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Simulated Moving Bed (SMB) chromatography is a well-known technique for the resolution of several high-value-added compounds. Parameters identification and model topology definition are arduous when one is dealing with complex systems such as a Simulated Moving Bed unit. Moreover, the large number of experiments necessary might be an expansive-long process. Hence, this work proposes a novel methodology for parameter estimation, screening the most suitable topology of the models sink-source (defined by the adsorption isotherm equation) and defining the minimum number of experiments necessary to identify the model. Therefore, a nested loop optimization problem is proposed with three levels considering the three main goals of the work: parameters estimation; topology screening by isotherm definition; minimum number of experiments necessary to yield a precise model. The proposed methodology emulated a real scenario by introducing noise in the data and using a Software-in-the-Loop (SIL) approach. Data reconciliation and uncertainty evaluation add robustness to the parameter estimation adding precision and reliability to the model. The methodology is validated considering experimental data from literature apart from the samples applied for parameter estimation, following a cross-validation. The results corroborate that it is possible to carry out trustworthy parameter estimation directly from an SMB unit with minimal system knowledge.

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) $\pi$ a given, general probability measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schr\"odinger-F\"ollmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process $\{X_t\}_{t\in [0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process is constructed such that, starting at $X_0=0$, one has $X_1\sim \pi$. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, \cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t.~$\pi$. We show that for this methodology to achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$, the associated cost is $\mathcal{O}(\epsilon^{-5})$. We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t.~$\pi$, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of $\mathcal{O}(\epsilon^2)$, the associated cost is, with high probability, $\mathcal{O}(\epsilon^{-2}|\log(\epsilon)|^{2+\delta})$, for any $\delta>0$. We implement our method on several examples including Bayesian inverse problems.

Investigators are increasingly using novel methods for extending (generalizing or transporting) causal inferences from a trial to a target population. In many generalizability and transportability analyses, the trial and the observational data from the target population are separately sampled, following a non-nested trial design. In practical implementations of this design, non-randomized individuals from the target population are often identified by conditioning on the use of a particular treatment, while individuals who used other candidate treatments for the same indication or individuals who did not use any treatment are excluded. In this paper, we argue that conditioning on treatment in the target population changes the estimand of generalizability and transportability analyses and potentially introduces serious bias in the estimation of causal estimands in the target population or the subset of the target population using a specific treatment. Furthermore, we argue that the naive application of marginalization-based or weighting-based standardization methods does not produce estimates of any reasonable causal estimand. We use causal graphs and counterfactual arguments to characterize the identification problems induced by conditioning on treatment in the target population and illustrate the problems using simulated data. We conclude by considering the implications of our findings for applied work.

Federated learning learns a neural network model by aggregating the knowledge from a group of distributed clients under the privacy-preserving constraint. In this work, we show that this paradigm might inherit the adversarial vulnerability of the centralized neural network, i.e., it has deteriorated performance on adversarial examples when the model is deployed. This is even more alarming when federated learning paradigm is designed to approximate the updating behavior of a centralized neural network. To solve this problem, we propose an adversarially robust federated learning framework, named Fed_BVA, with improved server and client update mechanisms. This is motivated by our observation that the generalization error in federated learning can be naturally decomposed into the bias and variance triggered by multiple clients' predictions. Thus, we propose to generate the adversarial examples via maximizing the bias and variance during server update, and learn the adversarially robust model updates with those examples during client update. As a result, an adversarially robust neural network can be aggregated from these improved local clients' model updates. The experiments are conducted on multiple benchmark data sets using several prevalent neural network models, and the empirical results show that our framework is robust against white-box and black-box adversarial corruptions under both IID and non-IID settings.

Reaching a consensus in a swarm of robots is one of the fundamental problems in swarm robotics, examining the possibility of reaching an agreement within the swarm members. The recently-introduced contamination problem offers a new perspective of the problem, in which swarm members should reach a consensus in spite of the existence of adversarial members that intentionally act to divert the swarm members towards a different consensus. In this paper, we search for a consensus-reaching algorithm under the contamination problem setting by taking a top-down approach: We transform the problem to a centralized two-player game in which each player controls the behavior of a subset of the swarm, trying to force the entire swarm to converge to an agreement on its own value. We define a performance metric for each players performance, proving a correlation between this metric and the chances of the player to win the game. We then present the globally optimal solution to the game and prove that unfortunately it is unattainable in a distributed setting, due to the challenging characteristics of the swarm members. We therefore examine the problem on a simplified swarm model, and compare the performance of the globally optimal strategy with locally optimal strategies, demonstrating its superiority in rigorous simulation experiments.

Using information-theoretic principles, we consider the generalization error (gen-error) of iterative semi-supervised learning (SSL) algorithms that iteratively generate pseudo-labels for a large amount of unlabelled data to progressively refine the model parameters. In contrast to most previous works that {\em bound} the gen-error, we provide an {\em exact} expression for the gen-error and particularize it to the binary Gaussian mixture model. Our theoretical results suggest that when the class conditional variances are not too large, the gen-error decreases with the number of iterations, but quickly saturates. On the flip side, if the class conditional variances (and so amount of overlap between the classes) are large, the gen-error increases with the number of iterations. To mitigate this undesirable effect, we show that regularization can reduce the gen-error. The theoretical results are corroborated by extensive experiments on the MNIST and CIFAR datasets in which we notice that for easy-to-distinguish classes, the gen-error improves after several pseudo-labelling iterations, but saturates afterwards, and for more difficult-to-distinguish classes, regularization improves the generalization performance.

Spatial autocorrelation measures such as Moran's index can be expressed as a pair of equations based on a standardized size variable and a globally normalized weight matrix. One is based on inner product, and the other is based on outer product of the size variable. The inner product equation is actually a spatial autocorrelation model. However, the theoretical basis of the inner product equation for Moran's index is not clear. This paper is devoted to revealing the antecedents and consequences of the inner product equation of Moran's index. The method is mathematical derivation and empirical analysis. The main results are as follows. First, the inner product equation is derived from a simple spatial autoregressive model, and thus the relation between Moran's index and spatial autoregressive coefficient is clarified. Second, the least squares regression is proved to be one of effective approaches for estimating spatial autoregressive coefficient. Third, the value ranges of the spatial autoregressive coefficient can be identified from three angles of view. A conclusion can be drawn that a spatial autocorrelation model is actually an inverse spatial autoregressive model, and Moran's index and spatial autoregressive models can be integrated into the same framework through inner product and outer product equations. This work may be helpful for understanding the connections and differences between spatial autocorrelation measurements and spatial autoregressive modeling.

We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.

This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential--algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a BDF discretization in time. Within this paper, we focus on the second-order case, resulting in a $3$-step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically.

For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality, which lives on the unbounded domain, and so cannot be treated numerically in a direct way. By boundary integral methods the problem is transformed and a novel hemivariational inequality (HVI) is obtained that lives on the interior domain and on the coupling boundary, only. Thus for discretization the coupling of finite elements and boundary elements is the method of choice. In addition smoothing techniques of nondifferentiable optimization are adapted and the nonsmooth part in the HVI is regularized. Thus we reduce the original variational problem to a finite dimensional problem that can be solved by standard optimization tools. We establish not only convergence results for the total approximation procedure, but also an asymptotic error estimate for the regularized HVI.