Space-time Adaptivity Research Articles

Space-time adaptivity for a multi-scale cancer invasion model

Space-time adaptivity for a multi-scale cancer invasion model

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loève expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers' conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.

Parallel and space-time adaptivity for the numerical simulation of cardiac action potentials

In this work, the study of parallel and space-time adaptivity for the numerical simulation of action potential propagation in cardiac electrophysiology is presented. The monodomain and bidomain models are employed for the numerical realization. This work is devoted to investigate the spatial adaptivity which is realized within multi finite element methods by using a gradient type a posteriori error estimator and the temporal adaptivity determined by a linearly implicit time integration techniques. In addition, the parallelization of such space-time adaptivity for the simulation of spiral wave dynamics based on non-overlapping domain decomposition techniques and dynamic load balancing is discussed. The numerical results demonstrate that the parallel simulations have been accelerated reasonably good using the adaptive space and time algorithm, with no significant loss in accuracy, for the single action potential wave propagation as well as reentry situations.

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations.

Enabling local time stepping in the parallel implicit solution of reaction–diffusion equations via space-time finite elements on shallow tree meshes

For many applications, local time stepping offers an interesting and worthwhile alternative to the by now well established global time step control. In fact, local time stepping can allow for a highly detailed resolution of localized features of the solution with strongly reduced computational cost, when compared to global time step control. However local time stepping is not applicable in a straight-forward manner in the context of fully implicit time-discretizations.Here, we present a method for the efficient parallel adaptive solution of (non-linear) partial differential equations, in particular reaction–diffusion equations, using spatially adapted time step sizes in the context of a fully implicit solution strategy. Our proposed method uses a discontinuous Galerkin method in-time approach within a full space-time approach. Moreover, it is designed from scratch for efficient parallel computation. We employ shallow tree-based mesh data structures in order to ensure a low memory footprint of the adaptive meshes. By solving the time-dependent partial differential equation on a (d+1)-dimensional non-conforming mesh, space-time adaptivity is naturally achieved. In combination with a discontinuous Galerkin method in-time the size of the arising systems can be precisely controlled.We additionally introduce and discuss a stabilization scheme for space-time mortar element methods that also has a highly positive impact on the efficiency of preconditioning techniques for the arising systems of equations. We present results from extensive numerical experiments that address the question of convergence and efficiency, linear and non-linear solver performance, parallel scalability up to 2048 cores as well as accuracy for the linear heat equation and a real world, non-linear reaction–diffusion equation from the field of computational electrocardiology.

A sequential-adaptive strategy in space–time with application to consolidation of porous media

Issues related to space-time adaptivity for a class of nonlinear and time-dependent problems are discussed. The dG(k)-methods are adopted for the time integration, and the a posteriori error control is based on the appropriate dual problem in space-time. One key ingredient is to decouple the error generation in space and time with a hierarchical decomposition of the discrete space of dual solutions. The main idea put forward in the paper is to increase the computational efficiency of the adaptive scheme by avoiding recursive adaptations of the whole time-mesh; rather, the space-mesh and the time-step defining each finite space-time slab are defined in a truly sequential fashion. The proposed adaptive strategy is applied to the coupled consolidation problem in geomechanics involving large deformations. Its performance is investigated with the aid of a numerical example in 2D.

Goal‐oriented space–time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow

SUMMARYThis paper presents a general strategy for designing adaptive space–time finite element discretizations of the nonstationary Navier–Stokes equations. The underlying framework is that of the dual weighted residual method for goal‐oriented a posteriori error estimation and automatic mesh adaptation. In this approach, the error in the approximation of certain quantities of physical interest, such as the drag coefficient, is estimated in terms of local residuals of the computed solution multiplied by sensitivity factors, which are obtained by numerically solving an associated dual problem. In the resulting local error indicators, the effects of spatial and temporal discretization are separated, which allows for the simultaneous adjustment of time step and spatial mesh size. The efficiency of the proposed method for the construction of economical meshes and the quantitative assessment of the error is illustrated by several test examples. Copyright © 2012 John Wiley & Sons, Ltd.

A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations

This paper presents a new resolution strategy for multi-scale streamer discharge simulations based on a second order time adaptive integration and space adaptive multiresolution. A classical fluid model is used to describe plasma discharges, considering drift–diffusion equations and the computation of electric field. The proposed numerical method provides a time-space accuracy control of the solution, and thus, an effective accurate resolution independent of the fastest physical time scale. An important improvement of the computational efficiency is achieved whenever the required time steps go beyond standard stability constraints associated with mesh size or source time scales for the resolution of the drift–diffusion equations, whereas the stability constraint related to the dielectric relaxation time scale is respected but with a second order precision. Numerical illustrations show that the strategy can be efficiently applied to simulate the propagation of highly nonlinear ionizing waves as streamer discharges, as well as highly multi-scale nanosecond repetitively pulsed discharges, describing consistently a broad spectrum of space and time scales as well as different physical scenarios for consecutive discharge/post-discharge phases, out of reach of standard non-adaptive methods.

A Variational Shape Optimization Approach for Image Segmentation with a Mumford–Shah Functional

We introduce a novel computational method for a Mumford–Shah functional, which decomposes a given image into smooth regions separated by closed curves. Casting this as a shape optimization problem, we develop a gradient descent approach at the continuous level that yields nonlinear PDE flows. We propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements. The method incorporates topological changes, such as splitting and merging for detection of multiple objects, space–time adaptivity, and a coarse-to-fine approach to process large images efficiently. We present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters.

Finite element method for time dependent scattering: nonreflecting boundary condition, adaptivity, and energy decay

An adaptive finite element method is developed for acoustic wave propagation in unbounded media. The efficiency and high accuracy of the method are achieved by combining an exact nonreflecting boundary condition [SIAM J. Appl. Math. 55 (1995) 280; J. Comput. Phys. 127 (1996) 52] with space–time adaptivity [East–West J. Numer. Math. 7(4) (1999) 263]. Hence the computational effort is concentrated where needed, while the artificial boundary can be brought as close as desired to the scatterer. Both features combined yield high accuracy and keep the number of unknowns to a minimum. An energy inequality is derived for the initial-boundary value problem at the continuous level. Together with an implicit second order time discretization it guarantees unconditional stability of the semi-discrete system. The resulting fully discrete linear system that needs to be solved every time step is unsymmetric but can be transformed into an equivalent sequence of small nonsymmetric and large symmetric positive definite systems, which are efficiently solved by conjugate gradient methods. Numerical examples illustrate the high accuracy of the method, in particular in the presence of complex geometry.

Implicit solution of hyperbolic equations with space-time adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.

Adaptive Techniques in Radar Signal Processing

Adaptive Techniques have come to stay in modern radar systems. From the simpler CFAR and the SLC devices of the last decade, attempts are now being made to incorporate full space-time adaptivity. T...