Oliver Röhrle

Andreas Hessenthaler, Ben S. Southworth, David Nordsletten et al.

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.

Jonas Kneifl, David Rosin, Oliver Röhrle et al.

In recent decades, the main focus of computer modeling has been on supporting the design and development of engineering prototyes, but it is now ubiquitous in non-traditional areas such as medical rehabilitation. Conventional modeling approaches like the finite element~(FE) method are computationally costly when dealing with complex models, making them of limited use for purposes like real-time simulation or deployment on low-end hardware, if the model at hand cannot be simplified in a useful manner. Consequently, non-traditional approaches such as surrogate modeling using data-driven model order reduction are used to make complex high-fidelity models more widely available anyway. They often involve a dimensionality reduction step, in which the high-dimensional system state is transformed onto a low-dimensional subspace or manifold, and a regression approach to capture the reduced system behavior. While most publications focus on one dimensionality reduction, such as principal component analysis~(PCA) (linear) or autoencoder (nonlinear), we consider and compare PCA, kernel PCA, autoencoders, as well as variational autoencoders for the approximation of a structural dynamical system. In detail, we demonstrate the benefits of the surrogate modeling approach on a complex FE model of a human upper-arm. We consider both the models deformation and the internal stress as the two main quantities of interest in a FE context. By doing so we are able to create a computationally low cost surrogate model which captures the system behavior with high approximation quality and fast evaluations.

David Rosin, Johannes Kässinger, Xingyao Yu et al.

This paper presents a novel neural network architecture for the purpose of pervasive visualisation of a 3D human upper limb musculoskeletal system model. Bringing simulation capabilities to resource-poor systems like mobile devices is of growing interest across many research fields, to widen applicability of methods and results. Until recently, this goal was thought to be out of reach for realistic continuum-mechanical simulations of musculoskeletal systems, due to prohibitive computational cost. Within this work we use a sparse grid surrogate to capture the surface deformation of the m.~biceps brachii in order to train a deep learning model, used for real-time visualisation of the same muscle. Both these surrogate models take 5 muscle activation levels as input and output Cartesian coordinate vectors for each mesh node on the muscle's surface. Thus, the neural network architecture features a significantly lower input than output dimension. 5 muscle activation levels were sufficient to achieve an average error of 0.97 +/- 0.16 mm, or 0.57 +/- 0.10 % for the 2809 mesh node positions of the biceps. The model achieved evaluation times of 9.88 ms per predicted deformation state on CPU only and 3.48 ms with GPU-support, leading to theoretical frame rates of 101 fps and 287 fps respectively. Deep learning surrogates thus provide a way to make continuum-mechanical simulations accessible for visual real-time applications.

Pouyan Asgharzadeh, Oliver Röhrle, Bettina M. Willie et al.

Throughout the process of aging, deterioration of bone macro- and micro-architecture, as well as material decomposition result in a loss of strength and therefore in an increased likelihood of fractures. To date, precise contributions of age-related changes in bone (re)modeling and (de)mineralization dynamics and its effect on the loss of functional integrity are not completely understood. Here, we present an image-based deep learning approach to quantitatively describe the dynamic effects of short-term aging and adaptive response to treatment in proximal mouse tibia and fibula. Our approach allowed us to perform an end-to-end age prediction based on $μ$CT images to determine the dynamic biological process of tissue maturation during a two week period, therefore permitting a short-term bone aging prediction with $95\%$ accuracy. In a second application, our radiomics analysis reveals that two weeks of in vivo mechanical loading are associated with an underlying rejuvenating effect of 5 days. Additionally, by quantitatively analyzing the learning process, we could, for the first time, identify the localization of the age-relevant encoded information and demonstrate $89\%$ load-induced similarity of these locations in the loaded tibia with younger bones. These data suggest that our method enables identifying a general prognostic phenotype of a certain bone age as well as a temporal and localized loading-treatment effect on this apparent bone age. Future translational applications of this method may provide an improved decision-support method for osteoporosis treatment at low cost.