In calculating expected information gain in optimal Bayesian experimental design, the computation of the inner loop in the classical double-loop Monte Carlo requires a large number of samples and suffers from underflow if the number of samples is small. These drawbacks can be avoided by using an importance sampling approach. We present a computationally efficient method for optimal Bayesian experimental design that introduces importance sampling based on the Laplace method to the inner loop. We derive the optimal values for the method parameters in which the average computational cost is minimized according to the desired error tolerance. We use three numerical examples to demonstrate the computational efficiency of our method compared with the classical double-loop Monte Carlo, and a more recent single-loop Monte Carlo method that uses the Laplace method as an approximation of the return value of the inner loop. The first example is a scalar problem that is linear in the uncertain parameter. The second example is a nonlinear scalar problem. The third example deals with the optimal sensor placement for an electrical impedance tomography experiment to recover the fiber orientation in laminate composites.
Tuberculosis and COVID-19 are among the diseases with major global public health concern and great socio-economic impact. Co-infection of these two diseases is inevitable due to their geographical overlap, a potential double blow as their clinical similarities could hamper strategies to mitigate their spread and transmission dynamics. To theoretically investigate the impact of control measures on their long-term dynamics, we formulate and analyze a mathematical model for the co-infection of COVID-19 and tuberculosis. Basic properties of the tuberculosis only and COVID-19 only sub-models are investigated as well as bifurcation analysis (possibility of the co-existence of the disease-free and endemic equilibria). The disease-free and endemic equilibria are globally asymptotically stable. The model is extended into an optimal control system by incorporating five control measures. These are: tuberculosis awareness campaign, prevention against COVID-19 (e.g., face mask, physical distancing), control against co-infection, tuberculosis and COVID-19 treatment. Five strategies which are combinations of the control measures are investigated. Strategy B which focuses on COVID-19 prevention, treatment and control of co-infection yields a better outcome in terms of the number of COVID-19 cases prevented at a lower percentage of the total cost of this strategy.
Finding the best setup for experiments is the main concern of Optimal Experimental Design (OED). We focus on the Bayesian experimental design problem of finding the setup that maximizes the Shannon expected information gain. We propose using the stochastic gradient descent and its accelerated counterpart, which employs Nesterov's method, to solve the optimization problem in OED. We improve the stochastic gradient acceleration with a restart technique, as O'Donoghue and Candes [9] originally proposed for deterministic optimization. We combine these optimization methods with three estimators of the objective function: the double-loop Monte Carlo estimator (DLMC), the Monte Carlo estimator using the Laplace approximation for the posterior distribution (MCLA) and the double-loop Monte Carlo estimator with Laplace-based importance sampling (DLMCIS). Using stochastic gradient methods and Laplace-based estimators together allows us to use expensive and complex models, such as those that require solving a partial differential equation (PDE). From a theoretical viewpoint, we derive an explicit formula to compute the stochastic gradient of the Monte Carlo methods including the Laplace approximation (MCLA) and the Laplace-based importance sampling (DLMCIS). Finally, from a computational standpoint, we study four examples: three based on analytical functions and one based on the finite element method solution to a PDE. The last example is an electrical impedance tomography experiment based on the complete electrode model. In these examples, the accelerated stochastic gradient for the MCLA approximation converges to local maxima in fewer model evaluations by up to five orders of magnitude than the gradient descent with DLMC.
An optimal experimental set-up maximizes the value of data for statistical inferences and predictions. An optimal set-up is particularly important for experiments that are time consuming or expensive to perform. In the context of partial differential equations (PDEs), multilevel methods have been proven to dramatically reduce the computational complexity of their single-level counterparts. Here, two multilevel methods, which efficiently compute the expected information gain using a Kullback-Leibler divergence measure in simulation-based Bayesian optimal experimental design, are proposed. The first method is a multilevel double loop Monte Carlo (MLDLMC) with importance sampling that greatly reduces the computational work of the inner loop. The second proposed method is a multilevel double loop stochastic collocation (MLDLSC) with importance sampling, which performs a high-dimensional integration by deterministic quadrature on sparse grids. In both methods, the Laplace approximation is used as an effective means of importance sampling, and the optimal values of the method parameters are determined by minimizing the average computational work, subject to a desired error tolerance. The computational efficiencies of the methods are demonstrated by computing the expected information gain from an electrical impedance tomography experiment where the fiber orientation in composite laminate materials are inferred through Bayesian inversion. MLDLSC performs better than MLDLMC when the regularity of the underlying computational model, with respect to the additive noise and the unknown parameters, can be exploited.
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