An efficient algorithm is proposed for Bayesian model calibration,which is commonly used to estimate the model parameters of non-linear,computationally expensive models using measurement data.The approach is based on ...An efficient algorithm is proposed for Bayesian model calibration,which is commonly used to estimate the model parameters of non-linear,computationally expensive models using measurement data.The approach is based on Bayesian statistics:using a prior distribution and a likelihood,the posterior distribution is obtained through application of Bayes’law.Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods.The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior.To determine such a nodal set an extension to weighted Leja nodes is introduced,based on a new weighting function.We prove that the convergence of the posterior has the same rate as the convergence of the model.If the convergence of the posterior is measured in the Kullback–Leibler divergence,the rate doubles.The algorithm and its theoretical properties are verified in three different test cases:analytical cases that confirm the correctness of the theoretical findings,Burgers’equation to show its applicability in implicit problems,and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computa-tionally expensive problems.展开更多
We present a brief survey on (Weakly) Admissible Meshes and corresponding Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points.These provide new computational tools for polynomial least s...We present a brief survey on (Weakly) Admissible Meshes and corresponding Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points.These provide new computational tools for polynomial least squares and interpolationon multidimensional compact sets, with different applications such as numerical cubature, digital filtering, spectral and high-order methods for PDEs.展开更多
文摘An efficient algorithm is proposed for Bayesian model calibration,which is commonly used to estimate the model parameters of non-linear,computationally expensive models using measurement data.The approach is based on Bayesian statistics:using a prior distribution and a likelihood,the posterior distribution is obtained through application of Bayes’law.Our novel algorithm to accurately determine this posterior requires significantly fewer discrete model evaluations than traditional Monte Carlo methods.The key idea is to replace the expensive model by an interpolating surrogate model and to construct the interpolating nodal set maximizing the accuracy of the posterior.To determine such a nodal set an extension to weighted Leja nodes is introduced,based on a new weighting function.We prove that the convergence of the posterior has the same rate as the convergence of the model.If the convergence of the posterior is measured in the Kullback–Leibler divergence,the rate doubles.The algorithm and its theoretical properties are verified in three different test cases:analytical cases that confirm the correctness of the theoretical findings,Burgers’equation to show its applicability in implicit problems,and finally the calibration of the closure parameters of a turbulence model to show the effectiveness for computa-tionally expensive problems.
文摘We present a brief survey on (Weakly) Admissible Meshes and corresponding Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points.These provide new computational tools for polynomial least squares and interpolationon multidimensional compact sets, with different applications such as numerical cubature, digital filtering, spectral and high-order methods for PDEs.
