Mauro Perego

Enhanced basal lubrication and the contribution of the Greenland ice sheet to future sea-level rise

We assess the effect of enhanced basal sliding on the flow and mass budget of the Greenland ice sheet, using a newly developed parameterization of the relation between meltwater runoff and ice flow. A wide range of observations suggest that water generated by melt at the surface of the ice sheet reaches its bed by both fracture and drainage through moulins. Once at the bed, this water is likely to affect lubrication, although current observations are insufficient to determine whether changes in subglacial hydraulics will limit the potential for the speedup of flow. An uncertainty analysis based on our best-fit parameterization admits both possibilities: continuously increasing or bounded lubrication. We apply the parameterization to four higher-order ice-sheet models in a series of experiments forced by changes in both lubrication and surface mass budget and determine the additional mass loss brought about by lubrication in comparison with experiments forced only by changes in surface mass balance. We use forcing from a regional climate model, itself forced by output from the European Centre Hamburg Model (ECHAM5) global climate model run under scenario A1B. Although changes in lubrication generate widespread effects on the flow and form of the ice sheet, they do not affect substantial net mass loss; increase in the ice sheet’s contribution to sea-level rise from basal lubrication is projected by all models to be no more than 5% of the contribution from surface mass budget forcing alone.

A model-based block-triangular preconditioner for the Bidomain system in electrocardiology

We introduce a preconditioner for the solution of the Bidomain system governing the propagation of action potentials in the myocardial tissue. The Bidomain model is a degenerate parabolic set of nonlinear reaction-diffusion equations. The nonlinear term describes the ion flux at the cellular level. The degenerate nature of the problem results in a severe ill conditioning of its discretization. Our preconditioning strategy is based on a suitable adaptation of the Monodomain model, a simplified version of the Bidomain one, which is by far simpler to solve, nevertheless is unable to capture significant features of the action potential propagation. The Monodomain preconditioner application to a non-symmetric formulation of the Bidomain system results at the algebraic level in a lower block-triangular preconditioner. We prove optimality of the preconditioner with respect to the mesh size, and corroborate our theoretical results with 3D numerical simulations both on idealized and real ventricle geometries.

A Variational Data Assimilation Procedure for the Incompressible Navier-Stokes Equations in Hemodynamics

We propose a data assimilation (DA) technique for including noisy measurements of the velocity field into the simulation of the Navier-Stokes equations (NSE) driven by hemodynamics applications. The technique is formulated as an inverse problem where we use a Discretize-then-Optimize approach to minimize the misfit between the recovered velocity field and the data, subject to the incompressible NSE. The DA procedure for this nonlinear problem is a combination of two approaches: the Newton method for the NSE and the DA procedure we designed and tested for the linearized problem. We discuss conditions on the location of velocity measurements that guarantee the well-posedness of the minimization process for the linearized problem. Numerical results, with both noise-free and noisy data, certify the theoretical analysis. Moreover, we consider 2D non-trivial geometries and 3D axisymmetric geometries. Also, we study the impact of noise on non-primitive variables of medical interest.

A Variational Approach for Estimating the Compliance of the Cardiovascular Tissue: An Inverse Fluid-Structure Interaction Problem

Estimation of the stiffness of a biological soft tissue is useful for the detection of pathologies such as tumors or atherosclerotic plaques. Elastography is a method based on the comparison between two images before and after a forced deformation of the tissue of interest. An inverse elasticity problem is then solved for Youngs modulus estimation. In the case of arteries, no forced deformation is required, since vessels naturally move under the action of blood. Youngs modulus can therefore be estimated by solving a coupled inverse fluid-structure interaction problem. In this paper we focus on the mathematical properties of this problem and its numerical solution. We give some well posedness analysis and some preliminary results based on a synthetic data set, i.e., test cases where the exact Youngs modulus is known and the displacement dataset is numerically generated by solving a forward fluid-structure interaction problem. We address the problem of the presence of the noise in the measured displacement and of the proper sampling frequency for obtaining reliable estimates.

Applications of variational data assimilation in computational hemodynamics

The development of new technologies for acquiring measures and images in order to investigate cardiovascular diseases raises new challenges in scientific computing. These data can be in fact merged with the numerical simulations for improving the accuracy and reliability of the computational tools. Assimilation of measured data and numerical models is well established in meteorology, whilst it is relatively new in computational hemodynamics. Different approaches are possible for the mathematical setting of this problem. Among them, we follow here a variational formulation, based on the minimization of the mismatch between data and numerical results by acting on a suitable set of control variables. Several modeling and methodological problems related to this strategy are open, such as the analysis of the impact of the noise affecting the data, and the design of effective numerical solvers. In this chapter we present three examples where a mathematically sound (variational) assimilation of data can significantly improve the reliability of the numerical models. Accuracy and reliability of computational models are increasingly important features in view of the progressive adoption of numerical tools in the design of new therapies and, more in general, in the decision making process of medical doctors.

Optimal initial conditions for coupling ice sheet models to Earth system models

We address complications in the coupling of a dynamic ice sheet model (ISM) and forcing from an Earth system model (ESM), which arise because of the unknown ISM initial conditions. Unless explicitly accounted for during ISM initialization, the ice sheet is far from thermomechanical equilibrium with the surface mass balance forcing from the ESM. Upon coupling to ESM forcing, the result is a shock and unphysical and undesirable transients in ice geometry and other state variables. Under the assumption of thermomechanical equilibrium, we present an approach for finding ISM initial conditions—characterized by optimization of the basal sliding coefficient and basal topography fields—that balance a best fit to surface velocity and basal topography observations against the minimization of unphysical transients when coupling to surface mass balance forcing. A quasi-Newton method is used to solve the resulting large-scale, partial differential equation-constrained optimization problem, where the cost function gradients with respect to the parameter fields are computed using adjoints. After studying properties of our approach on a synthetic test problem, we apply the method toward obtaining optimal initial conditions for a model of the Greenland ice sheet. Our results show that, in the presence of uncertainties in the basal topography, ice thickness should also be treated as an optimization variable. While the focus here is on the coupling between an ISM and ESM-derived surface mass balance, the method is easily extended to include optimal coupling to forcing from an ocean model through submarine melt rates.

On the scalability of the Albany/FELIX first-order Stokes approximation ice sheet solver for large-scale simulations of the Greenland and Antarctic ice sheets

We examine the scalability of the recently developed Albany/FELIX finite-element based code for the first-order Stokes momentum balance equations for ice flow. We focus our analysis on the performance of two possible preconditioners for the iterative solution of the sparse linear systems that arise from the discretization of the governing equations: (1) a preconditioner based on the incomplete LU (ILU) factorization, and (2) a recently-developed algebraic multigrid (AMG) preconditioner, constructed using the idea of semi-coarsening. A strong scalability study on a realistic, high resolution Greenland ice sheet problem reveals that, for a given number of processor cores, the AMG preconditioner results in faster linear solve times but the ILU preconditioner exhibits better scalability. A weak scalability study is performed on a realistic, moderate resolution Antarctic ice sheet problem, a substantial fraction of which contains floating ice shelves, making it fundamentally different from the Greenland ice sheet problem. Here, we show that as the problem size increases, the performance of the ILU preconditioner deteriorates whereas the AMG preconditioner maintains scalability. This is because the linear systems are extremely ill-conditioned in the presence of floating ice shelves, and the ill-conditioning has a greater negative effect on the ILU preconditioner than on the AMG preconditioner.

An ice sheet model validation framework for the Greenland ice sheet

We propose a new ice sheet model validation framework - the Cryospheric Model Comparison Tool (CmCt) - that takes advantage of ice sheet altimetry and gravimetry observations collected over the past several decades and is applied here to modeling of the Greenland ice sheet. We use realistic simulations performed with the Community Ice Sheet Model (CISM) along with two idealized, non-dynamic models to demonstrate the framework and its use. Dynamic simulations with CISM are forced from 1991 to 2013 using combinations of reanalysis-based surface mass balance and observations of outlet glacier flux change. We propose and demonstrate qualitative and quantitative metrics for use in evaluating the different model simulations against the observations. We find that the altimetry observations used here are largely ambiguous in terms of their ability to distinguish one simulation from another. Based on basin- and whole-ice-sheet scale metrics, we find that simulations using both idealized conceptual models and dynamic, numerical models provide an equally reasonable representation of the ice sheet surface (mean elevation differences of <1 m). This is likely due to their short period of record, biases inherent to digital elevation models used for model initial conditions, and biases resulting from firn dynamics, which are not explicitly accounted for in the models or observations. On the other hand, we find that the gravimetry observations used here are able to unambiguously distinguish between simulations of varying complexity, and along with the CmCt, can provide a quantitative score for assessing a particular model and/or simulation. The new framework demonstrates that our proposed metrics can distinguish relatively better from relatively worse simulations and that dynamic ice sheet models, when appropriately initialized and forced with the right boundary conditions, demonstrate predictive skill with respect to observed dynamic changes occurring on Greenland over the past few decades. An extensible design will allow for continued use of the CmCt as future altimetry, gravimetry, and other remotely sensed data become available for use in ice sheet model validation.

A Matrix Dependent/Algebraic Multigrid Approach for Extruded Meshes with Applications to Ice Sheet Modeling

A multigrid method is proposed that combines ideas from matrix dependent multigrid for structured grids and algebraic multigrid for unstructured grids. It targets problems where a three-dimensional mesh can be viewed as an extrusion of a two-dimensional, unstructured mesh in a third dimension. Our motivation comes from the modeling of thin structures via finite elements and, more specifically, the modeling of ice sheets. Extruded meshes are relatively common for thin structures and often give rise to anisotropic problems when the thin direction mesh spacing is much smaller than the broad direction mesh spacing. Within our approach, the first few multigrid hierarchy levels are obtained by applying matrix dependent multigrid to semicoarsen in a structured thin direction fashion. After sufficient structured coarsening, the resulting mesh contains only a single layer corresponding to a two-dimensional, unstructured mesh. Algebraic multigrid can then be employed in a standard manner to create further coarse le...

A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions

We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandias agile software components toolkit. The latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method.