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We consider large-scale implicit solvers for the numerical solution of partial differential equations (PDEs). The solvers require the high-bandwith networks of an HPC system for a fast time to solution. The increasing variability in performance of the HPC systems, most likely caused by variable communication latencies and network congestion, however, makes the execution time of solver algorithms unpredictable and hard to measure. In particular, the performance variability of the underlying system makes the reliable comparison of different algorithms and implementations difficult or impossible on HPC. We propose the use of statistical methods relying on hidden Markov models (HMM) to separate variable performance data into regimes corresponding to different levels of system latency. This allows us to, for ex- ample, identify and remove time periods when extremely high system latencies throttle application performance and distort performance measurements. We apply HMM to the careful analysis of implicit conjugate gradient solvers for finite-element discretized PDE, in particular comparing several new communication hiding methods for matrix-free operators of a PDE, which are critical for achieving peak performance in state-of-the-art PDE solvers. The HMM analysis allows us to overcome the strong performance variability in the HPC system. Our performance results for a model PDE problem discretized with 135 million degrees of freedom parallelized over 7168 cores of the Anvil supercomputer demonstrate that the communication hiding techniques can achieve up to a 10% speedup for the matrix-free matrix-vector product. 

Variable viscosity in Earth’s mantle exerts a fundamental control on mantle convection and plate tectonics, yet rigorously constraining the underlying parameters has remained a challenge. Inverse methods have not been sufficiently robust to handle the severe viscosity gradients and nonlinearities (arising from dislocation creep and plastic failure) while simultaneously resolving the megathrust and bending slabs globally. Using global plate motions as constraints, we overcome these challenges by combining a scalable nonlinear Stokes solver that resolves the key tectonic features with an adjoint-based Bayesian approach. Assuming plate cooling, variations in the thickness of continental lithosphere, slabs, and broad scale lower mantle structure as well as a constant grain size through the bulk of the upper mantle, a good fit to global plate motions is found with a nonlinear upper mantle stress exponent of 2.43 $\pm$0.25 (mean $\pm$SD). A relatively low yield stress of 151 $\pm$19 MPa is required for slabs to bend during subduction and transmit a slab pull that generates asymmetrical subduction. The recovered long-term strength of megathrusts (plate interfaces) varies between different subduction zones, with South America having a larger strength and Vanuatu and Central America having lower values with important implications for the stresses driving megathrust earthquakes. 

This work explored synaptic strengths in a computational neuroscience model of a controller for the hip joint of a rat which consists of Ia interneurons, Renshaw cells, and the associated motor neurons. This circuit has been referred to as the Canonical Motor Microcircuit (CMM). It is thought that the CMM acts to modulate motor neuron activity at the output stage. We first created a biomechanical model of a rat hindlimb consisting of a pelvis, femur, shin, foot, and flexor-extensor muscle pairs modeled with a Hill muscle model. We then modeled the CMM using non-spiking leaky-integrator neural models connected with conductance-based synapses. To tune the parameters in the network, we implemented an automated approach for parameter search using the Markov chain Monte Carlo (MCMC) method to solve a parameter estimation problem in a Bayesian inference framework. As opposed to traditional optimization techniques, the MCMC method identifies probability densities over the multidimensional space of parameters. This allows us to see a range of likely parameters that produce model outcomes consistent with animal data, determine if the distribution of likely parameters is uni- or multi-modal, as well as evaluate the significance and sensitivity of each parameter. This approach will allow for further analysis of the circuit, specifically, the function and significance of Ia feedback and Renshaw cells. 

One of the most common types of models that helps us to understand neuron behavior is based on the Hodgkin–Huxley ion channel formulation (HH model). A major challenge with inferring parameters in HH models is non-uniqueness: many different sets of ion channel parameter values produce similar outputs for the same input stimulus. Such phenomena result in an objective function that exhibits multiple modes (i.e., multiple local minima). This non-uniqueness of local optimality poses challenges for parameter estimation with many algorithmic optimization techniques. HH models additionally have severe non-linearities resulting in further challenges for inferring parameters in an algorithmic fashion. To address these challenges with a tractable method in high-dimensional parameter spaces, we propose using a particular Markov chain Monte Carlo (MCMC) algorithm, which has the advantage of inferring parameters in a Bayesian framework. The Bayesian approach is designed to be suitable for multimodal solutions to inverse problems. We introduce and demonstrate the method using a three-channel HH model. We then focus on the inference of nine parameters in an eight-channel HH model, which we analyze in detail. We explore how the MCMC algorithm can uncover complex relationships between inferred parameters using five injected current levels. The MCMC method provides as a result a nine-dimensional posterior distribution, which we analyze visually with solution maps or landscapes of the possible parameter sets. The visualized solution maps show new complex structures of the multimodal posteriors, and they allow for selection of locally and globally optimal value sets, and they visually expose parameter sensitivities and regions of higher model robustness. We envision these solution maps as enabling experimentalists to improve the design of future experiments, increase scientific productivity and improve on model structure and ideation when the MCMC algorithm is applied to experimental data. 

Abstract Strain localization and resulting plasticity and failure play an important role in the evolution of the lithosphere. These phenomena are commonly modeled by Stokes flows with viscoplastic rheologies. The nonlinearities of these rheologies make the numerical solution of the resulting systems challenging, and iterative methods often converge slowly or not at all. Yet accurate solutions are critical for representing the physics. Moreover, for some rheology laws, aspects of solvability are still unknown. We study a basic but representative viscoplastic rheology law. The law involves a yield stress that is independent of the dynamic pressure, referred to as von Mises yield criterion. Two commonly used variants, perfect/ideal and composite viscoplasticity, are compared. We derive both variants from energy minimization principles, and we use this perspective to argue when solutions are unique. We propose a new stress‐velocity Newton solution algorithm that treats the stress as an independent variable during the Newton linearization but requires solution only of Stokes systems that are of the usual velocity‐pressure form. To study different solution algorithms, we implement 2‐D and 3‐D finite element discretizations, and we generate Stokes problems with up to 7 orders of magnitude viscosity contrasts, in which compression or tension results in significant nonlinear localization effects. Comparing the performance of the proposed Newton method with the standard Newton method and the Picard fixed‐point method, we observe a significant reduction in the number of iterations and improved stability with respect to problem nonlinearity, mesh refinement, and the polynomial order of the discretization.