Corrected kriging update formulae for batch-sequential data assimilation

Recently, a lot of effort has been spent in the efficient computation of kriging predictors when observations are assimilated sequentially. In particular, kriging update formulae enabling significant computational savings were derived. Taking advantage of the previous kriging mean and variance computations helps avoiding a costly matrix inversion when adding one observation to the TeX already available ones. In addition to traditional update formulae taking into account a single new observation, Emery (2009) proposed formulae for the batch-sequential case, i.e. when TeX new observations are simultaneously assimilated. However, the kriging variance and covariance formulae given in Emery (2009) for the batch-sequential case are not correct. In this paper, we fix this issue and establish correct expressions for updated kriging variances and covariances when assimilating observations in parallel. An application in sequential conditional simulation finally shows that coupling update and residual substitution approaches may enable significant speed-ups.

Distance-based kriging relying on proxy simulations for inverse conditioning

Let us consider a large set of candidate parameter fields, such as hydraulic conductivity maps, on which we can run an accurate forward flow and transport simulation. We address the issue of rapidly identifying a subset of candidates whose response best match a reference response curve. In order to keep the number of calls to the accurate flow simulator computationally tractable, a recent distance-based approach relying on fast proxy simulations is revisited, and turned into a non-stationary kriging method where the covariance kernel is obtained by combining a classical kernel with the proxy. Once the accurate simulator has been run for an initial subset of parameter fields and a kriging metamodel has been inferred, the predictive distributions of misfits for the remaining parameter fields can be used as a guide to select candidate parameter fields in a sequential way. The proposed algorithm, Proxy-based Kriging for Sequential Inversion (ProKSI), relies on a variant of the Expected Improvement, a popular criterion for kriging-based global optimization. A statistical benchmark of ProKSI’s performances illustrates the efficiency and the robustness of the approach when using different kinds of proxies.

A benchmark of kriging-based infill criteria for noisy optimization

Responses of many real-world problems can only be evaluated perturbed by noise. In order to make an efficient optimization of these problems possible, intelligent optimization strategies successfully coping with noisy evaluations are required. In this article, a comprehensive review of existing kriging-based methods for the optimization of noisy functions is provided. In summary, ten methods for choosing the sequential samples are described using a unified formalism. They are compared on analytical benchmark problems, whereby the usual assumption of homoscedastic Gaussian noise made in the underlying models is meet. Different problem configurations (noise level, maximum number of observations, initial number of observations) and setups (covariance functions, budget, initial sample size) are considered. It is found that the choices of the initial sample size and the covariance function are not critical. The choice of the method, however, can result in significant differences in the performance. In particular, the three most intuitive criteria are found as poor alternatives. Although no criterion is found consistently more efficient than the others, two specialized methods appear more robust on average.

KrigInv: An efficient and user-friendly implementation of batch-sequential inversion strategies based on kriging

Several strategies relying on kriging have recently been proposed for adaptively estimating contour lines and excursion sets of functions under severely limited evaluation budget. The recently released R package KrigInv 3 is presented and offers a sound implementation of various sampling criteria for those kinds of inverse problems. KrigInv is based on the DiceKriging package, and thus benefits from a number of options concerning the underlying kriging models. Six implemented sampling criteria are detailed in a tutorial and illustrated with graphical examples. Different functionalities of KrigInv are gradually explained. Additionally, two recently proposed criteria for batch-sequential inversion are presented, enabling advanced users to distribute function evaluations in parallel on clusters or clouds of machines. Finally, auxiliary problems are discussed. These include the fine tuning of numerical integration and optimization procedures used within the computation and the optimization of the considered criteria.

Noisy kriging-based optimization methods: A unified implementation within the DiceOptim package

Kriging-based optimization relying on noisy evaluations of complex systems has recently motivated contributions from various research communities. Five strategies have been implemented in the DiceOptim package. The corresponding functions constitute a user-friendly tool for solving expensive noisy optimization problems in a sequential framework, while offering some flexibility for advanced users. Besides, the implementation is done in a unified environment, making this package a useful device for studying the relative performances of existing approaches depending on the experimental setup. An overview of the package structure and interface is provided, as well as a description of the strategies and some insight about the implementation challenges and the proposed solutions. The strategies are compared to some existing optimization packages on analytical test functions and show promising performances.

Radiographic and safety details of vertebral body stenting: results from a multicenter chart review

BACKGROUND Up to one third of BKP treated cases shows no appreciable height restoration due to loss of both restored height and kyphotic realignment after balloon deflation. This shortcoming has called for an improved method that maintains the height and realignment reached by the fully inflated balloon until stabilization of the vertebral body by PMMA-based cementation. Restoration of the physiological vertebral body height for pain relief and for preventing further fractures of adjacent and distant vertebral bodies must be the main aim for such a method. A new vertebral body stenting system (VBS) stabilizes the vertebral body after balloon deflation until cementation. The radiographic and safety results of the first 100 cases where VBS was applied are presented. METHODS During the planning phase of an ongoing international multicenter RCT, radiographic, procedural and followup details were retrospectively transcribed from charts and xrays for developing and testing the case report forms. Radiographs were centrally assessed at the institution of the first/senior author. RESULTS 100 patients (62 with osteoporosis) with a total of 103 fractured vertebral bodies were treated with the VBS system. 49 were females with a mean age of 73.2 years; males were 66.7 years old. The mean preoperative anterior-middle-posterior heights were 20.3-17.6-28.0 mm, respectively. The mean local kyphotic angle was 13.1[degree sign]. The mean preoperative Beck Index (anterior edge height/posterior edge height) was 0.73, the mean alternative Beck Index (middle height/posterior edge height) was 0.63. The mean postoperative heights were restored to 24.5-24.6-30.4 mm, respectively. The mean local kyphotic angle was reduced to 8.9[degree sign]. The mean postoperative Beck Index was 0.81, the mean alternative one was 0.82. The overall extrusion rate was 29.1%, the symptomatic one was 1%. In the osteoporosis subgroup there were 23.8% extrusions. Within the three months followup interval there were 9% of adjacent and 4% of remote new fractures, all in the osteoporotic group. CONCLUSIONS VBS showed its strengths especially in realignment of crush and biconcave fractures. Given that fracture mobility is present, the realignment potential is sound and increases with the severity of preoperative vertebral body deformation.

Fast parallel kriging-based stepwise uncertainty reduction with application to the identification of an excursion set

Stepwise uncertainty reduction (SUR) strategies aim at constructing a sequence of points for evaluating a function  f in such a way that the residual uncertainty about a quantity of interest progressively decreases to zero. Using such strategies in the framework of Gaussian process modeling has been shown to be efficient for estimating the volume of excursion of f above a fixed threshold. However, SUR strategies remain cumbersome to use in practice because of their high computational complexity, and the fact that they deliver a single point at each iteration. In this article we introduce several multipoint sampling criteria, allowing the selection of batches of points at which f can be evaluated in parallel. Such criteria are of particular interest when f is costly to evaluate and several CPUs are simultaneously available. We also manage to drastically reduce the computational cost of these strategies through the use of closed form formulas. We illustrate their performances in various numerical experiments, including a nuclear safety test case. Basic notions about kriging, auxiliary problems, complexity calculations, R code, and data are available online as supplementary materials.