# Adjoint-Based Optimization through Reduced-Order Subdomain Surrogate Modelling

### An Application in Reservoir Simulation

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## Abstract

Although a transition to more sustainable energy production is necessary, fossil fuels will remain a crucial contributor to the world's energy production in the near future. In numerically maximizing the production of these fuels, as well as in many other optimization problems, an objective function has to be optimized on the basis of an input vector, related through some underlying model. Although the adjoint methodology generally allows for efficient gradient-based optimization of such a problem, it quickly becomes infeasible when the model comprises a large-scale system, or access to this model is prohibited. To resolve this, we consider the application of a model order reduction scheme, in combination with subdomain surrogate modeling, to perform the optimization using a reduced-order approximation of the model. In particular, we employ principal orthogonal decomposition (POD), efficiently reducing the size of the underlying system of equations, on the basis of a limited number of samples of the full-order model. Applying a domain decomposition, this number of samples can be reduced even further, although at a decreased efficiency. Next, radial basis function (RBF) interpolation is applied, using the samples to construct a trajectory piecewise linear approximation of the reduced-order model in each subdomain. Combining these different techniques, an optimization algorithm is constructed, applying the adjoint methodology to the reduced-order model in order to compute an approximate gradient. The accuracy of this approximation was found to be poor, but comparable to alternative sample-based methods.

An improved accuracy could be achieved by reducing the number of input parameters, allowing for more efficient optimization when applying the algorithm to a particular reservoir model. For another model, however, the algorithm performed worse upon reducing the size of the input, as a result of the fewer degrees of freedom in the optimization procedure. Reacquiring this freedom during the optimization, improved results could be attained, but the number of iterations and thus the cost of the method increased drastically. Comparing the full input implementation to another sample-based method, specifically a straight gradient ensemble algorithm, it was found to produce comparable results. Each method was able to surpass the other, dependent on the particular situation, though the reduced-adjoint algorithm generally expended more effort to attain similar results. This suggests that further study is necessary, for example improving the RBF interpolation or input reduction techniques, to fully exploit the benefits of the POD-TPWL methodology.