Model Variational Inverse Problems Governed by Partial Differential Equations

Petra, Noémi; Stadler, Georg

doi:10.21236/ada555315

Cited by 44 publications

(45 citation statements)

References 25 publications

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“…In light of these observations, we do not present any detailed results using the continuous adjoint formulation since we do not see any advantages to this approach for our problem. We note that these finding are consistent with those reported in [19] and [42] for general Runge-Kutta time stepping methods, in [27], where the discrete and continuous adjoint approaches were applied to automatic aerodynamic optimization, and in [29] for general variational inverse problems governed by partial differential equations. A similar gradient discrepancy between discretization/optimization versus optimization/discretization can also occur with respect to the spatial discretization -see the discussion for a shape optimization problem in [18].…”

Section: Continuous Versus Discrete Adjoint Formulationsupporting

confidence: 90%

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

et al. 2014

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An adjoint formulation for the gradient-based optimization of oil-gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford's AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil-water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented -a formal treatment within the optimizer, and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. duced) relative to reference solutions range from 4.2% to 11.6%. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are in some cases better than those obtained using the formal constraint handling procedure.

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Section: Continuous Versus Discrete Adjoint Formulationsupporting

confidence: 90%

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

et al. 2014

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“…We use Taylor-Hood finite elements (Elman et al 2005), that is, continuous second-order elements for the velocity components, and continuous first-order elements for the pressure. Our implementation is in MATLAB 1 , and we use COMSOL v3.5 2 for meshing and for the assembly of finite element matrices, similar to the model problems in Petra & Stadler (2011).…”

Section: O D E L S E T U P a N D N U M E R I C A L S O L U T I O Nmentioning

confidence: 99%

Adjoint-based estimation of plate coupling in a non-linear mantle flow model: theory and examples

Ratnaswamy

Stadler

Gurnis

2015

Geophysical Journal International

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S U M M A R YWe develop and validate a systematic approach to infer plate boundary strength and rheological parameters in models of mantle flow from surface velocity observations. Based on a realistic rheological model that includes yielding and strain rate weakening from dislocation creep, we formulate the inverse problem in a Bayesian inference framework. To study the distribution of parameters that are consistent with the observations, we compute the maximum a posteriori (MAP) point, Gaussian approximations of the parameter distribution around that MAP point, and employ Markov Chain Monte Carlo (MCMC) sampling methods. The computation of the MAP point and the Gaussian approximation require first and second derivatives of an objective function subject to non-linear Stokes equations; these derivatives are computed efficiently using adjoint Stokes equations. We set up 2-D numerical experiments with many of the elements expected in a global geophysical inversion. This setup incorporates three subduction zones with slab and weak zone (interplate fault) geometry consistent with average seismic characteristics. With these experiments, we demonstrate that when the temperature field is known, we can recover the strength of plate boundaries, the yield stress and strain rate exponent in the upper mantle. When the number of uncertain parameters increases, there are trade-offs between the inferred parameters. These trade-offs depend on how well the observational data represents the surface velocities, and on the weakness of plate boundaries. As the plate boundary coupling drops below a threshold, the uncertainty of the inferred parameters increases due to insensitivity of plate motion to plate coupling. Comparing the trade-offs between inferred rheological parameters found from the Gaussian approximation of the parameter distribution and from MCMC sampling, we conclude that the Gaussian approximation-which is significantly cheaper to compute-is often a good approximation, in particular locally around the MAP point. Thus, the method can be applied to the global problem of inferring non-linear constitutive parameters and plate coupling factors for each subduction zone in a global geophysical inversion with known slab structure.

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“…The boundary ∂D includes both the external boundary and the building walls. The velocity field v, shown in Figure 2 (right), is obtained by solving a steady Navier-Stokes equation as detailed in [58,59].…”

Section: : End Functionmentioning

confidence: 99%

Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems

2018

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We develop a framework for goal-oriented optimal design of experiments (GOODE) for largescale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g., a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A-and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an 1 -norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.

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Model Variational Inverse Problems Governed by Partial Differential Equations

Cited by 44 publications

References 25 publications

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

Adjoint-based estimation of plate coupling in a non-linear mantle flow model: theory and examples

Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems

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