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This letter addresses the sensor placement problem for monitoring spatial phenomena by employing an estimation/prediction metric, i.e., noisy observations from a limited number of sensors are used to estimate the phenomena over the whole region. To solve the formulated problem, we propose a random search-based simulation optimization algorithm to iteratively select the sensor locations out of a possibly countably infinite subset of candidates. We further consider the sensor placement problem given a constraint on the energy consumption, and we propose a scheme which superimposes the Lagrange multiplier method for nonlinear programming upon our proposed discrete simulation optimization algorithm. We present numerical examples to demonstrate the fast convergence as well as the effectiveness of this simulation based algorithm.