A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems

Neitzel, Ira; Pieper, Konstantin; Vexler, Boris; Walter, Daniel

doi:10.1007/s00211-019-01073-3

Cited by 14 publications

(29 citation statements)

References 33 publications

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“…In this case, there are various different "information criteria" F to evaluate the quality of the overall measurement setup u, which are usually convex, smooth, but extended real valued functionals (allowing for the value +∞). G is often chosen to be a convex indicator function to enforce u M(Ω) ≤ 1, but a cost term as in (P source ) can also be considered; see, e.g., [50].…”

Section: Introductionmentioning

confidence: 99%

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Pieper

Walter

2021

ESAIM: COCV

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A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear O(1/k) rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear O (\zeta^k) convergence rate is obtained locally.

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Section: Introductionmentioning

confidence: 99%

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Pieper

Walter

2021

ESAIM: COCV

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“…We build on previous work [3,13,15], which developed efficient methods for computing A-optimal designs for high-or infinite-dimensional linear inverse problems using either a Bayesian or a frequentist approach. Other approaches for computing optimal experimental designs for high/infinite-dimensional linear inverse problems are explored, for example, in [2,5,22]. In [22], the authors propose a measure-based OED formulation that does not choose sensor locations from a finite number of candidate locations but allows sensors to be placed anywhere on a closed subset of the domain.…”

Section: Arxiv:191208915v1 [Mathoc] 18 Dec 2019mentioning

confidence: 99%

“…Other approaches for computing optimal experimental designs for high/infinite-dimensional linear inverse problems are explored, for example, in [2,5,22]. In [22], the authors propose a measure-based OED formulation that does not choose sensor locations from a finite number of candidate locations but allows sensors to be placed anywhere on a closed subset of the domain. The articles [2,5] explore an alternate OED criterion-D-optimality-for infinite-dimensional Bayesian linear inverse problems.…”

Section: Arxiv:191208915v1 [Mathoc] 18 Dec 2019mentioning

confidence: 99%

“…In this section, we characterize the irreducible uncertainties we must take into account when computing optimal sensor locations for the inverse problem presented in section 6.2. The sources of irreducible uncertainty are the advection velocity v and the initial time T 0 in (22). As explained further below, the uncertainty in the velocity field (24) stems from the log-permeability field θ(x) of the aquifer being uncertain.…”

Section: Subsurface Flow Example: Irreducible Uncertaintiesmentioning

confidence: 99%

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Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs

2020

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We present a method for computing A-optimal sensor placements for infinitedimensional Bayesian linear inverse problems governed by PDEs with irreducible model uncertainties. Here, irreducible uncertainties refers to uncertainties in the model that exist in addition to the parameters in the inverse problem, and that cannot be reduced through observations. Specifically, given a statistical distribution for the model uncertainties, we compute the optimal design that minimizes the expected value of the posterior covariance trace. The expected value is discretized using Monte Carlo leading to an objective function consisting of a sum of trace operators and a binary-inducing penalty. Minimization of this objective requires a large number of PDE solves in each step. To make this problem computationally tractable, we construct a composite low-rank basis using a randomized range finder algorithm to eliminate forward and adjoint PDE solves. We also present a novel formulation of the A-optimal design objective that requires the trace of an operator in the observation rather than the parameter space. The binary structure is enforced using a weighted regularized 0 -sparsification approach. We present numerical results for inference of the initial condition in a subsurface flow problem with inherent uncertainty in the flow fields and in the initial times.

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“…We specifically focus on experiments where sensors need to be positioned and inputs must be chosen in order to achieve a maximum information gain for the estimated values of the model parameters. Optimal sensor placement has been addressed within the PDE-context in [1,2,23] and optimal input configuration has been extensively analyzed for both linear and nonlinear ordinary differential equations in various engineering applications [6,17,21,22,28]. However, in these cases the problem dimension is small compared to a (discretized) time-variant PDE and thus, gradient-based optimization with a sensitivity approach, as suggested by [4] and [19], works fine.…”

Section: Introductionmentioning

confidence: 99%

Detection of Model Uncertainty in the Dynamic Linear-Elastic Model of Vibrations in a Truss

Matei

Ulbrich

2021

Lecture Notes in Mechanical Engineering

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Dynamic processes have always been of profound interest for scientists and engineers alike. Often, the mathematical models used to describe and predict time-variant phenomena are uncertain in the sense that governing relations between model parameters, state variables and the time domain are incomplete. In this paper we adopt a recently proposed algorithm for the detection of model uncertainty and apply it to dynamic models. This algorithm combines parameter estimation, optimum experimental design and classical hypothesis testing within a probabilistic frequentist framework. The best setup of an experiment is defined by optimal sensor positions and optimal input configurations which both are the solution of a PDE-constrained optimization problem. The data collected by this optimized experiment then leads to variance-minimal parameter estimates. We develop efficient adjoint-based methods to solve this optimization problem with SQP-type solvers. The crucial test which a model has to pass is conducted over the claimed true values of the model parameters which are estimated from pairwise distinct data sets. For this hypothesis test, we divide the data into k equally-sized parts and follow a k-fold cross-validation procedure. We demonstrate the usefulness of our approach in simulated experiments with a vibrating linear-elastic truss.

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A sparse control approach to optimal sensor placement in PDE-constrained parameter estimation problems

Cited by 14 publications

References 33 publications

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs

Detection of Model Uncertainty in the Dynamic Linear-Elastic Model of Vibrations in a Truss

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