In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method is combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e, the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order $(p \leq 4)$ XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal $(p + 1)$ and super $(p + 2)$ convergence properties of HDG while removing the fitting mesh restriction.
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Time-to-event estimands are central to many oncology clinical trials. The estimand framework (addendum to the ICH E9 guideline) calls for precisely defining the treatment effect of interest to align with the clinical question of interest and requires predefining the handling of intercurrent events that occur after treatment initiation and either preclude the observation of an event of interest or impact the interpretation of the treatment effect. We discuss a practical problem in clinical trial design and execution, i.e. in some clinical contexts it is not feasible to systematically follow patients to an event of interest. Loss to follow-up in the presence of intercurrent events can affect the meaning and interpretation of the study results. We provide recommendations for trial design, stressing the need for close alignment of the clinical question of interest and study design, impact on data collection and other practical implications. When patients cannot be systematically followed, compromise may be necessary to select the best available estimand that can be feasibly estimated under the circumstances. We discuss the use of sensitivity and supplementary analyses to examine assumptions of interest.
With computational models becoming more expensive and complex, surrogate models have gained increasing attention in many scientific disciplines and are often necessary to conduct sensitivity studies, parameter optimization etc. In the scientific discipline of uncertainty quantification (UQ), model input quantities are often described by probability distributions. For the construction of surrogate models, space-filling designs are generated in the input space to define training points, and evaluations of the computational model at these points are then conducted. The physical parameter space is often transformed into an i.i.d. uniform input space in order to apply space-filling training procedures in a sensible way. Due to this transformation surrogate modeling techniques tend to suffer with regard to their prediction accuracy. Therefore, a new method is proposed in this paper where input parameter transformations are applied to basis functions for universal kriging. To speed up hyperparameter optimization for universal kriging, suitable expressions for efficient gradient-based optimization are developed. Several benchmark functions are investigated and the proposed method is compared with conventional methods.
The number of modes in a probability density function is representative of the model's complexity and can also be viewed as the number of existing subpopulations. Despite its relevance, little research has been devoted to its estimation. Focusing on the univariate setting, we propose a novel approach targeting prediction accuracy inspired by some overlooked aspects of the problem. We argue for the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view blending global and local density properties. Our method builds upon a combination of flexible kernel estimators and parsimonious compositional splines. Feature exploration, model selection and mode testing are implemented in the Bayesian inference paradigm, providing soft solutions and allowing to incorporate expert judgement in the process. The usefulness of our proposal is illustrated through a case study in sports analytics, showcasing multiple companion visualisation tools. A thorough simulation study demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, our method emerges as a top-tier alternative offering innovative solutions for analysts.
The present work presents a stable POD-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-B\'enard convection in a square geometry for three Rayleigh numbers: 10^4 (steady state), 3*10^5 (periodic), and 6*10^6 (chaotic). Stability is obtained through a particular (staggered-grid) FOM discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy for the non-chaotic cases. However, for the chaotic case, convergence with increasing numbers of modes is not evident, and additional measures are required to represent the effect of the smallest (neglected) scales.
Channel estimation (CE) plays a key role in reconfigurable intelligent surface (RIS)-aided multiple-input multiple-output (MIMO) communication systems, while it poses a challenging task due to the passive nature of RIS and the cascaded channel structures. In this paper, a partially decoupled atomic norm minimization (PDANM) framework is proposed for CE of RIS-aided MIMO systems, which exploits the three-dimensional angular sparsity of the channel. In particular, PDANM partially decouples the differential angles at the RIS from other angles at the base station and user equipment, reducing the computational complexity compared with existing methods. A reweighted PDANM (RPDANM) algorithm is proposed to further improve CE accuracy, which iteratively refines CE through a specifically designed reweighing strategy. Building upon RPDANM, we propose an iterative approach named RPDANM with adaptive phase control (RPDANM-APC), which adaptively adjusts the RIS phases based on previously estimated channel parameters to facilitate CE, achieving superior CE accuracy while reducing training overhead. Numerical simulations demonstrate the superiority of our proposed approaches in terms of running time, CE accuracy, and training overhead. In particular, the RPDANM-APC approach can achieve higher CE accuracy than existing methods within less than 40 percent training overhead while reducing the running time by tens of times.
Dimension reduction techniques have long been an important topic in statistics, and active subspaces (AS) have received much attention this past decade in the computer experiments literature. The most common approach towards estimating the AS is to use Monte Carlo with numerical gradient evaluation. While sensible in some settings, this approach has obvious drawbacks. Recent research has demonstrated that active subspace calculations can be obtained in closed form, conditional on a Gaussian process (GP) surrogate, which can be limiting in high-dimensional settings for computational reasons. In this paper, we produce the relevant calculations for a more general case when the model of interest is a linear combination of tensor products. These general equations can be applied to the GP, recovering previous results as a special case, or applied to the models constructed by other regression techniques including multivariate adaptive regression splines (MARS). Using a MARS surrogate has many advantages including improved scaling, better estimation of active subspaces in high dimensions and the ability to handle a large number of prior distributions in closed form. In one real-world example, we obtain the active subspace of a radiation-transport code with 240 inputs and 9,372 model runs in under half an hour.
We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which, we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.
Designing efficient and rigorous numerical methods for sequential decision-making under uncertainty is a difficult problem that arises in many applications frameworks. In this paper we focus on the numerical solution of a subclass of impulse control problem for piecewise deterministic Markov process (PDMP) when the jump times are hidden. We first state the problem as a partially observed Markov decision process (POMDP) on a continuous state space and with controlled transition kernels corresponding to some specific skeleton chains of the PDMP. Then we proceed to build a numerically tractable approximation of the POMDP by tailor-made discretizations of the state spaces. The main difficulty in evaluating the discretization error come from the possible random or boundary jumps of the PDMP between consecutive epochs of the POMDP and requires special care. Finally we extensively discuss the practical construction of discretization grids and illustrate our method on simulations.
Since their introduction in Abadie and Gardeazabal (2003), Synthetic Control (SC) methods have quickly become one of the leading methods for estimating causal effects in observational studies in settings with panel data. Formal discussions often motivate SC methods by the assumption that the potential outcomes were generated by a factor model. Here we study SC methods from a design-based perspective, assuming a model for the selection of the treated unit(s) and period(s). We show that the standard SC estimator is generally biased under random assignment. We propose a Modified Unbiased Synthetic Control (MUSC) estimator that guarantees unbiasedness under random assignment and derive its exact, randomization-based, finite-sample variance. We also propose an unbiased estimator for this variance. We document in settings with real data that under random assignment, SC-type estimators can have root mean-squared errors that are substantially lower than that of other common estimators. We show that such an improvement is weakly guaranteed if the treated period is similar to the other periods, for example, if the treated period was randomly selected. While our results only directly apply in settings where treatment is assigned randomly, we believe that they can complement model-based approaches even for observational studies.
In this paper, we study a priori error estimates for the finite element approximation of the nonlinear Schr\"{o}dinger-Poisson model. The electron density is defined by an infinite series over all eigenvalues of the Hamiltonian operator. To establish the error estimate, we present a unified theory of error estimates for a class of nonlinear problems. The theory is based on three conditions: 1) the original problem has a solution $u$ which is the fixed point of a compact operator $\Ca$, 2) $\Ca$ is Fr\'{e}chet-differentiable at $u$ and $\Ci-\Ca'[u]$ has a bounded inverse in a neighborhood of $u$, and 3) there exists an operator $\Ca_h$ which converges to $\Ca$ in the neighborhood of $u$. The theory states that $\Ca_h$ has a fixed point $u_h$ which solves the approximate problem. It also gives the error estimate between $u$ and $u_h$, without assumptions on the well-posedness of the approximate problem. We apply the unified theory to the finite element approximation of the Schr\"{o}dinger-Poisson model and obtain optimal error estimate between the numerical solution and the exact solution. Numerical experiments are presented to verify the convergence rates of numerical solutions.
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