Troubled-Cell Indicator Based on Mean Value Property for Hybrid WENO Schemes

In this paper, we introduce a new type of troubled-cell indicator to improve hybrid weighted essentially non-oscillatory (WENO) schemes for solving the hyperbolic conservation laws. The hybrid WENO schemes selectively adopt the high-order linear upwind scheme or the WENO scheme to avoid the local characteristic decompositions and calculations of the nonlinear weights in smooth regions. Therefore, they can reduce computational cost while maintaining non-oscillatory properties in non-smooth regions. Reliable troubled-cell indicators are essential for efficient hybrid WENO methods. Most of troubled-cell indicators require proper parameters to detect discontinuities precisely, but it is very difficult to determine the parameters automatically. We develop a new troubled-cell indicator derived from the mean value theorem that does not require any variable parameters. Additionally, we investigate the characteristics of indicator variable; one of the conserved properties or the entropy is considered as indicator variable. Detailed numerical tests for 1D and 2D Euler equations are conducted to demonstrate the performance of the proposed indicator. The results with the proposed troubled-cell indicator are in good agreement with pure WENO schemes. Also the new indicator has advantages in the computational cost compared with the other indicators.     

Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

Conservation laws are considered to be fundamental laws of nature. It has broad applications in many fields, including physics, chemistry, biology, geology, and engineering. Solving the differential equations associated with conservation laws is a major branch in computational mathematics. The recent success of machine learning, especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on 1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner. We will show that the problem of solving conservation laws can be naturally viewed as a sequential decision-making process, and the numerical schemes learned in such a way can easily enforce long-term accuracy. Furthermore, the learned policy network is carefully designed to determine a good local discrete approximation based on the current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some state-of-the-art numerical solvers such as WENO schemes, and supervised learning based approach L3D and PINN, and how well it generalizes.     

Fisher information matrix for nonlinear mixed effects multiple response models: Evaluation of the appropriateness of the first order linearization using a pharmacokinetic/pharmacodynamic model

We focus on the Fisher information matrix used for design evaluation and optimization in nonlinear mixed effects multiple response models. We evaluate the appropriateness of its expression computed by linearization as proposed for a single response model. Using a pharmacokinetic–pharmacodynamic (PKPD) example, we first compare the computation of the Fisher information matrix with approximation to one derived from the observed matrix on a large simulation using the stochastic approximation expectation–maximization algorithm (SAEM). The expression of the Fisher information matrix for multiple responses is also evaluated by comparison with the empirical information obtained through a replicated simulation study using the first‐order linearization estimation methods implemented in the NONMEM software (first‐order (FO), first‐order conditional estimate (FOCE)) and the SAEM algorithm in the MONOLIX software. The predicted errors given by the approximated information matrix are close to those given by the information matrix obtained without linearization using SAEM and to the empirical ones obtained with FOCE and SAEM. The simulation study also illustrates the accuracy of both FOCE and SAEM estimation algorithms when jointly modelling multiple responses and the major limitations of the FO method. This study highlights the appropriateness of the approximated Fisher information matrix for multiple responses, which is implemented in PFIM 3.0, an extension of the R function PFIM dedicated to design evaluation and optimization. It also emphasizes the use of this computing tool for designing population multiple response studies, as for instance in PKPD studies or in PK studies including the modelling of the PK of a drug and its active metabolite. Copyright © 2009 John Wiley & Sons, Ltd.     

ITERATIVE DYNAMIC PROGRAMMING FOR MINIMUM ENERGY CONTROL PROBLEMS WITH TIME DELAY

This paper presents the use of iterative dynamic programming employing exact penalty functions for minimum energy control problems. We show that exact continuously non-differentiable penalty functions are superior to continuously differentiable penalty functions in terms of satisfying final state constraints. We also demonstrate that the choice of an appropriate penalty function factor depends on the relative size of the time delay with respect to the final time and on the expected value of the energy consumption. A quadratic approximation (QA) of the delayed variables is much better than a linear approximation (LA) of the same for relatively large time delays. The QA improves the rate of convergence and avoids the formation of ‘kinks‘. A more general way of selecting appropriate penalty function factors is given and the results obtained using four illustrative examples of varying complexity corroborate the efficacy of the method.     

A comprehensive non‐invasive framework for automated evaluation of acute renal transplant rejection using DCE‐MRI

The objective was to develop a novel and automated comprehensive framework for the non‐invasive identification and classification of kidney non‐rejection and acute rejection transplants using 2D dynamic contrast‐enhanced magnetic resonance imaging (DCE‐MRI). The proposed approach consists of four steps. First, kidney objects are segmented from the surrounding structures with a geometric deformable model. Second, a non‐rigid registration approach is employed to account for any local kidney deformation. In the third step, the cortex of the kidney is extracted in order to determine dynamic agent delivery, since it is the cortex that is primarily affected by the perfusion deficits that underlie the pathophysiology of acute rejection. Finally, we use an analytical function‐based model to fit the dynamic contrast agent kinetic curves in order to determine possible rejection candidates. Five features that map the data from the original data space to the feature space are chosen with a k‐nearest‐neighbor (KNN) classifier to distinguish between acute rejection and non‐rejection transplants. Our study includes 50 transplant patients divided into two groups: 27 patients with stable kidney function and the remainder with impaired kidney function. All of the patients underwent DCE‐MRI, while the patients in the impaired group also underwent ultrasound‐guided fine needle biopsy. We extracted the kidney objects and the renal cortex from DCE‐MRI for accurate medical evaluation with an accuracy of 0.97 ± 0.02 and 0.90 ± 0.03, respectively, using the Dice similarity metric. In a cohort of 50 participants, our framework classified all cases correctly (100%) as rejection or non‐rejection transplant candidates, which is comparable to the gold standard of biopsy but without the associated deleterious side‐effects. Both the 95% confidence interval (CI) statistic and the receiver operating characteristic (ROC) analysis document the ability to separate rejection and non‐rejection groups. The average plateau (AP) signal magnitude and the gamma‐variate model functional parameter α have the best individual discriminating characteristics. Copyright © 2013 John Wiley & Sons, Ltd.     

Re-examination of the potential step chronoamperometry method through numerical inversion of Laplace transforms. II. Application examples

In the first part of this work, we proposed the so-called GS-, FE- and FT-PSCA algorithms to investigate a wide range of electrode geometry and chemical, electrochemical and one-dimensional mass-transport processes by the potential-step chronoamperometry (PSCA) method. Each algorithm provides a full explicit formulation of Faradaic current with respect to time, applied potential and electrochemical parameters (initial concentrations, geometrical parameter(s), standard potential, electrochemical and chemical rate constants and diffusion coefficients). Application examples relative to diffusion equations with spherical electrode geometry are presented in this second article by way of illustration of the potentialities of GS- and FT-PSCA algorithms. First, the case of one-step electrochemical reactions at spherical and hemispherical electrodes of any radius is investigated when both implicated species are soluble in the electrolytic solution. The numerical solution obtained from GS-PSCA algorithm is favourably compared to previous results in the electrochemical literature. Next, we use the FT-PSCA algorithm for investigating spherical diffusion with amalgamation reaction. Finally, the characteristic features of chronoamperograms and sampled-current voltammograms pertaining to the above reaction are examined and discussed.     

Structure detection of semiparametric structural equation models with Bayesian adaptive group lasso

Structural equation models (SEMs) are widely recognized as the most important statistical tool for assessing the interrelationships among latent variables. This study develops a Bayesian adaptive group least absolute shrinkage and selection operator procedure to perform simultaneous model selection and estimation for semiparametric SEMs, wherein the structural equation is formulated using the additive nonparametric functions of observed and latent variables. We propose the use of basis expansions to approximate the unknown functions. By introducing adaptive penalties to the groups of basis expansions, the nonlinear, linear, or non‐existent effects of observed and latent variables in the structural equation can be automatically detected. A simulation study demonstrates that the proposed method performs satisfactorily. This paper presents an application of revealing the observed and latent risk factors of diabetic kidney disease. Copyright © 2015 John Wiley & Sons, Ltd.     

Global Analysis of Methylation Profiles From High Resolution CpG Data

New high throughput technologies are now enabling simultaneous epigenetic profiling of DNA methylation at hundreds of thousands of CpGs across the genome. A problem of considerable practical interest is identification of large scale, global changes in methylation that are associated with environmental variables, clinical outcomes, or other experimental conditions. However, there has been little statistical research on methods for global methylation analysis using technologies with individual CpG resolution. To address this critical gap in the literature, we develop a new strategy for global analysis of methylation profiles using a functional regression approach wherein we approximate either the density or the cumulative distribution function (CDF) of the methylation values for each individual using B‐spline basis functions. The spline coefficients for each individual are allowed to summarize the individual's overall methylation profile. We then test for association between the overall distribution and a continuous or dichotomous outcome variable using a variance component score test that naturally accommodates the correlation between spline coefficients. Simulations indicate that our proposed approach has desirable power while protecting type I error. The method was applied to detect methylation differences, both genome wide and at LINE1 elements, between the blood samples from rheumatoid arthritis patients and healthy controls and to detect the epigenetic changes of human hepatocarcinogenesis in the context of alcohol abuse and hepatitis C virus infection. A free implementation of our methods in the R language is available in the Global Analysis of Methylation Profiles (GAMP) package at http://research.fhcrc.org/wu/en.html .