A Transfer Operator Methodology for Optimal Sensor Placement Accounting for Uncertainty

Sensors in buildings are used for a wide variety of applications such as monitoring air quality, contaminants, indoor temperature, and relative humidity. These are used for accessing and ensuring indoor air quality, and also for ensuring safety in the event of chemical and biological attacks. It follows that optimal placement of sensors become important to accurately monitor contaminant levels in the indoor environment. However, contaminant transport inside the indoor environment is governed by the indoor flow conditions which are affected by various uncertainties associated with the building systems including occupancy and boundary fluxes. Therefore, it is important to account for all associated uncertainties while designing the sensor layout. The transfer operator based framework provides an effective way to identify optimal placement of sensors. Previous work has been limited to sensor placements under deterministic scenarios. In this work we extend the transfer operator based approach for optimal sensor placement while accounting for building systems uncertainties. The methodology provides a probabilistic metric to gauge coverage under uncertain conditions. We illustrate the capabilities of the framework with examples exhibiting boundary flux uncertainty

Gradient-Based Estimation of Uncertain Parameters for Elliptic Partial Differential Equations

This paper addresses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called finite noise problem in the space of functions with bounded mixed derivatives. We prove convergence of finite noise minimizers to the appropriate infinite dimensional ones, and devise a stochastic augmented Lagrangian method for locating these numerically. Lastly, we illustrate our method with three numerical examples

Propagation of two independent sources of uncertainty in the electrocardiography imaging inverse solution

International audienceThis work investigates the effects of the inputs parameters uncertainties (organs conductivities, boundary data) on the electrocardiography (ECG) imaging problem. These inputs are very important for the construction of the torso potential for the forward problem and for the non-invasive electrical potential on the heart surface in the case of the inverse problem. We propose a new stochastic formulation allowing to combine both sources of errors. We formulate the forward and the inverse stochastic problems by considering the inputs parameters as random fields and a sto-chastic optimal control formulation. In order to quantify multiple independent sources of uncertainties on the forward and inverse solutions, we attribute suitable probability density functions for each randomness source, and apply stochastic finite elements based on generalized polynomial chaos method. The efficiency of this approach to solve the forward and inverse ECG problem and the usability to quantify the effect of organs conductivity and epicardial boundary data uncertainties in the torso are demonstrated through a number of numerical simulations on a 2D computational mesh of a realistic torso geometry

Interactive uncertainty allocation and trade-off for early-stage design of complex systems

A common probabilistic approach to perform uncertainty allocation is to assign acceptable variability in the sources of uncertainty, such that prespecified probabilities of meeting performance constraints are satisfied. However, the computational cost of obtaining the associated tradeoffs increases significantly when more sources of uncertainty and more outputs are considered. Consequently, visualizing and exploring the decision (trade) space becomes increasingly difficult, which, in turn, makes the decision-making process cumbersome for practicing designers. To address this problem, proposed is a parameterization of the input probability distribution functions, to account for several statistical moments. This, combined with efficient uncertainty propagation and inverse computation techniques, results in a computational system that performs order(s) of magnitude faster than a state-of-the-art optimization technique. The approach is demonstrated by means of an illustrative example and a representative aircraft thermal system integration example

Stochastic collocation for optimal control problems with stochastic pde constraints

pre-printWe discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations (SPDE). Thereby the constraining SPDE depends on data which is not deterministic but random. Assuming a deterministic control, randomness within the states of the input data will propagate to the states of the system. For the solution of SPDEs there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos. Modal-based stochastic Galerkin and nodal-based stochastic collocation versions of this methodology exist, both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates. In this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program (SQP) for the minimization of objective functions constrained by an SPDE. The stochastic function involves several higher-order moments of the random states of the system as well as classical regularization of the control. In particular we discuss several objective functions of tracking type. Numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach

A moment-matching method to study the variability of phenomena described by partial differential equations

International audienceMany phenomena are modeled by deterministic differential equations , whereas the observation of these phenomena, in particular in life sciences, exhibits an important variability. This paper addresses the following question: how can the model be adapted to reflect the observed variability? Given an adequate model, it is possible to account for this variability by allowing some parameters to adopt a stochastic behavior. Finding the parameters probability density function that explains the observed variability is a difficult stochastic inverse problem, especially when the computational cost of the forward problem is high. In this paper, a non-parametric and non-intrusive procedure based on offline computations of the forward model is proposed. It infers the probability density function of the uncertain parameters from the matching of the statistical moments of observable degrees of freedom (DOFs) of the model. This inverse procedure is improved by incorporating an algorithm that selects a subset of the model DOFs that both reduces its computational cost and increases its robustness. This algorithm uses the pre-computed model outputs to build an approximation of the local sensitivities. The DOFs are selected so that the maximum information on the sensitivities is conserved. The proposed approach is illustrated with elliptic and parabolic PDEs. In the Appendix, an nonlinear ODE is considered and the strategy is compared with two existing ones