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Register a new userAdaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification
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Added by
Tim Sullivan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design.
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Classical a posteriori error analysis for differential equations quantifies the error in a Quantity of Interest (QoI) which is represented as a bounded linear functional of the solution. In this work we consider a posteriori error estimates of a quantity of interest that cannot be represented in this fashion, namely the time at which a threshold is crossed for the first time. We derive two representations for such errors and use an adjoint-based a posteriori approach to estimate unknown terms that appear in our representation. The first representation is based on linearizations using Taylors Theorem. The second representation is obtained by implementing standard root-finding techniques. We provide several examples which demonstrate the accuracy of the methods. We then embed these error estimates within a framework to provide error bounds on a cumulative distribution function when parameters of the differential equations are uncertain.
This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) method and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF) algorithm. The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier-Stokes (RANS) multi-grid solver $chi$navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementations of both algorithms, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBFs seem to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.
This paper analyses the following question: let $mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $ ablacdot (A_j abla u_j) + k^2 n_j u_j= -f$. How small must $|A_1 -A_2|_{L^q}$ and $|{n_1} - {n_2}|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(mathbf{A}_1)^{-1}mathbf{A}_2$ or $mathbf{A}_2(mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $mathbf{A}_1$ to be a good left- or right-preconditioner for $mathbf{A}_2$?). We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients $A$ and $n$. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different $A$ and $n$, and the answer to the question above dictates to what extent a previously-calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.
We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared to a set of benchmark methods including Kriging and Inverse distance weighting. Random Fourier features is a linear model $beta(pmb x) = sum_{k=1}^K beta_k e^{iomega_k pmb x}$ approximating the velocity field, with frequencies $omega_k$ randomly sampled and amplitudes $beta_k$ trained to minimize a loss function. We include a physically motivated divergence penalty term $| abla cdot beta(pmb x)|^2$, as well as a penalty on the Sobolev norm. We derive a bound on the generalization error and derive a sampling density that minimizes the bound. Following (arXiv:2007.10683 [math.NA]), we devise an adaptive Metropolis-Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.
We propose a novel $hp$-multilevel Monte Carlo method for the quantification of uncertainties in the compressible Navier-Stokes equations, using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes while simultaneously increasing the polynomial degree of the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-multilevel Monte Carlo method to compute the mean field with prescribed accuracy is, in best-case, of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overestimating these quantities. The method is in particular designed for application on queue-based computing systems, where it is desirable to compute a large number of samples during one iteration, without overestimating the optimal number of samples. Our theoretical results are verified by numerical experiments for the two-dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.
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