Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

Saibaba, Arvind K.; Hart, Joseph; Waanders, Bart Gustaaf van Bloemen

doi:10.1002/nla.2364

Cited by 22 publications

(17 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach is efficient for large problems which admit low rank structure, as is commonly observed in engineering applications. We refer the reader to [11,22] for more details.…”

Section: Discretizationmentioning

confidence: 99%

“…Hyper-Differential Sensitivity Analysis (HDSA) utilizes advanced numerical linear algebra for efficient computational of these sensitivities. The algorithmic generality of HDSA has been demonstated on a range of applications problems [11,22,23]. In the context of trajectory planning for hypersonic vehicles, a higher order numerical model is sampled for aerodynamic coefficients in the open-loop formulation to determine solutions that avoid saturating feedback controllers.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sensitivity driven experimental design to facilitate control of dynamical systems

Hart¹,

Waanders²,

Hood³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

Control of nonlinear dynamical systems is a complex and multifaceted process. Essential elements of many engineering systems include high fidelity physics-based modeling, offline trajectory planning, feedback control design, and data acquisition strategies to reduce uncertainties. This article proposes an optimization centric perspective which couples these elements in a cohesive framework. We introduce a novel use of hyper-differential sensitivity analysis to understand the sensitivity of feedback controllers to parametric uncertainty in physics-based models used for trajectory planning. These sensitivities provide a foundation to define an optimal experimental design which seeks to acquire data most relevant in reducing demand on the feedback controller. Our proposed framework is illustrated on the Zermelo navigation problem and a hypersonic trajectory control problem using data from NASA's X-43 hypersonic flight tests.

show abstract

“…This approach is efficient for large problems which admit low rank structure, as is commonly observed in engineering applications. We refer the reader to [11,22] for more details.…”

Section: Discretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Sensitivity driven experimental design to facilitate control of dynamical systems

Hart¹,

Waanders²,

Hood³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…we conduct all the numerical experiments with A = U ΣV . This test case is directly inspired from [22,23]. We consider two different target ranks k (k = 5 and k = 15, respectively) to allow a variety of results and comments.…”

Section: 2mentioning

confidence: 99%

A general error analysis for randomized low-rank approximation methods

Diouane¹,

Gürol²,

Perrotolo³

et al. 2022

Preprint

View full text Add to dashboard Cite

We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we derive error bounds in expectation in the non-standard Gaussian case, assuming a nontrivial mean and a general covariance matrix for the random matrix variable. The proposed analysis generalizes and improves the error bounds for spectral and Frobenius norms proposed by Halko, Martinsson and Tropp. Third, we consider the Randomized Singular Value Decomposition and specialize our error bounds in expectation in this setting. Numerical experiments on an instructional synthetic test case demonstrate the tightness of the new error bounds.

show abstract

“…Each matrix-vector product with the misfit Hessian requires two PDE solves (assuming that adjoints are used [30]). Randomized methods afford greater parallelism than iterative methods when adequate computational resources are available [16,27]. Algorithm 5.2 adapts Algorithm 6 in [28] and provides a scalable (in inversion parameter dimension) approach to solve the Hessian GEVP by iterating on the number of desired eigenvalues (defined by the while loop at Line 5 of Algorithm 5.2) whose termination criteria comes from the interpretation of the eigenvalues as a ratio of contributions from the likelihood and prior (4.2).…”

Section: Proof (→)mentioning

confidence: 99%

“…Hyper-differential sensitivity analysis. Through a combination of tools from post-optimality sensitivity analysis, PDE-constrained optimization, and numerical linear algebra, HDSA has provided unique and valuable insights for optimal control and deterministic inverse problems [16,27,30]. This subsection provides essential background to prepare for our extension of HDSA to Bayesian inverse problems which follows.…”

mentioning

confidence: 99%

Enabling hyper-differential sensitivity analysis for ill-posed inverse problems

Hart¹,

Waanders²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and using techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always feasible or appropriate as a result of ill-posedness in the inverse problem. We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates. Our proposed framework is demonstrated on a nonlinear multi-physics inverse problem motivated by estimation of spatially heterogenous material properties in the presence of spatially distributed parametric modeling uncertainties.

show abstract

Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

Cited by 22 publications

References 30 publications

Sensitivity driven experimental design to facilitate control of dynamical systems

Sensitivity driven experimental design to facilitate control of dynamical systems

A general error analysis for randomized low-rank approximation methods

Enabling hyper-differential sensitivity analysis for ill-posed inverse problems

Contact Info

Product

Resources

About