Multisecant quasi-Newton methods have been shown to be particularly suited to solve nonlinear fixed-point equations that arise from partitioned multiphysics simulations where the exact Jacobian is inaccessible. In all these methods, the underdetermined multisecant equation for the approximate (inverse) Jacobian is enhanced by a norm minimization condition. The standard choice is the minimization of the Frobenius norm of the approximate inverse Jacobian. In this setting, it is well known that transient fluid-structure simulations typically require the use of secant information also from previous time steps to achieve a small enough number of iterations per implicit time step. The number of these time steps highly depends on the application, the physical parameters, the used solvers, and the mesh resolution. Using too few leads to a relatively high number of iterations, while using too many leads not only to a computational overhead but also to an increase in the number of iterations as well. Determining the optimal number requires a costly trial-and-error process. In this paper, we present results for two different approaches to overcome this issue: The first approach is to use a modified method (presented in [F. Lindner, M. Mehl, K. Scheufele, and B. Uekermann, "A Comparison of Various Quasi-Newton Schemes for Partitioned Fluid-Structure Interaction," in Proceedings of ECCOMAS Coupled Problems, Venice, 2015, pp. 1-12]) that minimizes the Frobenius norm of the difference between the current approximate (inverse) Jacobian and that of the previous time step. Thus, previous time step information is taken into account in an implicit and automatized way without magic parameters. The second approach is to use an unrestricted number of previous time steps in combination with a suitable filtering algorithm automatically removing secant information that is outdated (thus, slowing down convergence) or contradicting newer information (deteriorating the condition of the multisecant equation system). We present a novel algorithm to realize the first idea with linear complexity in the number of coupling surface unknowns (note that already storing the approximate inverse Jacobian would induce quadratic complexity) and the efficient parallel implementation for both approaches. This results in highly efficient, parallelizable, and robust iterative solvers applicable for surface coupling in many types of multiphysics simulations using black-box solver software. In addition, our numerical results for the fluid-structure benchmark (FSI3) from Turek and Hron ["Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow," in Fluid-Structure Interaction, Springer, Berlin, 2006, pp. 371-385] and for a flow through a flexible tube with a large variety of parameter settings prove the robustness and numerical efficiency of the first approach in particular. The second approach can be shown to be highly sensitive to the choice of the filtering method for the secant information a...
We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of Magnetic Resonance Imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver. Related work.Although there has been a lot of work on forward problems for tumor growth, there has been less work on inverse problems. The latter has different aspects. The first is the underlying biophysical model. The second is the inverse problem setup, observation operators and the existence of scans at multiple points, the noise models, inversion parameters and constraints. And the third one is the solution algorithm.Regarding the underlying model, like us, most researchers focus on parameter calibration of a handful of model parameters using single-species reaction-diffusion equations [7,22,24,28,33,39,51,58,59]. While more complex models describing processes like mass effect, angiogenesis and chemotaxis [23,26,48,54,60] exist, they have not been considered for calibration due to theoretical and computational challenges. However, several groups, including ours, are working to address these challenges.Regarding the inverse problem setup, in most studies t...
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