This work presents a computationally efficient probabilistic framework that enables the identification of model parameters from noisy measurements of the response. We consider transient PDE-based models, where the parameters correspond to physical properties. An efficient and reliable procedure for estimation of those unknown parameters is pursued. The proposed framework uses a Bayesian approach, an efficient sequential Monte Carlo sampling scheme, and adaptive reduced-order models (ROMs). The Bayesian approach has several advantages including the ability to provide not only point estimates of the quantities of interest, but also measures of credibility and robustness concerning those estimates. The associated sequential Monte Carlo method sampling scheme is embarrassingly parallelizable, as well as efficient in terms of the number of calls to the forward solver (e.g. a finite element code) used in evaluating the likelihood function. We propose to use a ROM adaptation procedure where projection-based ROMs are seen as points on a certain Riemannian manifold and are 'tracked' and interpolated during the sampling process using a database of precomputed ROMs. This approach ensures that an appropriate ROM is used for the likelihood evaluation in every region of the parameter space, thus leading to computational savings while conserving sufficient accuracy. Using numerical examples, we illustrate the capabilities of the proposed framework, and show that it leads to quality estimates with a quantified predictive uncertainty.

The present paper discusses a sampling framework that enables the optimization of complex systems characterized by high-dimensional uncertainties and design variables. We are especially concerned with problems relating to random heterogeneous materials where uncertainties arise from the stochastic variability of their properties. In particular, we reformulate topology optimization problems to account for the design of truly random composites. In addition, we address the optimal perscription of input loads/excitations in order to achieve a target response by the random material system. The methodological advances proposed in this paper consist of an adaptive Sequential Monte Carlo scheme that economizes the number of runs of the forward solver and allows the analyst to identify several local maxima that provide important information with regards to the robustness of the design. We further propose a principled manner of introducing information from approximate models that can ultimately lead to further reductions in computational cost.

We have developed an efficient convex optimization strategy enabling the simultaneous attenuation of random and erratic noise with interpolation in prestack seismic data. For a particular analysis window, frequency slice spatial data were reorganized into a block Toeplitz matrix with Toeplitz blocks as in Cadzow/singular spectrum analysis methods. The signal and erratic noise were, respectively, modeled as low-rank and sparse components of this matrix, and then a joint low-rank and sparse inversion (JLRSI) enabled us to recover the low-rank signal component from noisy and incomplete data thanks to joint minimization of a nuclear norm term and an L 1-norm term. The convex optimization framework, related to recent developments in the field of compressed sensing, enabled the formulation of a well-posed problem as well as the use of state-of-the-art algorithms. We proposed an alternating directions method of multipliers scheme associated with an efficient singular value thresholding kernel. Numerical results on field data illustrated the effectiveness of the JLRSI approach at recovering missing data and increasing the signal-to-noise ratio.