arXiv subjects

Rolf Krause

Publications and source records attributed to Rolf Krause.

At least 37 records · Page 2Linked to original sources

arXiv2024

An eikonal model with re-excitability for fast simulations in cardiac electrophysiology

Precision cardiology based on cardiac digital twins requires accurate simulations of cardiac arrhythmias. However, detailed models, such as the monodomain model, are computationally costly and have limited applicability in practice. Thus, it desirable to have fast models that can still represent the main physiological features presented during cardiac arrhythmias. The eikonal model is an approximation of the monodomain model that is widely used to describe the arrival times of the electrical wave. However, the standard eikonal model does not generalize to the complex re-entrant dynamics that characterize the cardiac arrhythmias. In this work, we propose an eikonal model that includes the tissue re-excitability, which allows to describe re-entries. The re-excitability properties are inferred from the monodomain model. Our eikonal model also handles the tissue anisotropy and heterogeneity. We compare the eikonal model to the monodomain model in various numerical experiments in the atria and the ventricles. The eikonal model is qualitatively accurate in the simulation of re-entries and can be potentially ran in real-time, opening the door to its clinical applicability.

cs.CE↗

arXiv2024

Symbol-based multilevel block $τ$ preconditioners for multilevel block Toeplitz systems: GLT-based analysis and applications

In recent years, there has been a renewed interest in preconditioning for multilevel Toeplitz systems, a research field that has been extensively explored over the past several decades. This work introduces novel preconditioning strategies using multilevel $τ$ matrices for both symmetric and nonsymmetric multilevel Toeplitz systems. Our proposals constitute a general framework, as they are constructed solely based on the generating function of the multilevel Toeplitz coefficient matrix, when it can be defined. We begin with nonsymmetric systems, where we employ a symmetrization technique by permuting the coefficient matrix to produce a real symmetric multilevel Hankel structure. We propose a multilevel $τ$ preconditioner tailored to the symmetrized system and prove that the eigenvalues of the preconditioned matrix sequence cluster at $\pm 1$, leading to rapid convergence when using the preconditioned minimal residual method. The high effectiveness of this approach is demonstrated through its application in solving space fractional diffusion equations. Next, for symmetric systems we introduce another multilevel $τ$ preconditioner and show that the preconditioned conjugate gradient method can achieve an optimal convergence rate, namely a rate that is independent of the matrix size, when employed for a class of ill-conditioned multilevel Toeplitz systems. Numerical examples are provided to critically assess the effectiveness of our proposed preconditioners compared to several leading existing preconditioned solvers, highlighting their superior performance.

math.NA↗

arXiv2024

A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $\ell^2$--norm on the fidelity term.

math.NA↗

arXiv2024

Two-level trust-region method with random subspaces

We introduce a two-level trust-region method (TLTR) for solving unconstrained nonlinear optimization problems. Our method uses a composite iteration step, which is based on two distinct search directions. The first search direction is obtained through minimization in the full/high-resolution space, ensuring global convergence to a critical point. The second search direction is obtained through minimization in the randomly generated subspace, which, in turn, allows for convergence acceleration. The efficiency of the proposed TLTR method is demonstrated through numerical experiments in the field of machine learning

math.NA↗

arXiv2024

The limitations of a standard phase-field model in reproducing jointing in sedimentary rock layers

Geological applications of phase-field methods for fracture are notably scarce. This work conducts a numerical examination of the applicability of standard phase-field models in reproducing jointing within sedimentary layers. We explore how the volumetric-deviatoric split alongside the AT1 and AT2 phase-field formulations have several advantages in simulating jointing, but also have intrinsic limitations that prevent a reliable quantitative analysis of rock fracture. The formulations qualitatively reproduce the process of joint saturation, along with the negative correlation between joint spacing and the height of the sedimentary layer. However, in quantitative comparison to alternative numerical methods and outcrop observations, the phase-field method overestimates joint spacings by a factor of 2 and induces unrealistic compressive fractures in the AT1 model, alongside premature shearing at layer interfaces for the AT2 model. The causes are identified to be intrinsic to the phase-field lengthscale and the unsuitable strength envelope arising from the Volumetric-Deviatoric split. Finally, our analysis elucidates on the phase-field lengthscale's distortion of the stress field around dilating fractures, causing the sedimentary layer to reach joint saturation prematurely, thereby stopping the nucleation of new fractures and leading to larger joint spacings than in natural examples. Decreasing the lengthscale results in gradual improvement but becomes prohibitively computationally expensive for very small lengthscales such that the limit of ``natural'' behavior was not reached in this study. Overall, our results motivate the development of constitutive phase-field models that are more suitable for geological applications and their benchmarking against geological observations.

physics.geo-ph↗

arXiv2024

Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $μ$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

math.NA↗

arXiv2024

Explicit stabilized multirate methods for the monodomain model in cardiac electrophysiology

Fully explicit stabilized multirate (mRKC) methods are well-suited for the numerical solution of large multiscale systems of stiff ordinary differential equations thanks to their improved stability properties. To demonstrate their efficiency for the numerical solution of stiff, multiscale, nonlinear parabolic PDE's, we apply mRKC methods to the monodomain equation from cardiac electrophysiology. In doing so, we propose an improved version, specifically tailored to the monodomain model, which leads to the explicit exponential multirate stabilized (emRKC) method. Several numerical experiments are conducted to evaluate the efficiency of both mRKC and emRKC, while taking into account different finite element meshes (structured and unstructured) and realistic ionic models. The new emRKC method typically outperforms a standard implicit-explicit baseline method for cardiac electrophysiology. Code profiling and strong scalability results further demonstrate that emRKC is faster and inherently parallel without sacrificing accuracy.

math.NA↗

arXiv2024

Fractional differential problems with numerical anti-reflective boundary conditions: a computational/precision analysis and numerical results

Twenty years ago the anti-reflective numerical boundary conditions (BCs) were introduced in a context of signal processing and imaging, for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here for quality of reconstruction we mean a better accuracy and the elimination of boundary artifacts, called ringing effects. Now we propose numerical anti-reflective BCs in the context of nonlocal problems of fractional differential type: the goals are the same i.e. a smaller approximation error and the reduction of boundary artifacts. In the latter setting, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflective BCs, we compare them from the perspective of computational efficiency, by considering nontruncated and truncated versions and also other standard numerical BCs such as reflective/Neumann. Several numerical tests, tables, and visualizations are provided and critically discussed. The conclusion is that the truncated numerical anti-reflective BCs perform better, both in terms of accuracy and low computational cost.

math.NA↗

arXiv2024

High-order parallel-in-time method for the monodomain equation in cardiac electrophysiology

Simulation of the monodomain equation, crucial for modeling the heart's electrical activity, faces scalability limits when traditional numerical methods only parallelize in space. To optimize the use of large multi-processor computers by distributing the computational load more effectively, time parallelization is essential. We introduce a high-order parallel-in-time method addressing the substantial computational challenges posed by the stiff, multiscale, and nonlinear nature of cardiac dynamics. Our method combines the semi-implicit and exponential spectral deferred correction methods, yielding a hybrid method that is extended to parallel-in-time employing the PFASST framework. We thoroughly evaluate the stability, accuracy, and robustness of the proposed parallel-in-time method through extensive numerical experiments, using practical ionic models such as the ten-Tusscher-Panfilov. The results underscore the method's potential to significantly enhance real-time and high-fidelity simulations in biomedical research and clinical applications.

math.NA↗

arXiv2024

Shape of my heart: Cardiac models through learned signed distance functions

The efficient construction of anatomical models is one of the major challenges of patient-specific in-silico models of the human heart. Current methods frequently rely on linear statistical models, allowing no advanced topological changes, or requiring medical image segmentation followed by a meshing pipeline, which strongly depends on image resolution, quality, and modality. These approaches are therefore limited in their transferability to other imaging domains. In this work, the cardiac shape is reconstructed by means of three-dimensional deep signed distance functions with Lipschitz regularity. For this purpose, the shapes of cardiac MRI reconstructions are learned to model the spatial relation of multiple chambers. We demonstrate that this approach is also capable of reconstructing anatomical models from partial data, such as point clouds from a single ventricle, or modalities different from the trained MRI, such as the electroanatomical mapping (EAM).

eess.IV↗

arXiv2023

Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies

We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative preconditioners significantly improve the convergence of the standard L-BFGS optimizer, while providing more accurate solutions of the underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism.

math.NA↗

arXiv2023

Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods

We propose to train neural networks (NNs) using a novel variant of the ``Additively Preconditioned Trust-region Strategy'' (APTS). The proposed method is based on a parallelizable additive domain decomposition approach applied to the neural network's parameters. Built upon the TR framework, the APTS method ensures global convergence towards a minimizer. Moreover, it eliminates the need for computationally expensive hyper-parameter tuning, as the TR algorithm automatically determines the step size in each iteration. We demonstrate the capabilities, strengths, and limitations of the proposed APTS training method by performing a series of numerical experiments. The presented numerical study includes a comparison with widely used training methods such as SGD, Adam, LBFGS, and the standard TR method.

math.NA↗

arXiv2023

Assessment of the CRD approximation for the observer's frame RIII redistribution matrix

Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer frame (RIII CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. In this work, we aim at quantitatively assessing the impact and the range of validity of the RIII CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII CRD in realistic 1D atmospheric models. We focus on the chromospheric Ca i line at 4227 A and on the photospheric Sr i line at 4607 A.

astro-ph.SR↗

arXiv2023

Boundary Integral Formulation of the Cell-by-Cell Model of Cardiac Electrophysiology

We propose a boundary element method for the accurate solution of the cell-by-cell bidomain model of electrophysiology. The cell-by-cell model, also called Extracellular-Membrane-Intracellular (EMI) model, is a system of reaction-diffusion equations describing the evolution of the electric potential within each domain: intra- and extra-cellular space and the cellular membrane. The system is parabolic but degenerate because the time derivative is only in the membrane domain. In this work, we adopt a boundary-integral formulation for removing the degeneracy in the system and recast it to a parabolic equation on the membrane. The formulation is also numerically advantageous since the number of degrees of freedom is sensibly reduced compared to the original model. Specifically, we prove that the boundary-element discretization of the EMI model is equivalent to a system of ordinary differential equations, and we consider a time discretization based on the multirate explicit stabilized Runge-Kutta method. We numerically show that our scheme convergences exponentially in space for the single-cell case. We finally provide several numerical experiments of biological interest.

math.NA↗

arXiv2022

Nonlinear Schwarz preconditioning for nonlinear optimization problems with bound constraints

We propose a nonlinear additive Schwarz method for solving nonlinear optimization problems with bound constraints. Our method is used as a "right-preconditioner" for solving the first-order optimality system arising within the sequential quadratic programming (SQP) framework using Newton's method. The algorithmic scalability of this preconditioner is enhanced by incorporating a solution-dependent coarse space, which takes into account the restricted constraints from the fine level. By means of numerical examples, we demonstrate that the proposed preconditioned Newton methods outperform standard active-set methods considered in the literature.

math.OC↗

arXiv2022

A Multigrid Preconditioner for Jacobian-free Newton-Krylov Methods

In this work, we propose a multigrid preconditioner for Jacobian-free Newton-Krylov (JFNK) methods. Our multigrid method does not require knowledge of the Jacobian at any level of the multigrid hierarchy. As it is common in standard multigrid methods, the proposed method also relies on three building blocks: transfer operators, smoothers, and a coarse level solver. In addition to the restriction and prolongation operator, we also use a projection operator to transfer the current Newton iterate to a coarser level. The three-level Chebyshev semi-iterative method is employed as a smoother, as it has good smoothing properties and does not require the representation of the Jacobian matrix. We replace the direct solver on the coarsest level with a matrix-free Krylov subspace method, thus giving rise to a truly Jacobian-free multigrid preconditioner. We will discuss all building blocks of our multigrid preconditioner in detail and demonstrate the robustness and the efficiency of the proposed method using several numerical examples.

math.NA↗

arXiv2022

AnyDijkstra, an algorithm to compute shortest paths on images with anytime properties

Images conveniently capture the result of physical processes, representing rich source of information for data driven medicine, engineering, and science. The modeling of an image as a graph allows the application of graph-based algorithms for content analysis. Amongst these, one of the most used is the Dijkstra Single Source Shortest Path algorithm (DSSSP), which computes the path with minimal cost from one starting node to all the other nodes of the graph. However, the results of DSSSP remains unknown for nodes until they are explored. Moreover, DSSSP execution is associated to frequent jumps between distant locations in the graph, which results in non-optimal memory access, reduced parallelization, and finally increased execution time. Therefore, we propose AnyDijkstra, an iterative implementation of the Dijkstra SSSP algorithm optimized for images, that retains anytime properties while accessing memory following a cache-friendly scheme and maximizing parallelization.

cs.DS↗

arXiv2022

Nonlinear Field-split Preconditioners for Solving Monolithic Phase-field Models of Brittle Fracture

One of the state-of-the-art strategies for predicting crack propagation, nucleation, and interaction is the phase-field approach. Despite its reliability and robustness, the phase-field approach suffers from burdensome computational cost, caused by the non-convexity of the underlying energy functional and a large number of unknowns required to resolve the damage gradients. In this work, we propose to solve such nonlinear systems in a monolithic manner using the Schwarz preconditioned inexact Newton's (SPIN) method. The proposed SPIN method leverages the field split approach and minimizes the energy functional separately with respect to displacement and the phase-field, in an additive and multiplicative manner. In contrast to the standard alternate minimization, the result of this decoupled minimization process is used to construct a preconditioner for a coupled linear system, arising at each Newton's iteration. The overall performance and the convergence properties of the proposed additive and multiplicative SPIN methods are investigated by means of several numerical examples. Comparison with widely-used alternate minimization is also performed and we show a reduction in the execution time up to a factor of 50. Moreover, we also demonstrate that this reduction grows even further with increasing problem size and larger loading increments.

math.NA↗