Interpolating the Trace of the Inverse of Matrix A+tB\mathbf{A} + t \mathbf{B}

We develop heuristic interpolation methods for the function $t \mapsto \operatorname{trace}\left( (\mathbf{A} + t \mathbf{B})^{-1} \right)$, where the matrices $\mathbf{A}$ and $\mathbf{B}$ are symmetric and positive definite and $t$ is a real variable. This function is featured in many applications in statistics, machine learning, and computational physics. The presented interpolation functions are based on the modification of a sharp upper bound that we derive for this function, which is a new trace inequality for matrices. We demonstrate the accuracy and performance of the proposed method with numerical examples, namely, the marginal maximum likelihood estimation for linear Gaussian process regression and the estimation of the regularization parameter of ridge regression with the generalized cross-validation method

Estimating the trace of the matrix inverse by interpolating from the diagonal of an approximate inverse

A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the evaluation of the function is expensive, the task is computationally challenging because the standard approach is based on a Monte Carlo method which converges slowly. We present a different approach that exploits the pattern correlation, if present, between the diagonal of the inverse of the matrix and the diagonal of some approximate inverse that can be computed inexpensively. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to the diagonal of the inverse. Depending on the quality of the approximate inverse, our method may serve as a standalone kernel for providing a fast trace estimate with a small number of samples. Furthermore, the method can be used as a variance reduction method for Monte Carlo in some cases. This is decided dynamically by our algorithm. An extensive set of experiments with various technique combinations on several matrices from some real applications demonstrate the potential of our method. (C) 2016 Published by Elsevier Inc

Data analytic UQ cascade

International audienceThis contribution gathers some of the ingredients presented at Erice during the third workshop on 'Variational Analysis and Aerospace Engineering'. It is a collection of several previous publications on how to set up an uncertainty quantification (UQ) cascade with ingredients of growing computational complexity for both forward and reverse uncertainty propagation. It uses data analysis ingredients in a context of existing deterministic simulation platforms. It starts with a complexity-based splitting of the independent variables and the definition of a parametric optimization problem. Geometric characterization of global sensitivity spaces through their dimensions and relative positions through principal angles between vector spaces bring a first set of information on the impact of uncertainties of the functioning parameters on the optimal solution. Joining the multi-point descent direction and Probability Density Function (PDF) quantiles of the optimization parameters permits to define the notion of Directional Extreme Scenarios (DES) without sampling of large dimension design spaces. One goes beyond DES with Ensemble Kalman Filters (EnKF) after the multi-point optimization algorithm is cast into an ensemble simulation environment. This formulation accounts for the variability in large dimension. The UQ cascade continues with the joint application of the EnKF and DES leading to the concept of Ensemble Directional Extreme Scenarios (EDES) which provides a more exhaustive description of the possible extreme scenarios. The different ingredients developed for this cascade also permits to quantify the impact of state uncertainties on the design and provide confidence bounds for the optimal solution. This is typical of inverse designs where the target should be assumed uncertain. Our proposal uses the previous DES strategy applied this time to the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using available adjoint-based gradients of the functional. Eventually , we go beyond inverse design and apply the method to general optimization problems. The ingredients of the paper have been applied to constrained aerodynamic performance analysis problems. 1 Context Our domain of interest is aerodynamic shape optimization. The questions of interest are:-can we propose an aircraft shape designed to have similar performances over a given range of some functioning parameters (to be formulated through the moments of a functional) ?-can we do that modifying as less as possible an existing mono-point optimization shape design loop ?-is it possible for the time-to-solution cost of this parametric shape design to remain comparable to the mono-point situation ? We consider a generic situation where the simulation aims at predicting a given quantity of interest j(x, α) and there are a few functioning or operating parameters α and several design parameters x involved. The ranges of the functioning parameters define the global operating/functioning conditions of a given design. This splitting of the independent variables in two sets is important for the sequel