Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from synthetic surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian approximation error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example, the basal sliding coefficient field is the parameter of primary interest which we seek to infer, and the rheology parameters (e.g., the flow rate factor or the Glen's flow law exponent coefficient field) represent so-called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.

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Inferring the basal sliding coefficient field for the Stokes ice sheet model under rheological uncertainty

10.5194/tc-2020-229 ◽

2020 ◽

Author(s):

Olalekan Babaniyi ◽

Ruanui Nicholson ◽

Umberto Villa ◽

Noémi Petra

Keyword(s):

Model Uncertainty ◽

Approximation Error ◽

Ice Sheet ◽

Forward Model ◽

Three Dimensions ◽

Low Rank ◽

Surface Velocity ◽

Low Rank Approximation ◽

Basal Sliding ◽

Bayesian Approximation

Abstract. We consider the problem of inferring the basal sliding coefficient field for an uncertain Stokes ice sheet forward model from surface velocity measurements. The uncertainty in the forward model stems from unknown (or uncertain) auxiliary parameters (e.g., rheology parameters). This inverse problem is posed within the Bayesian framework, which provides a systematic means of quantifying uncertainty in the solution. To account for the associated model uncertainty (error), we employ the Bayesian Approximation Error (BAE) approach to approximately premarginalize simultaneously over both the noise in measurements and uncertainty in the forward model. We also carry out approximative posterior uncertainty quantification based on a linearization of the parameter-to-observable map centered at the maximum a posteriori (MAP) basal sliding coefficient estimate, i.e., by taking the Laplace approximation. The MAP estimate is found by minimizing the negative log posterior using an inexact Newton conjugate gradient method. The gradient and Hessian actions to vectors are efficiently computed using adjoints. Sampling from the approximate covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. We study the performance of the BAE approach in the context of three numerical examples in two and three dimensions. For each example the basal sliding coefficient field is the parameter of primary interest, which we seek to infer, and the rheology parameters (e.g., the flow rate factor, or the Glen's flow law exponent coefficient field) represent so called nuisance (secondary uncertain) parameters. Our results indicate that accounting for model uncertainty stemming from the presence of nuisance parameters is crucial. Namely our findings suggest that using nominal values for these parameters, as is often done in practice, without taking into account the resulting modeling error, can lead to overconfident and heavily biased results. We also show that the BAE approach can be used to account for the additional model uncertainty at no additional cost at the online stage.

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Swarm Intelligence for Non-Negative Matrix Factorization

International Journal of Swarm Intelligence Research ◽

10.4018/jsir.2011100102 ◽

2011 ◽

Vol 2 (4) ◽

pp. 12-34 ◽

Cited By ~ 44

Author(s):

Andreas Janecek ◽

Ying Tan

Keyword(s):

Swarm Intelligence ◽

Matrix Factorization ◽

Classification Accuracy ◽

Approximation Error ◽

Synthetic Data ◽

Low Rank ◽

Low Rank Approximation ◽

Approximation Quality ◽

Initialization Strategy ◽

Non Negative Matrix Factorization

The Non-negative Matrix Factorization (NMF) is a special low-rank approximation which allows for an additive parts-based and interpretable representation of the data. This article presents efforts to improve the convergence, approximation quality, and classification accuracy of NMF using five different meta-heuristics based on swarm intelligence. Several properties of the NMF objective function motivate the utilization of meta-heuristics: this function is non-convex, discontinuous, and may possess many local minima. The proposed optimization strategies are two-fold: On the one hand, a new initialization strategy for NMF is presented in order to initialize the NMF factors prior to the factorization; on the other hand, an iterative update strategy is proposed, which improves the accuracy per runtime for the multiplicative update NMF algorithm. The success of the proposed optimization strategies are shown by applying them on synthetic data and data sets coming from the areas of spam filtering/email classification, and evaluate them also in their application context. Experimental results show that both optimization strategies are able to improve NMF in terms of faster convergence, lower approximation error, and better classification accuracy. Especially the initialization strategy leads to significant reductions of the runtime per accuracy ratio for both, the NMF approximation as well as the classification results achieved with NMF.

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An inexact Gauss-Newton method for inversion of basal sliding and rheology parameters in a nonlinear Stokes ice sheet model

Journal of Glaciology ◽

10.3189/2012jog11j182 ◽

2012 ◽

Vol 58 (211) ◽

pp. 889-903 ◽

Cited By ~ 53

Author(s):

Noemi Petrat ◽

Hongyu Zhu ◽

Georg Stadler ◽

Thomas J.R. Hughes ◽

Omar Ghattas

Keyword(s):

Inverse Problem ◽

Newton Method ◽

Ice Sheet ◽

Nonlinear Least Squares ◽

Surface Velocity ◽

Inexact Newton Method ◽

Observation Error ◽

Nonlinear Rheology ◽

Cost Functional ◽

Basal Sliding

AbstractWe propose an infinite-dimensional adjoint-based inexact Gauss-Newton method for the solution of inverse problems governed by Stokes models of ice sheet flow with nonlinear rheology and sliding law. The method is applied to infer the basal sliding coefficient and the rheological exponent parameter fields from surface velocities. The inverse problem is formulated as a nonlinear least-squares optimization problem whose cost functional is the misfit between surface velocity observations and model predictions. A Tikhonov regularization term is added to the cost functional to render the problem well-posed and account for observational error. Our findings show that the inexact Newton method is significantly more efficient than the nonlinear conjugate gradient method and that the number of Stokes solutions required to solve the inverse problem is insensitive to the number of inversion parameters. The results also show that the reconstructions of the basal sliding coefficient converge to the exact sliding coefficient as the observation error (here, the noise added to synthetic observations) decreases, and that a nonlinear rheology makes the reconstruction of the basal sliding coefficient more difficult. For the inversion of the rheology exponent field, we find that horizontally constant or smoothly varying parameter fields can be reconstructed satisfactorily from noisy observations.

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SPAN: A Stochastic Projected Approximate Newton Method

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i02.5511 ◽

2020 ◽

Vol 34 (02) ◽

pp. 1520-1527

Author(s):

Xunpeng Huang ◽

Xianfeng Liang ◽

Zhengyang Liu ◽

Lei Li ◽

Yue Yu ◽

...

Keyword(s):

Theoretical Analysis ◽

Convergence Rate ◽

Newton Method ◽

Hessian Matrix ◽

Approximation Error ◽

Optimization Methods ◽

Second Order ◽

Low Rank ◽

Low Rank Approximation ◽

Benchmark Datasets

Second-order optimization methods have desirable convergence properties. However, the exact Newton method requires expensive computation for the Hessian and its inverse. In this paper, we propose SPAN, a novel approximate and fast Newton method. SPAN computes the inverse of the Hessian matrix via low-rank approximation and stochastic Hessian-vector products. Our experiments on multiple benchmark datasets demonstrate that SPAN outperforms existing first-order and second-order optimization methods in terms of the convergence wall-clock time. Furthermore, we provide a theoretical analysis of the per-iteration complexity, the approximation error, and the convergence rate. Both the theoretical analysis and experimental results show that our proposed method achieves a better trade-off between the convergence rate and the per-iteration efficiency.

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Swarm Intelligence for Dimensionality Reduction

Nature-Inspired Computing ◽

10.4018/978-1-5225-0788-8.ch060 ◽

2016 ◽

pp. 1564-1589 ◽

Cited By ~ 1

Author(s):

Andreas Janecek ◽

Ying Tan

Keyword(s):

Swarm Intelligence ◽

Classification Accuracy ◽

Approximation Error ◽

Low Rank ◽

Data Sets ◽

Low Rank Approximation ◽

Approximation Quality ◽

Compact Representations ◽

Low Rank Approximations ◽

Update Strategy

Low-rank approximations allow for compact representations of data with reduced storage and runtime requirements and reduced redundancy and noise. The Non-Negative Matrix Factorization (NMF) is a special low-rank approximation that allows for additive parts-based, interpretable representation of the data. Various properties of NMF are similar to Swarm Intelligence (SI) methods: indeed, most NMF objective functions and most SI fitness functions are non-convex, discontinuous, and may possess many local minima. This chapter summarizes efforts on improving convergence, approximation quality, and classification accuracy of NMF using five different meta-heuristics based on SI and evolutionary computation. The authors present (1) new initialization strategies for NMF, and (2) an iterative update strategy for NMF. The applicability of the approach is illustrated on data sets coming from the areas of spam filtering and email classification. Experimental results show that both optimization strategies are able to improve NMF in terms of faster convergence, lower approximation error, and/or better classification accuracy.

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