here, we explore a different approach to adaptivity based on algebraic degree of freedom subset selection during spectral deferred correction sweeps, which realizes a kind of multirate higher order integration. Numerical experience indicates a significant performance increase compared to uniform simulations.
This C++ code implements a cell-by-cell model of cardiac excitation using a piecewise-continuous finite element discretization and spectral deferred correction time stepping. The code is based on the Kaskade 7 finite element toolbox and forms a prototype for the µCarp code to be implemented in the Microcard project.
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.
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.
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.
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.
Following axon pathfinding, growth cones transition from stochastic filopodial exploration to the formation of a limited number of synapses. How the interplay of filopodia and synapse assembly ensures robust connectivity in the brain has remained a challenging problem. Here, we developed a new 4D analysis method for filopodial dynamics and a data-driven computational model of synapse formation for R7 photoreceptor axons in developing Drosophila brains. Our live data support a 'serial synapse formation' model, where at any time point only a single 'synaptogenic' filopodium suppresses the synaptic competence of other filopodia through competition for synaptic seeding factors. Loss of the synaptic seeding factors Syd-1 and Liprin-α leads to a loss of this suppression, filopodial destabilization and reduced synapse formation, which is sufficient to cause the destabilization of entire axon terminals. Our model provides a filopodial 'winner-takes-all' mechanism that ensures the formation of an appropriate number of synapses.
Following axon pathfinding, growth cones transition from stochastic filopodial exploration to the formation of a limited number of synapses. How the interplay of filopodia and synapse assembly ensures robust connectivity in the brain has remained a challenging problem. Here, we developed a new 4D analysis method for filopodial dynamics and a data-driven computational model of synapse formation for R7 photoreceptor axons in developing Drosophila brains. Our live data support a 'serial synapse formation' model, where at any time point only a single 'synaptogenic' filopodium suppresses the synaptic competence of other filopodia through competition for synaptic seeding factors. Loss of the synaptic seeding factors Syd-1 and Liprin-α leads to a loss of this suppression, filopodial destabilization and reduced synapse formation, which is sufficient to cause the destabilization of entire axon terminals. Our model provides a filopodial 'winner-takes-all' mechanism that ensures the formation of an appropriate number of synapses.
The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.
We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.
Rectal temperature measurement (RTM) from crime scenes is an important parameter for temperature-based time of death estimation (TDE). Various influential variables exist in TDE methods like the uncertainty in thermal and environmental parameters. Although RTM depends in particular on the location of measurement position, this relationship has never been investigated separately. The presented study fills this gap using Finite Element (FE) simulations of body cooling. A manually meshed coarse human FE model and an FE geometry model developed from the CT scan of a male corpse are used for TDE sensitivity analysis. The coarse model is considered with and without a support structure of moist soil. As there is no clear definition of ideal rectal temperature measurement location for TDE, possible variations in RTM location (RTML) are considered based on anatomy and forensic practice. The maximum variation of TDE caused by RTML changes is investigated via FE simulation. Moreover, the influence of ambient temperature, of FE model change and of the models positioning on a wet soil underground are also discussed. As a general outcome, we notice that maximum TDE deviations of up to ca. 2-3 h due to RTML deviations have to be expected. The direction of maximum influence of RTML change on TDE generally was on the line caudal to cranial.
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.
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.
Temperature-based time of death estimation (TTDE) using simulation methods such as the finite element (FE) method promises higher accuracy and broader applicability in nonstandard cooling scenarios than established phenomenological methods. Their accuracy depends crucially on the simulation model to capture the actual situation. The model fidelity in turn hinges on the representation of the corpse’s anatomy in form of computational meshes as well as on the thermodynamic parameters.
While inaccuracies in anatomy representation due to coarse mesh resolution are known to have a minor impact on the estimated time of death, the sensitivity with respect to larger differences in the anatomy has so far not been studied. We assess this sensitivity by comparing four independently generated and vastly different anatomical models in terms of the estimated time of death in an identical cooling scenario. In order to isolate the impact of shape variation, the models are scaled to a reference size, and the possible impact of measurement location variation is excluded explicitly, which gives a lower bound on the impact of anatomy on the estimated time of death.
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.
Cardiac electrograms are an important tool to study the spread of excitation waves inside the heart, which in turn underlie muscle contraction. Electrograms can be used to analyse the dynamics of these waves, e.g. in fibrotic tissue. In computational models, these analyses can be done with greater detail than during minimally invasive in vivo procedures. Whilst homogenised models have been used to study electrogram genesis, such analyses have not yet been done in cellularly resolved models. Such high resolution may be required to develop a thorough understanding of the mechanisms behind abnormal excitation patterns leading to arrhythmias. In this study, we derived electrograms from an excitation propagation simulation in the Extracellular, Membrane, Intracellular (EMI) model, which represents these three domains explicitly in the mesh. We studied the effects of the microstructural excitation dynamics on electrogram genesis and morphology. We found that electrograms are sensitive to the myocyte alignment and connectivity, which translates into micro-fractionations in the electrograms.
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.
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 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.
Conduction velocity in cardiac tissue is a crucial electrophysiological parameter for arrhythmia vulnerability. Pathologically reduced conduction velocity facilitates arrhythmogenesis because such conduction velocities decrease the wavelength with which re-entry may occur. Computational studies on CV and how it changes regionally in models at spatial scales multiple times larger than actual cardiac cells exist. However, microscopic conduction within cells and between them have been studied less in simulations. In this work, we study the relation of microscopic conduction patterns and clinically observable macroscopic conduction using an extracellular-membrane-intracellular model which represents cardiac tissue with these subdomains at subcellular resolution. By considering cell arrangement and non-uniform gap junction distribution, it yields anisotropic excitation propagation. This novel kind of model can for example be used to understand how discontinuous conduction on the microscopic level affects fractionation of electrograms in healthy and fibrotic tissue. Along the membrane of a cell, we observed a continuously propagating activation wavefront. When transitioning from one cell to the neighbouring one, jumps in local activation times occurred, which led to lower global conduction velocities than locally within each cell.
Epidemiological models can not only be used to forecast the course of a pandemic like COVID-19, but also to propose and design non-pharmaceutical interventions such as school and work closing. In general, the design of optimal policies leads to nonlinear optimization problems that can be solved by numerical algorithms. Epidemiological models come in different complexities, ranging from systems of simple ordinary differential equations (ODEs) to complex agent-based models (ABMs). The former allow a fast and straightforward optimization, but are limited in accuracy, detail, and parameterization, while the latter can resolve spreading processes in detail, but are extremely expensive to optimize. We consider policy optimization in a prototypical situation modeled as both ODE and ABM, review numerical optimization approaches, and propose a heterogeneous multilevel approach based on combining a fine-resolution ABM and a coarse ODE model. Numerical experiments, in particular with respect to convergence speed, are given for illustrative examples.
Adaptive Gaussian Process Regression for Efficient Building of Surrogate Models in Inverse Problems
(2023)
In a task where many similar inverse problems must be solved, evaluating costly simulations is impractical. Therefore, replacing the model y with a surrogate model y(s) that can be evaluated quickly leads to a significant speedup. The approximation quality of the surrogate model depends strongly on the number, position, and accuracy of the sample points. With an additional finite computational budget, this leads to a problem of (computer) experimental design. In contrast to the selection of sample points, the trade-off between accuracy and effort has hardly been studied systematically. We therefore propose an adaptive algorithm to find an optimal design in terms of
position and accuracy. Pursuing a sequential design by incrementally appending the computational budget leads to a convex and constrained optimization problem. As a surrogate, we construct a Gaussian process regression model. We measure the global approximation error in terms of its impact on the accuracy of the identified parameter and aim for a uniform absolute tolerance, assuming that y(s) is computed by finite element calculations. A priori error estimates and a coarse estimate of computational effort relate the expected improvement of the surrogate model error to computational effort, resulting in the most efficient combination of sample point and evaluation tolerance. We also allow for improving the accuracy of already existing sample points by continuing previously truncated finite element solution procedures.