We consider the following inverse problem: to infer the unknown basal geothermal heat flux field given surface velocity observations and a non-Newtonian full Stokes ice sheet flow model governed by thermomechanically coupled mass, momentum, and energy equations. Grid-based discretization of the basal heat flux field leads to a high-dimensional inverse problem. The main aim of this paper is to present an efficient method for solving this large-scale coupled-physics inverse problem and to use model problems to study the prospects for, and limitations of, inferring the geothermal heat flux from surface ice velocities.

Ice sheet models are characterized by unknown or uncertain parameters stemming from the lack of direct observations of the interior and the base of the ice sheet. Unknown parameters include those that represent basal friction, basal topography, rheology, geothermal heat flux, and ice thickness. The geothermal heat flux parameter field, in particular, has a strong influence on the thermal state of the ice and hence plays a critical role in understanding the dynamics of the ice sheet through its effect on basal and internal ice temperatures . The direct measurement of the geothermal heat flux is only locally available . Estimates of this parameter field have been obtained via inference using global seismic tomographic models , satellite magnetic data models , or tectonic models . However, these inferred basal heat flux fields do not agree with one another in large regions. While sensitivity studies show that the resulting uncertainties in the geothermal heat flux have an impact on the ice flow, especially in the slow flow regions , inversion for the geothermal heat flux from surface velocity observations has not been addressed previously to the best of our knowledge.

When formulating the thermomechanically coupled inverse problem, we must assume an appropriate thermal regime, which depends critically on the geothermal heat flux. Ice sheets and glaciers can be in one of the following four thermal states: (1) all of the ice is below the melting point; (2) the melting point is reached only at the bed; (3) a basal layer of finite thickness is at melting point; or (4) all of the ice is at the melting point except for a surface layer p. 205. While in the first case the thermal basal boundary condition is simply characterized by the geothermal heat flux, in the other three cases this condition must be modified to include the heat generated by friction at the base and to account for melting. Due to the unknown basal state of the ice, these latter three cases lead to more complex inverse problems. These inverse problems typically involve variational inequalities in the forward problem since the thermal regime depends on the geothermal heat flux. Here, we assume that all of the ice is below the melting point. In the model problems studied in this paper we ensure this by providing a moderate (about the average of the geothermal heat flux at the base of Antarctica) geothermal heat flux. In addition, we assume the ice flow is in a steady state. These assumptions result in a tractable inverse problem that allows us to study the sensitivity of the ice flow velocity with respect to the geothermal heat flux and thus the characteristics of the inverse problem.

The inverse problem is formulated as a regularized nonlinear least-squares minimization problem governed by thermomechanically coupled nonlinear Stokes and thermal advection–diffusion equations. The cost functional we minimize represents the sum of the squared differences between observed and predicted surface velocities and a regularization term that renders this ill-posed inverse problem well posed. Discretizing the infinite-dimensional geothermal heat flux field and the governing partial differential equations (PDEs) leads to a large-scale numerical optimization problem; as such, derivative-based optimization methods offer the best hope for its efficient solution . In , we presented an infinite-dimensional adjoint-based inexact Gauss–Newton method for the inference of basal friction and rheology parameters from surface velocity observations and a nonlinear Stokes model of ice sheet flow. Here, we extend our previous work to the present inverse problem of inferring the geothermal heat flux in a thermomechanically coupled ice flow model. This problem also serves as a prototype for a broader class of multiphysics inverse problems.

We systematically study how well finite-amplitude variations of the geothermal heat flux can be recovered from noisy surface velocity observations. To be precise, we invert for geothermal heat flux fields that contain large and short-wavelength variations using velocity observations with various degrees of error. Our results show that the quality of the reconstructed geothermal heat flux deteriorates with shorter-wavelength variations and with increasing noise level in the observations. In addition, we study the influence of the number of observations and find that the reconstruction improves as the number of observation points increases, provided the discretization of the model equations is sufficiently fine to capture the additional information from a larger number of observations. To analyze prospects and limitations of the inversion, we also investigate the spectrum of the Hessian of the data misfit part of the cost functional, which provides information about directions in parameter space that can be recovered from observations.

A common approach to the numerical solution of multiphysics problems uses operator splitting; namely, motivated by the difficulty of either solving a two-way coupled system or computing the Jacobian of a coupling term, one discards certain coupling terms in the Jacobian of the forward problem to reduce the two-way coupled problem to one that is coupled in one direction. The coupled problem is then solved by iterating back and forth between the solution of single physics components. This approach, which we term “one-way coupled”, can often yield convergence to the solution of the fully coupled multiphysics problem, depending on the spectral radius of a certain iteration matrix. The one-way coupled approach has been used successfully for the solution of thermomechanically coupled ice sheet forward problems in , , , , and .

However, when solving the corresponding multiphysics inverse problem using gradient-based methods, the use of such a one-way coupled approach may be problematic. In particular, sacrificing coupling terms in the Jacobian (while often acceptable for the forward problem) will lead to an incorrect adjoint operator, since this operator is given by the transpose of the Jacobian. This approximate adjoint operator leads to an incorrect adjoint solution, which then leads to an incorrect gradient. Since the necessary optimality condition for the inverse problem states that the gradient must vanish, an incorrect gradient leads to the wrong solution of the inverse problem. Moreover, since line search methods require descent on the cost functional in a direction based on the gradient, the inconsistency between the cost functional and its gradient can lead to failure of the line search and thus lack of convergence. Thus, sacrificing coupling terms as is commonly done for the forward problem may not lead to convergent inverse iterations, and if the inverse iterations do converge, they will converge to the wrong inverse solution.

In general, how much of a difference this will make to the solution of the inverse problem will depend on the strength of the coupling terms that have been neglected in the adjoint problem. In particular, despite a gradient that has been computed from an incorrect adjoint equation, and early termination of optimization iterations, one might still obtain a reasonable approximation of the correct inverse solution. To illustrate these issues in the context of a thermomechanically coupled ice sheet inverse problem, we neglect certain coupling terms in the Jacobian (as might be done in a forward solver), leading to an incorrect adjoint operator. We then compare inversion results obtained using an approximate gradient based on a one-way coupled adjoint operator – which we refer to as a “one-way coupled gradient” – with inversions that use the correct gradient (i.e., based on the fully coupled adjoint). The results indicate that using this one-way coupled gradient instead of the correct gradient leads to a deterioration in the convergence rate of the inverse solver and eventual failure of the line search, but the resulting inverse solution for the geothermal flux does not differ substantially from the correct inverse solution.

The remaining sections of this paper are organized as follows. In Sect. , we describe the forward ice sheet problem and the corresponding inverse problem for the geothermal heat flux. In Sect. , we give expressions for the adjoint-based gradient and action of the Hessian of the cost functional. Then, in Sect.  we present the discretization of the forward problem, which involves a stabilization technique applied to prevent oscillatory solutions when the heat equation is advection-dominated, and discuss the optimize-then-discretize (OTD) and discretize-then-optimize (DTO) approaches for computing the gradient of the cost functional. In Sect. , we present inversion results for two- and three-dimensional model problems and in Sect.  we discuss the fully coupled versus one-way coupled approaches to computing the gradient for thermomechanically coupled ice sheet inverse problem.

We first state the forward problem and then formulate the inverse problem to infer the geothermal heat flux from surface velocity observations.

To compute the minimizer for the large-scale optimization problem Eq. (), we employ a derivative-based descent method and thus require derivatives of the nonlinear least-squares optimization problem Eq. () with respect to the parameter G. To improve over linearly convergent methods (such as the nonlinear conjugate gradients (CG) method) or superlinearly convergent methods (such as limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method), here we advocate a Newton method, which employs Hessian information (i.e., second derivatives) to provide an (asymptotic) quadratic convergence rate. Moreover, typically Newton's method converges in a number of iterations that is independent of the parameter dimension/mesh size, which is typically not true of gradient-only methods . Beyond the faster convergence, in our experience the use of Hessian information for highly nonlinear inverse problems such as those involving ice sheet models is typically able to obtain a reduction of another 1 or 2 orders of magnitude in the norm of the gradient, leading to the extraction of additional details in the reconstructed parameter field.

Starting with an initial guess for the parameter field G, Newton's method iteratively updates the parameter field based on minimizing a sequence of quadratic approximations of the cost functional, J, using gradient and Hessian information of J with respect to G. That is, the parameter is updated by Gnew=G+αG̃, where G is the current model parameter field, α is the step length, appropriately chosen so that the cost functional J is sufficiently decreased at each iteration, and G̃ is the direction which is obtained by solving the linear system H(G)(G̃)=-G(G). Here, G(G) and H(G) denote the gradient and the Hessian of the least-squares cost functional J, respectively, evaluated at the current parameter field G.

In this section, we provide expressions for the gradient G(G) and Hessian H(G). For the efficient computation of gradient and Hessian operators, we employ adjoint methods (see, for example, ). All expressions in this section are given in infinite-dimensional form, which has several advantages compared to discretizing the optimization problem first and then differentiating. First, one avoids differentiating through artifacts of the discretization or solver, which may not even be differentiable. Second, it is much easier and “cleaner” to derive gradient and Hessian information at the infinite-dimensional level. Third, the resulting expressions are in weak form, which provides a natural and systematic path to discretization by Galerkin finite elements. The downside to differentiating at the infinite-dimensional level is that the resulting gradient and Hessian expressions may not be “consistent”. These issues will be discussed in the next section.

In what follows, we use the formal Lagrange approach, which computes the gradient by taking variations of a Lagrangian functional. The Lagrangian functional L combines the cost functional Eq. () with the weak form Eq. () of the forward problem, with test functions v^, q^, and λ^. These test functions act as Lagrange multipliers and become the adjoint velocity v, adjoint pressure q, and adjoint temperature λ: L(u,p,θ;v,q,λ;G):=J(G)+a(u,p,θ;v,q,λ)-〈ρg,v〉Ω-〈G,λ〉Γb.

The gradient of J with respect to the unknown heat flux G is found by requiring that variations of the Lagrangian L with respect to the forward and adjoint variables vanish. The gradient G(G) is then found by taking the variation of L with respect to G. In strong form, the gradient evaluated at G, G(G), is then given by G(G):=-∇⋅(γT∇G)-λonΓb,(γT∇G)⋅n‾on∂Γb, where n‾ is the outer normal vector on ∂Γb. Variations of L with respect to the adjoint variables (v, q, λ) simply recover the forward problem. However, variations of L with respect to the forward variables (u, p, θ) yield the so-called “adjoint problem”, which is given in strong form by ∇⋅v=0inΩ,-∇⋅σv=-ρcλ∇θinΩ,-ρcu⋅∇λ-∇⋅(K∇λ)=FλinΩ,λ=0,σvn=B*uobs-BuonΓt,v=0,K∇λ⋅n=0onΓb,+additionallateralBCs. Here, B* is the adjoint of the observation operator, which maps point observations on Γt to a function. In particular, if B corresponds to the evaluation of a sufficiently smooth function at points, the adjoint B* maps a finite-dimensional vector to the sum of Dirac delta functions corresponding to these points, weighted by the input vector. The adjoint stress σv in Eq. () depends on the forward velocity u and the temperature θ and is given by σv:=2η(u,θ)I+1-n2nε˙u⊗ε˙uε˙II+ϵε˙v-1+nnλε˙u-Iq, where ε˙v is the adjoint strain rate tensor and is given by 12(∇v + ∇vT), I is the fourth-order identity tensor, and ⊗ represents the tensor (or outer) product between second-order tensors. Note that σv coincides with the adjoint stress derived by for the isothermal Stokes model except the term proportional to the adjoint temperature λ, which arises due to the strain heating term on the right-hand side of Eq. (). The right-hand side in the adjoint energy equation (Eq. ) is given by Fλ:=2QnRθ2η(u,θ)ε˙u:ε˙v-λε˙u:ε˙u. As can be seen from Eqs. ()–(), the adjoint problem is driven by the misfit between observed and predicted surface velocity on the top boundary, i.e., B*(uobs - Bu). Since the observations are of ice velocity on the top surface Γt, the data misfit shows up in the adjoint problem as a source term for the Neumann boundary condition on Γt, Eq. (). Observations in the interior of Ω would amount to a similar contribution on the right-hand side of Eq. (). Since the adjoint equation depends on (u, p, θ), each gradient computation also requires the solution of the forward problem (Eqs. –). Solution of the adjoint problem (Eqs. –) provides the adjoint temperature λ needed to evaluate the gradient in Eq. ().

Now that the computation of the gradient, which forms the right-hand side of the Newton system Eq. () has been described, we present the computation of the Hessian operator, H, on the left-hand side of the Newton system. We note that explicitly forming and storing the Hessian matrix resulting upon discretization is not an option, since computing each column would require at least a linearized forward solve. Instead, we solve the Newton system Eq. () using the linear CG method, which requires not the explicit Hessian but rather only the action of the Hessian on a vector at each CG iteration. We next present expressions for this Hessian action on vectors in terms of the solution of a pair of linearized forward and adjoint problems. These expressions are simply stated here; an analogous derivation, for the isothermal case, is presented in . The action of the Hessian operator in a given CG direction G̃, evaluated at the current iterate, G, can be expressed in strong form as H(G)(G̃):=-∇⋅(γT∇G̃)-λ̃onΓb,(γT∇G̃)⋅n‾on∂Γb. Beyond the forward and adjoint equations that must be solved to evaluate the gradient, the Hessian action requires two additional forward-like equations: the “incremental forward” and “incremental adjoint' equations. These can be derived using second derivatives of the Lagrangian functional . The resulting incremental forward problem, which is to be solved for the incremental forward velocity, pressure, and temperature variables (ũ, p̃, θ̃), is given by ∇⋅ũ=0inΩ,-∇⋅σũ=0inΩ, ρcu⋅∇θ̃-∇⋅(K∇θ̃)=Fθ̃inΩ,θ̃=0,σũn=0onΓt,ũ=0,K∇θ̃⋅n=G̃onΓb,+additionallateralBCs with σũ:=2η(u,θ)I+1-n2nε˙u⊗ε˙uε˙II+ϵε˙ũ-QnRθ2θ̃ε˙u-Ip̃,Fθ̃:=-ρcũ⋅∇θ+2η1+nnε˙u:ε˙u-QnRθ2θ̃ε˙u:ε˙u. Note that the incremental forward problem Eqs. ()–() resembles the forward problem and in fact corresponds to a linearized (with respect to all variables) version of it. Both the operator and the right-hand side depend on the forward variables (u, p, θ), and the right-hand side also depends on the CG direction G̃.

The resulting incremental adjoint problem, to be solved for the incremental adjoint velocity, pressure, and temperature (ṽ, q̃, λ̃), is then given by ∇⋅ṽ=0inΩ,-∇⋅σṽ+ρcλ̃∇θ=∇⋅τv-ρcλ∇θ̃inΩ,-ρcu⋅∇λ̃-∇⋅(K∇λ̃)=Fλ̃+F^inΩ,λ̃=0,σṽn=-B*Bũ-τvnonΓt,ṽ=0,K∇λ̃⋅n=0onΓb,+additionallateralBCs with σṽ:=2η(u,θ)I+1-n2nε˙u⊗ε˙uε˙II+ϵε˙ṽ-1+nnλ̃ε˙u-Iq̃, τv:=2η(u,θ)1-n2nε˙u:ε˙vε˙II+ϵI+ε˙u⊗ε˙v+ε˙v⊗ε˙uε˙II+ϵε˙ũ+1-n2n1-3n2nε˙u:ε˙vε˙u⊗ε˙uε˙II+ϵ2ε˙ũ-1+nnλI+1-n2nε˙u⊗ε˙uε˙II+ϵε˙ũ+QnRθ2θ̃1+nnλε˙u-I+1-n2nε˙u⊗ε˙uε˙II+ϵε˙v, Fλ̃:=2QnRθ2η(u,θ)ε˙u:ε˙ṽ-λ̃ε˙u:ε˙u, F^:=ρcũ⋅∇λ+2Qη(u,θ)nRθ2ε˙ũ:ε˙v+2Qη(u,θ)nRθ21-n2nε˙u:ε˙ũε˙II+ϵε˙u:ε˙v-1+nnλε˙u:ε˙ũ-2Qθ̃η(u,θ)nRθ2QnRθ2+2θ2ε˙u:ε˙v-λε˙u:ε˙u. Note that the incremental adjoint problem Eqs. ()–() resembles the adjoint problem and is in fact its linearization with respect to the forward and adjoint variables and the unknown geothermal heat flux. Its operator depends on the forward variables only (as does the incremental forward operator), while its right-hand-side source terms depend not just on the forward variables (as does the incremental forward problem) but also on the adjoint and incremental forward variables.

In conclusion, to evaluate the expression for the gradient Eq. () for a given value of the geothermal heat flux G, we first solve the forward problem Eqs. ()–(), followed by the adjoint problem Eqs. ()–() (given the forward solution). To then evaluate the Hessian action Eq. () in a given direction G̃ at each CG iteration, we solve the incremental forward problem Eqs. ()–() (given the forward solution) and then solve the incremental adjoint equation (Eqs. –) (given the forward, adjoint, and incremental forward solutions).

It is well known that the Newton update direction computed by solving Eq. () is a descent direction only if the Hessian is positive definite, which is only guaranteed close to a minimizer . The remedy we apply here (for more details, see ) is to neglect terms in the Hessian expression that involve the adjoint variable, that is, the terms highlighted in blue in Eqs. ()–(). This leads to the so-called Gauss–Newton approximation of the Hessian , which (with appropriate regularization) is guaranteed to be positive definite. Moreover, since accurate solution of the Newton system Eq. () is needed only close to the minimum of the regularized data misfit functional J, we terminate the CG iterations early for iterates that are far from the converged solution. This so-called “inexact Newton” method terminates the CG iterations when the norm of the residual of the linear system Eq. () drops below a tolerance that is proportional to the norm of the gradient; i.e., we terminate at CG iteration i when |H(G)G̃i+G(G)|≤β|G(G)|, where the so-called forcing term β itself can depend on |G(G)|. Far from the minimum – when the relative gradient is large – the tolerance is also large, and the CG iterations are terminated early to prevent over-solving. As the minimum is approached, the norm of the gradient decreases, thereby enforcing an increasingly more accurate solution of the Newton system Eq. (). The criterion above is often able to significantly reduce the overall number of CG iterations – and thus the required number of incremental forward/adjoint solves – while still maintaining fast local convergence. When β is taken as the order of square root of the gradient, the inexact Newton method retains superlinear convergence .

It is critical that the total number of CG iterations be as small as possible, since as mentioned above, each iteration requires a pair of forward/adjoint incremental problem solves. Despite the reduction in overall number of CG iterations provided by inexact solution of the Newton step, the number can still be large when a good preconditioner is not used. An effective preconditioner is simply the inverse of the regularization operator, which amounts to a Laplacian solve on the basal surface. This is because the Hessian of the data misfit operator, like many ill-posed infinite-dimensional inverse operators, has eigenvalues that decay to zero; preconditioning by an inverse Laplacian simply increases the rate of decay. Thus the resulting preconditioned Hessian behaves like a compact perturbation of the identity with smooth dominant eigenfunctions, for which CG converges rapidly and in a number of iterations that is independent of the mesh size; see, for example, . This is the preconditioner we use in the numerical examples below.

Once a descent direction is computed by inexact solution of the Newton step equation (Eq. ), we must guarantee that sufficient decrease in J is obtained in that direction so that convergence of the iterations can be assured. This is achieved by a line search that finds a step size α satisfying the so-called Armijo condition , which has the attractive property that it requires only cost functional evaluations and not gradient information. The Newton iterations are repeated until the norm of the gradient of J is sufficiently small. The inexact Newton method is summarized in Algorithm , in which we use μ = 0.5 and ν = 10-4 for the line search.

In this section, we describe the discretization of the forward and the inverse problems and discuss a stabilization technique required to avoid oscillations for advection-dominated problems. We compare two approaches for computing the gradient of the cost functional, namely the OTD and the DTO approaches.

In this section, we study properties of the inverse problem to infer the unknown geothermal heat flux field from surface velocity observations. In particular, we study the limits of our ability to invert for the heat flux as a function of the length scales of the heat flux and of the noise level in the velocity observations.

We consider a two- and a three-dimensional ice slab, Ω, of length L = 80 km and the surface elevation s given by s(x)=H-H0cos⁡πx2L+H0, where x ∈ [0, L], H = 2 km is the maximum ice thickness, and H0 = 0.1 km is the ice thickness at the outflow boundary Γo. The coordinate system and the ice slab domain for the two-dimensional problem are shown in Fig. ; the three-dimensional geometry is an extrusion in the y direction of this geometry.

In all model problems, we assume that the surface temperature increases as the elevation decreases as follows: θs(x)=θ0+a(H-s(x)), where θ0 = -50 ∘C is the temperature at x = 0, and a is the lapse rate, taken to be a = 6.5 ∘C km-1.

The boundary conditions are as follows.

On the top surface, Γt, we assume zero traction for the velocity and the surface temperature θs defined in Eq. (), i.e.,σun=0,θ=θs.

Multiphysics forward problems are commonly solved using so-called “one-way coupled” or “operator-split” approaches. For example, for a coupled problem with two physics components, the first physics subproblem would be solved assuming the state variables of the second physics subproblem remain fixed, after which the second physics subproblem is solved using the just-computed first physics state variables. One then iterates until convergence, which is guaranteed only if the spectral radius of a certain iteration matrix is less than unity. If the iteration converges, it converges to the correct solution. Such one-way coupled solvers have been used successfully for ice flow forward problems , in which case the solver iterates back and forth between Stokes and energy equation solves, passing velocities from the former to the latter, and temperatures from the latter to the former. The convergence rate is only linear, as opposed to quadratic for a fully coupled Newton forward solver, but one might still prefer the one-way coupled approach due to its ability to capitalize on existing single-physics solvers and codes, its avoidance of computing Jacobians of coupling terms, and the difficulties of designing preconditioners for the fully coupled Jacobian. Therefore, it is tempting to use the same operator from a one-way coupled forward solver to also solve the adjoint problem during inversion. However, this also leads to an incorrect adjoint operator, since it discards some of the coupling blocks within the operator. This in turn leads to an incorrect gradient, which can lead to inaccurate or incorrect solutions of the inverse problem, depending on how strong the coupling terms in the Jacobian of the fully coupled problem are. In this section we illustrate this issue using the multiphysics inverse problem given by the coupled system consisting of the Stokes equations (Eqs.  and ) and the energy equation (Eq. ). In the rest of this section, to simplify the notation, we drop the h superscripts on discrete variables.

In the following discussion, we express the forward problem Eqs. ()–() in terms of the residuals of the discretized equations, as follows: ru(u,p,θ)=0,rp(u)=0,rθ(u,θ)=0, where u, p, and θ denote the discretized velocity, pressure, and temperature, respectively, and ru, rp, and rθ are the discrete residuals of the momentum, mass, and energy equations, respectively. The discrete adjoint system corresponding to Eqs. ()–() can be written as BuuTBupBθuTBupT00BuθT0BθθTvqλ=fv00, where v, q, and λ denote the discretized adjoint velocity, pressure, and temperature, respectively, and fv is the right-hand side of the discrete adjoint momentum equation corresponding to the misfit term in Eq. (). Here, Buu = ∂ru/∂u, Bup = ∂ru/∂p, Buθ = ∂ru/∂θ, Bθu = ∂rθ/∂u, and Bθθ = ∂rθ/∂θ. We note that the submatrix [BuuT, Bup; BupT, 0] is the transpose of the linearized discrete Stokes operator (which is in fact symmetric), and Buθ and Bθu are Jacobians of the coupling terms between the Stokes and the energy equations. In particular, Buθ is the term corresponding to the derivative of the momentum residual ru with respect to the temperature, and Bθu is the term corresponding to the derivative of the energy residual rθ with respect to the velocity. We call the gradients obtained when neglecting either of these coupling matrices in the adjoint systems “one-way coupled gradients” and denote these by Gowc. Next, we study the consequences of the use of one-way coupled gradients on the inversion.

As an illustration of neglecting Jacobians of coupling terms in the adjoint equation, we neglect BθuT in Eq. (). Note that we retain BuθT in the adjoint operator. This allows us to uncouple the adjoint Stokes equation from the adjoint energy equation, leading to a block triangular system. This can be solved by first solving for the adjoint velocity and pressure and then computing the adjoint temperature using the just-computed adjoint velocity. Because we have neglected a block within the adjoint operator, we obtain an incorrect adjoint solution, which then leads to an incorrect gradient. How incorrect the gradient is depends on the “magnitude” of Bθu. To study the implications of using the resulting one-way coupled gradient Gowc in an inverse problem, we compare the inverse solution based on the one-way coupled gradient with the solution obtained with the exact gradient Gexact. We summarize our findings in Fig. . Since the one-way coupled gradient is not the correct gradient of the cost functional J, a descent direction is not guaranteed, and as a result the Gauss–Newton method for solving the inverse problem Eq. () terminates when the search direction based on the one-way coupled gradient is not a descent direction for J. Thus the solution of the inverse problem based on the one-way coupled gradient can differ from the correct solution not only because the incorrect optimality condition is being satisfied but also because the search direction can terminate prematurely due to inconsistency of the gradient and cost functional. However, despite the fact that we attempt to solve the wrong optimality conditions (i.e., vanishing of the one-way coupled gradient rather than the exact gradient) and despite the premature termination, one can still obtain a reasonable approximation of the inverse solution. This is depicted in Fig. a, which shows the inferred geothermal heat flux based both on the exact gradient and on the one-way coupled gradient. As can be seen, they are close to each other.

In Fig. b we show the convergence of the norm of the gradient for two iterations, one corresponding to the correct gradient and one corresponding to the one-way coupled gradient. Note that the one-way coupled gradient iteration terminates prematurely after 11 iterations. Figure c explores why the one-way coupled iteration terminated early.

First, we plot the angle between the exact gradient Gexact and the one-way coupled gradient Gowc, i.e, cos⁡ϕ1=〈Gexact,Gowc〉Γb|Gexact|L2|Gowc|L2. As can be seen, initially the one-way coupled gradient direction coincides with the exact gradient direction, but the angle between them increases substantially in the later iterations. Beyond this incorrectness, the Gauss–Newton search direction based on the one-way coupled gradient is not even a descent direction for the cost function J, leading to the premature termination of the Gauss–Newton iterations. This is because the one-way coupled gradient is not consistent with the contours of J (which are computed using the correct forward model). Note that a search direction G̃ is a descent direction only if its angle ϕ2 with respect to the negative gradient direction -Gexact is less than π/2, i.e., cos⁡(ϕ2) > 0 p. 21. The cosine of ϕ2 is thus given by cos⁡ϕ2=〈-Gexact,G̃〉Γb|Gexact|L2|G̃|L2, where G̃ is the Newton search direction. Figure c plots the values of cos⁡ϕ2. As can be seen, the line search with the search direction based on the one-way coupled approach fails at iteration 11, when cos⁡(ϕ2) < 0. In other words, not only is the computed search direction incorrect (relative to that of a correct Gauss–Newton step), but it does not even point downhill!

These results illustrate several important characteristics of approximations made in inverse problems governed by multiphysics forward models. First, discarding the Jacobians of coupling terms within the adjoint operator can result in substantially incorrect gradients. This could lead to incorrect solution of the inverse problem due to the fact that the vanishing of the gradient constitutes the first-order necessary condition for solution of the inverse problem. It could also lead to premature termination of the iterations due to the loss of a descent direction stemming from inconsistency of the gradient with the contours of the cost function. Second, despite the incorrect gradient, it may still be possible to obtain a reasonable solution to the inverse problem, particularly when the discrepancy between exact and approximate gradients remains small for a sufficient number of iterations to provide a good approximate inverse solution.

We have formulated an inverse problem for estimating the uncertain geothermal heat flux at the base of an ice sheet or glacier in a thermomechanically coupled nonlinear Stokes model from surface velocity observations. Since the forward problem involves an advection-dominated energy equation, a SUPG stabilization was used to suppresses non-physical oscillations in the temperature field. This required use of a discretize-then-optimize approach to compute adjoint-based gradients and Hessians. We advocated an inexact Newton method to solve the discretized inverse problem. Using two- and three-dimensional model problems, we studied the identifiability of the geothermal heat flux field on the basal boundary. We found that the quality of the reconstruction deteriorates with shorter-wavelength variations of this heat flux and with increasing noise in the observations. In particular, a geothermal heat flux with a mean value of 0.06 W m-2 can be reconstructed accurately from observations that contain 1 % noise (SNR = 100) when the wavelength-to-ice-thickness ratio is ∼20 and from observations that contain 5 % noise (SNR = 20) for a wavelength-to-ice-thickness ratio of ∼ 40. In addition, we studied the influence of the number of observations and the mesh resolution on the reconstruction and found that the reconstruction improves substantially as the number of observation points increases, provided the discretization is fine enough.

Moreover, we derived expressions for the gradient and the Hessian of the cost functional for a fully thermomechanically coupled Stokes forward model. We discussed problems that can occur when the gradient is approximated by a so-called one-way coupled approach, in which the two-way coupling of Stokes and the energy equations is replaced by one-way coupling, as is frequently done within forward solvers. The results show that the inversion based on a one-way coupled approach can fail to converge due to the inconsistency of the gradient and the cost functional, leading to the loss of a descent direction. Nevertheless, one might still obtain a reasonable approximate inverse solution, particularly if important features of the reconstructed solution emerge early in optimization iterations, before the iterations terminate prematurely.

We have used synthetic observations on idealized geometries to probe the limits of invertibility for the geothermal heat flux field. We have assumed that the ice is cold everywhere and thus enforced a no-slip boundary condition at the base. In reality, the ice may reach the pressure melting point at some basal locations. This requires a different set of boundary conditions, which account for ice either below or at the melting point. Solution of thermomechanically coupled ice flow models with such variational inequality boundary conditions is the subject of our current work.