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This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a \textsl{highly nonlinear} way on a parameter $y\in [-1,1]^{\mathbb{N}}$. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove $\ell^p$-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs.

The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods applied to non-linear problems or for numerical approaches that make use of regularization techniques to cure low smoothness of the solution. This result is obtained through an estimate of the derivatives of composite functions in weighted uniform norm.

We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational map is based on partial differential equations and, consequently, is computationally expensive to evaluate. A-optimality is a widely used and easy-to-interpret criterion for Bayesian experimental design. This criterion seeks the optimal experimental design by minimizing the expected conditional variance, which is also known as the expected posterior variance. This study presents a novel likelihood-free approach to the A-optimal experimental design that does not require sampling or integrating the Bayesian posterior distribution. The expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, and we take advantage of the orthogonal projection property to approximate the conditional expectation. We derive an asymptotic error estimation for the proposed estimator of the expected conditional variance and show that the intractability of the posterior distribution does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation in the implementation of our method. We then extend our approach for dealing with the case that the domain of experimental design parameters is continuous by integrating the training process of the ANN into minimizing the expected conditional variance. Through numerical experiments, we demonstrate that our method greatly reduces the number of observation model evaluations compared with widely used importance sampling-based approaches. This reduction is crucial, considering the high computational cost of the observational models. Code is available at //github.com/vinh-tr-hoang/DOEviaPACE.

Selective inference methods are developed for group lasso estimators for use with a wide class of distributions and loss functions. The method includes the use of exponential family distributions, as well as quasi-likelihood modeling for overdispersed count data, for example, and allows for categorical or grouped covariates as well as continuous covariates. A randomized group-regularized optimization problem is studied. The added randomization allows us to construct a post-selection likelihood which we show to be adequate for selective inference when conditioning on the event of the selection of the grouped covariates. This likelihood also provides a selective point estimator, accounting for the selection by the group lasso. Confidence regions for the regression parameters in the selected model take the form of Wald-type regions and are shown to have bounded volume. The selective inference method for grouped lasso is illustrated on data from the national health and nutrition examination survey while simulations showcase its behaviour and favorable comparison with other methods.

During the process of robot-assisted ultrasound(US) puncture, it is important to estimate the location of the puncture from the 2D US images. To this end, the calibration of the US image becomes an important issue. In this paper, we proposed a depth camera-based US calibration method, where an easy-to-deploy device is designed for the calibration. With this device, the coordinates of the puncture needle tip are collected respectively in US image and in the depth camera, upon which a correspondence matrix is built for calibration. Finally, a number of experiments are conducted to validate the effectiveness of our calibration method.

This work puts forth low-complexity Riemannian subspace descent algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. Different from the existing Riemannian gradient descent variants, the proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix. The resulting updates avoid the costly matrix operations like matrix exponentiation and dense matrix multiplication, which are generally required in almost all other Riemannian optimization algorithms on SPD manifold. We further identify a broad class of functions, arising in diverse applications, such as kernel matrix learning, covariance estimation of Gaussian distributions, maximum likelihood parameter estimation of elliptically contoured distributions, and parameter estimation in Gaussian mixture model problems, over which the Riemannian gradients can be calculated efficiently. The proposed uni-directional and multi-directional Riemannian subspace descent variants incur per-iteration complexities of $\mathcal{O}(n)$ and $\mathcal{O}(n^2)$ respectively, as compared to the $\mathcal{O}(n^3)$ or higher complexity incurred by all existing Riemannian gradient descent variants. The superior runtime and low per-iteration complexity of the proposed algorithms is also demonstrated via numerical tests on large-scale covariance estimation problems.

A variant of the standard notion of branching bisimilarity for processes with discrete relative timing is proposed which is coarser than the standard notion. Using a version of ACP (Algebra of Communicating Processes) with abstraction for processes with discrete relative timing, it is shown that the proposed variant allows of both the functional correctness and the performance properties of the PAR (Positive Acknowledgement with Retransmission) protocol to be analyzed. In the version of ACP concerned, the difference between the standard notion of branching bisimilarity and its proposed variant is characterized by a single axiom schema.

In this work, we analyze the relation between reparametrizations of gradient flow and the induced implicit bias on general linear models, which encompass various basic classification and regression tasks. In particular, we aim at understanding the influence of the model parameters - reparametrization, loss, and link function - on the convergence behavior of gradient flow. Our results provide user-friendly conditions under which the implicit bias can be well-described and convergence of the flow is guaranteed. We furthermore show how to use these insights for designing reparametrization functions that lead to specific implicit biases like $\ell_p$- or trigonometric regularizers.

The aim of the current research is to analyse and discover, in a real context, behaviours, reactions and modes of interaction of social actors (people) with the humanoid robot Pepper. Indeed, we wanted to observe in a real, highly frequented context, the reactions and interactions of people with Pepper, placed in a shop window, through a systematic observation approach. The most interesting aspects of this research will be illustrated, bearing in mind that this is a preliminary analysis, therefore, not yet definitively concluded.

We consider an unknown multivariate function representing a system-such as a complex numerical simulator-taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than a given threshold. This problem, which we call Quantile Set Inversion (QSI), occurs for instance in the context of robust (reliability-based) optimization problems, when looking for the set of solutions that satisfy the constraints with sufficiently large probability. To solve the QSI problem, we propose a Bayesian strategy based on Gaussian process modeling and the Stepwise Uncertainty Reduction (SUR) principle, to sequentially choose the points at which the function should be evaluated to efficiently approximate the set of interest. We illustrate the performance and interest of the proposed SUR strategy through several numerical experiments.