DOI: 10.5194/tc-10-1477-2016
Scopus记录号: 2-s2.0-84978405272
论文题名: Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model
作者: Zhu H ; , Petra N ; , Stadler G ; , Isaac T ; , Hughes T ; J ; R ; , Ghattas O
刊名: Cryosphere
ISSN: 19940416
出版年: 2016
卷: 10, 期: 4 起始页码: 1477
结束页码: 1494
语种: 英语
英文关键词: advection-diffusion equation
; boundary condition
; geothermal gradient
; heat flux
; ice sheet
; optimization
; prediction
; reconstruction
; steady-state equilibrium
; Stokes formula
; thermomechanics
; three-dimensional modeling
英文摘要: We address the inverse problem of inferring the basal geothermal heat flux from surface velocity observations using a steady-state thermomechanically coupled nonlinear Stokes ice flow model. This is a challenging inverse problem since the map from basal heat flux to surface velocity observables is indirect: the heat flux is a boundary condition for the thermal advection diffusion equation, which couples to the nonlinear Stokes ice flow equations; together they determine the surface ice flow velocity. This multiphysics inverse problem is formulated as a nonlinear least-squares optimization problem with a cost functional that includes the data misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to render the problem well posed. We derive adjoint-based gradient and Hessian expressions for the resulting partial differential equation (PDE)-constrained optimization problem and propose an inexact Newton method for its solution. As a consequence of the Petrov Galerkin discretization of the energy equation, we show that discretization and differentiation do not commute; that is, the order in which we discretize the cost functional and differentiate it affects the correctness of the gradient. Using two-and three-dimensional model problems, we study the prospects for and limitations of the inference of the geothermal heat flux field from surface velocity observations. The results show that the reconstruction improves as the noise level in the observations decreases and that short-wavelength variations in the geothermal heat flux are difficult to recover. We analyze the ill-posedness of the inverse problem as a function of the number of observations by examining the spectrum of the Hessian of the cost functional. Motivated by the popularity of operator-split or staggered solvers for forward multiphysics problems i.e., those that drop two-way coupling terms to yield a one-way coupled forward Jacobian we study the effect on the inversion of a one-way coupling of the adjoint energy and Stokes equations. We show that taking such a one-way coupled approach for the adjoint equations can lead to an incorrect gradient and premature termination of optimization iterations. This is due to loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost functional. Nevertheless, one may still obtain a reasonable approximate inverse solution particularly if important features of the reconstructed solution emerge early in optimization iterations, before the premature termination. © 2016 Author(s). Citation statistics:
资源类型: 期刊论文
标识符: http://119.78.100.158/handle/2HF3EXSE/75114
Appears in Collections: 影响、适应和脆弱性 气候变化与战略
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作者单位: Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, United States; Applied Mathematics, School of Natural Sciences, University of California, Merced, CA, United States; Courant Institute of Mathematical Sciences, New York University, New York, NY, United States; Computation Institute, University of Chicago, Chicago, IL, United States; Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, TX, United States; Jackson School of Geosciences, University of Texas at Austin, Austin, TX, United States; Department of Mechanical Engineering, University of Texas at Austin, Austin, TX, United States
Recommended Citation:
Zhu H,, Petra N,, Stadler G,et al. Inversion of geothermal heat flux in a thermomechanically coupled nonlinear Stokes ice sheet model[J]. Cryosphere,2016-01-01,10(4)