## Design optimization under uncertainty

In the simplest possible context (assuming a single objective and risk-neutrality), the design optimization problem that our methodologies address is:

\begin{equation}

\mathbf{x} = \arg\max_{\mathbf{x}}\mathbb{E}\left[f(\mathbf{x})\right],

\label{eq:problem}

\end{equation}

where \(f(\mathbf{x})\) is the

Our research objective is to lay the foundations a versatile and pragmatic framework for design optimization under uncertainty that addresses all (most) of the key problems we described above. The central hypothesis is that such a framework should be posed in terms of a fully Bayesian formalism that is adaptively refined using value-based information acquisition criteria. The Bayesian formalism is adopted due to its inherent ability to represent states of knowledge. This ability provides the key towards the formulation of generic value-based information acquisition policies capable of dealing with multiple noisy information sources with parametric uncertainties. We reinforce the pragmatic aspect of this work by incorporating several technical developments deemed necessary for the efficient numerical implementation of the proposed framework in realistic design problems, e.g., high-dimensional design/stochastic spaces, and arbitrarily correlated information sources. The primary research objective will be accomplished by addressing three research questions:

\begin{equation}

\mathbf{x} = \arg\max_{\mathbf{x}}\mathbb{E}\left[f(\mathbf{x})\right],

\label{eq:problem}

\end{equation}

where \(f(\mathbf{x})\) is the

*objective*function of the application. We are particularly interested in problems that exhibit any combinations of the following difficulties.- Evaluating \(f(\mathbf{x})\) is very time consuming (hours-days in supercomputers).
- Observations of \(f(\mathbf{x})\) are uncertain either because of intrinsic
*(aleatoric)*uncertainties or because of inability to fully resolve all the parameters of the underlying physical models (*epistemic*uncertainty). - The random variables that describe the uncertainty in the measurement of \(f(\mathbf{x})\) are
*high-dimensional.* - The design variables \(\mathbf{x}\) are
*high-dimensional*. - The optimization problem defined in equation \eqref{eq:problem} exhibits multiple local maxima and, thus, requires a
*global*approach. - The evaluation of \(f(\mathbf{x})\) might depend on multiple
*information sources,*e.g., physical models of varying fidelity, computer simulations and physical experiments, etc. - and more...

Our research objective is to lay the foundations a versatile and pragmatic framework for design optimization under uncertainty that addresses all (most) of the key problems we described above. The central hypothesis is that such a framework should be posed in terms of a fully Bayesian formalism that is adaptively refined using value-based information acquisition criteria. The Bayesian formalism is adopted due to its inherent ability to represent states of knowledge. This ability provides the key towards the formulation of generic value-based information acquisition policies capable of dealing with multiple noisy information sources with parametric uncertainties. We reinforce the pragmatic aspect of this work by incorporating several technical developments deemed necessary for the efficient numerical implementation of the proposed framework in realistic design problems, e.g., high-dimensional design/stochastic spaces, and arbitrarily correlated information sources. The primary research objective will be accomplished by addressing three research questions:

- How can we make information acquisition decisions for design optimization problems under uncertainty?
- How can we make the proposed paradigm applicable to realistic design optimization problems?
- How can resource allocation decisions be made in design problems involving multiple distributed teams?