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Alternating directions implicit integration in a general linear method framework
Sarshar, Arash; Roberts, Steven; Sandu, Adrian (Elsevier, 2021-05-15)
Alternating Directions Implicit (ADI) integration is an operator splitting approach to solve parabolic and elliptic partial differential equations in multiple dimensions based on solving sequentially a set of related one-dimensional equations. Classical ADI methods have order at most two, due to the splitting errors. Moreover, when the time discretization of stiff one-dimensional problems is based on Runge–Kutta schemes, additional order reduction may occur. This work proposes a new ADI approach based on the partitioned General Linear Methods framework. This approach allows the construction of high order ADI methods. Due to their high stage order, the proposed methods can alleviate the order reduction phenomenon seen with other schemes. Numerical experiments are shown to provide further insight into the accuracy, stability, and applicability of these new methods.
Linearly implicit GARK schemes
Sandu, Adrian; Guenther, Michael; Roberts, Steven (Elsevier, 2021-03-01)
Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge–Kutta (GARK) framework, which constructs multimethods based on Runge–Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.
A multi-attribute supply chain network resilience assessment framework based on SNA-inspired indicators
Kazemian, Iman; Torabi, S. Ali; Zobel, Christopher W.; Li, Yuhong; Baghersad, Milad (Springer, 2021-07-22)
This study proposes a supply chain resilience assessment framework at the network (i.e. structural) level based on quantifying supply chain networks’ structural factors and their relationships to different resilience strategies, by using a hybrid DEMATEL-ANP approach. DEMATEL is used to quantify interdependencies between the structural resilience factors, and between the resilience strategies. ANP is then used to quantify the outer-dependencies among these elements and to construct the limit super-matrix from which the global weights of all the decision network’s elements are estimated. To create the structural resilience factors, different network factors are selected and adopted from the social network analysis and supply chain resilience literatures. A case study is then performed to assess the performance of the proposed approach and to derive important observations to support future decision making. According to the results, the proposed approach can suitably measure the resilience performance of a supply chain network and help decision makers plan for more effective resilience improvement actions.
Multirate implicit Euler schemes for a class of differential-algebraic equations of index-1
Hachtel, Christoph; Bartel, Andreas; Guenther, Michael; Sandu, Adrian (Elsevier, 2021-05-15)
Systems of differential equations which consist of subsystems with widely different dynamical behaviour can be integrated by multirate time integration schemes to increase the efficiency. These schemes allow the usage of inherent step sizes according to the dynamical properties of the subsystem. In this paper, we extend the multirate implicit Euler method to semi-explicit differential–algebraic equations of index-1 where the algebraic constraints only occur in the slow changing subsystem. We discuss different coupling approaches and show that consistency and convergence order 1 can be reached. Numerical experiments validate the analytical results.
Multirate linearly-implicit GARK schemes
Guenther, Michael; Sandu, Adrian (Springer, 2021-12-28)
Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.
Neural network based pore flow field prediction in porous media using super resolution
Zhou, Xu-Hui X.; McClure, James; Chen, Cheng; Xiao, Heng (2021)
Previous works have demonstrated using the geometry of the microstructure of porous media to predict the ow velocity fields therein based on neural networks. However, such schemes are purely based on geometric information without accounting for the physical constraints on the velocity fields such as that due to mass conservation. In this work, we propose using a super-resolution technique to enhance the velocity field prediction by utilizing coarse-mesh velocity fields, which are often available inexpensively but carry important physical constraints. We apply our method to predict velocity fields in complex porous media. The results demonstrate that incorporating the coarse-mesh flow field significantly improves the prediction accuracy of the fine-mesh flow field as compared to predictions that rely on geometric information alone. This study highlights the merits of including coarse-mesh flow field with physical constraints embedded in it.
Partitioned exponential methods for coupled multiphysics systems
Narayanamurthi, Mahesh; Sandu, Adrian (Elsevier, 2021-03-01)
Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. The traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.
Regret in the Newsvendor Model with Demand and Yield Randomness
Chen, Zhi; Xie, Weijun (Wiley, 2021-07-28)
We study the fundamental stochastic newsvendor model that considers both demand and yield randomness. It is usually difficult in practice to describe precisely the joint demand and yield distribution, although partial statistical information and empirical data about this ambiguous distribution are often accessible. We combat the issue of distributional ambiguity by taking a data-driven distributionally robust optimization approach to hedge against all distributions that are sufficiently close to a uniform and discrete distribution of empirical data, where closeness is measured by the type-∞ Wasserstein distance. We adopt the minimax regret decision criterion to assess the optimal order quantity that minimizes the worst-case regret. Several properties about the minimax regret model, including optimality condition, regret bound, and the worst-case distribution, are presented. The optimal order quantity can be determined via an efficient golden section search. We extend the analysis to the Hurwicz criterion model, which generalizes the popular albeit pessimistic maximin model (maximizing the worst-case expected profit) and its (less noticeable) more optimistic counterpart—the maximax model (maximizing the best-case expected profit). Finally, we present numerical comparisons of our data-driven minimax regret model with data-driven models based on the Hurwicz criterion and with a minimax regret model based on partial statistical information on moments.
Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs
Bansal, Manish; Zhang, Yingqiu (Springer, 2021-01-22)
In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1–20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of lp-norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. p= 1 and p= 2 , respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts.
Self-Supervised Graph Learning With Hyperbolic Embedding for Temporal Health Event Prediction
Lu, Chang; Reddy, Chandan K.; Ning, Yue (IEEE, 2021-09-20)
Electronic health records (EHRs) have been heavily used in modern healthcare systems for recording patients' admission information to health facilities. Many data-driven approaches employ temporal features in EHR for predicting specific diseases, readmission times, and diagnoses of patients. However, most existing predictive models cannot fully utilize EHR data, due to an inherent lack of labels in supervised training for some temporal events. Moreover, it is hard for the existing methods to simultaneously provide generic and personalized interpretability. To address these challenges, we propose Sherbet, a self-supervised graph learning framework with hyperbolic embeddings for temporal health event prediction. We first propose a hyperbolic embedding method with information flow to pretrain medical code representations in a hierarchical structure. We incorporate these pretrained representations into a graph neural network (GNN) to detect disease complications and design a multilevel attention method to compute the contributions of particular diseases and admissions, thus enhancing personalized interpretability. We present a new hierarchy-enhanced historical prediction proxy task in our self-supervised learning framework to fully utilize EHR data and exploit medical domain knowledge. We conduct a comprehensive set of experiments on widely used publicly available EHR datasets to verify the effectiveness of our model. Our results demonstrate the proposed model's strengths in both predictive tasks and interpretable abilities.
Subspace adaptivity in Rosenbrock-Krylov methods for the time integration of initial value problems
Tranquilli, Paul; Glandon, Ross; Sandu, Adrian (Elsevier, 2021-03-15)
The Rosenbrock–Krylov family of time integration schemes is an extension of Rosenbrock-W methods that employs a specific Krylov based approximation of the linear system solutions arising within each stage of the integrator. This work proposes an extension of Rosenbrock–Krylov methods to address stability questions which arise for methods making use of inexact linear system solution strategies. Two approaches for improving the stability and efficiency of Rosenbrock–Krylov methods are proposed, one through direct control of linear system residuals and the second through a novel extension of the underlying Krylov space to include stage right hand side vectors. Rosenbrock–Krylov methods employing the new approaches show a substantial improvement in computational efficiency relative to prior implementations.
A Theoretical and Empirical Investigation of Feedback in Ideation Contests
Jiang, Juncai; Wang, Yu (Wiley, 2019-11-11)
Ideation contests are commonly used across public and private sectors to generate new ideas for solving problems, creating designs, and improving products or processes. In such a contest, a firm or an organization (the seeker) outsources an ideation task online to a distributed population of independent agents (solvers) in the form of an open call. Solvers compete to exert efforts and the one with the best solution wins a bounty. In evaluating solutions, the seeker typically has subjective taste that is unobservable to solvers. In practice, the seeker often provides solvers with feedback, which discloses useful information about her private taste. In this study, we develop a game-theoretic model of feedback in unblind ideation contests, where solvers’ solutions and the seeker's feedback are publicly visible by all. We show that feedback plays an informative role in mitigating the information asymmetry between the seeker and solvers, thereby inducing solvers to exert more efforts in the contest. We also show that some key contest and solver characteristics (CSC, including contest reward, contest duration, solver expertise, and solver population) have a direct effect on solver effort. Interestingly, by endogenizing the seeker's feedback decision, we find that the optimal feedback volume increases with contest reward, contest duration, solver expertise, but decreases with solver population. Thus, CSC elements also have an indirect effect on solvers’ effort level, with feedback volume mediating this effect. Employing a dataset from Zhubajie.com, a leading online ideation platform in China, we find empirical evidence that is consistent with these theoretical predictions.

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