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At least 1,153 records · Page 64

Direct methanol feed fuel cell and system

Improvements to non acid methanol fuel cells include new formulations for materials. The platinum and ruthenium are more exactly mixed together. Different materials are substituted for these materials. The backing material for the fuel cell electrode is specially treated to improve its characteristics. A special sputtered electrode is formed which is extremely porous. The fuel cell system also comprises a fuel supplying part including a meter which meters an amount of fuel which is used by the fuel cell, and controls the supply of fuel based on said metering.

Surampudi, Subbarao↗

Surface meteorology and Solar Energy

The Release 5.1 Surface meteorology and Solar Energy (SSE) data contains parameters formulated for assessing and designing renewable energy systems. Parameters fall under 11 categories including: Solar cooking, solar thermal applications, solar geometry, tilted solar panels, energy storage systems, surplus product storage systems, cloud information, temperature, wind, other meteorological factors, and supporting information. This latest release contains new parameters based on recommendations by the renewable energy industry and it is more accurate than previous releases. On-line plotting capabilities allow quick evaluation of potential renewable energy projects for any region of the world. The SSE data set is formulated from NASA satellite- and reanalysis-derived insolation and meteorological data for the 10-year period July 1983 through June 1993. Results are provided for 1 degree latitude by 1 degree longitude grid cells over the globe. Average daily and monthly measurements for 1195 World Radiation Data Centre ground sites are also available. [Mission Objectives] The SSE project contains insolation and meteorology data intended to aid in the development of renewable energy systems. Collaboration between SSE and technology industries such as the Hybrid Optimization Model for Electric Renewables ( HOMER ) may aid in designing electric power systems that employ some combination of wind turbines, photovoltaic panels, or diesel generators to produce electricity. [Temporal_Coverage: Start_Date=1983-07-01; Stop_Date=1993-06-30] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180].

Stackhouse, Paul W.↗

Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

We develop a numerically stable sequential formulation of thermoporomechanics for largely deformable gas hydrate deposits, extended from the fixed stress split of infinitesimal transformation. Constitutive equations are based on the total Lagrangian approach for both flow and geomechanics, including dynamic full tensor permeability and thermal conductivity updated from the deformation gradient. For space discretization, we take the cell-centered finite volume and node-based finite element method for flow and geomechanics, respectively. Then, we propose a sequential implicit method for all-way coupled thermoporomechanics, where the nonisothermal multiphase flow problem of gas hydrates is solved implicitly first and then the geomechanics problem is solved implicitly at the next step. During solution of the flow problem, we fix the rate of first Pioal total stress for numerical stability as well as apply porosity correction and entropy correction to account for geomechanical effects. We test numerical examples where flow and geomechanics parameters are based on deep oceanic gas hydrate deposits. When applying depressurization, even though the results between the infinitesimal transformation and finite strain geomechanics are similar in the early stages due to small deformation, we find differences between them in the late times as deformation becomes large. Accordingly, permeability and thermal conductivity tensors become nonisotropic full tensors although they are initially isotropic. Furthermore, we identify numerical stability of the developed sequential method from the test cases that exhibit the highly complex coupled gas hydrate systems with large deformation. Thus, the proposed sequential formulation can be applied in largely deformable gas hydrate systems.

42 ENGINEERING↗

Clay-Based Biohybrid Materials for Biomedical and Pharmaceutical Applications

Clays have traditionally been linked to health care, being used for centuries in the fight against infections and diseases. Similarly, biohybrids produced by combinations of clays and biological species through 'bottom-up' approaches have been evaluated over the past decade for biomedical and pharmaceutical uses. These biohybrids show interesting features such as biocompatibility and biodegradability which make them suitable for health care applications. The aim of the present communication was to review recent research contributions describing progress and the role of biohybrid materials based on clays in biomedicine and pharmacy disciplines. Emphasis will be on the authors' own experience of this topic, particularly on aspects related to controlled drug-delivery systems, adjuvants of vaccines, and vectors for non-viral gene transfection. Bionanocomposites offer several advantages for use in the design of new and efficient pharmacological formulations for cutaneous and oral administration. In these systems, the drug is typically entrapped in the clay and protected by a biopolymer matrix, and both components contribute to a gradual release of the drug. Clay-based hybrids have also shown their efficacy in vaccines as they can act as nanocarriers of viral particles, due to the biomimetic interface created on the clay surface after adsorption of suitable biomolecules such as phospholipids, while the clay acts as an adjuvant to increase the efficacy of the vaccine. Finally, a new application of clays as non-viral vectors for controlled gene delivery is attracting increasing interest in the treatment of diverse diseases; clays such as sepiolite have demonstrated their ability to act as nanocarriers of nucleic acids and facilitate their transfection in mammalian cells. (author)

36 MATERIALS SCIENCE↗

Trends in Human Spaceflight: Failure Tolerance, High Reliability and Correlated Failure History

In a half century of human spaceflight, NASA has continuously refined agency safety and reliability requirements in response to mission demands, critical failures, and technology development. Early spacecraft, including Mercury, Gemini and Apollo vehicles, were highly reliant on dissimilar redundancy and demonstrated test margins. Later programs, such as the reusable Space Transportation System (STS) and International Space Station (ISS), introduced probabilistic studies and isolated two-failure tolerance to improve robustness at the expense of added complexity. More recently, the Orion Multi-Program Crew Vehicle (MPCV) program adopted universal single-failure tolerance with two categorical exceptions; Zero-Failure Tolerant (0FT) and Design for Minimum Risk (DFMR) hardware. Failure tolerance variances are defined and managed in accordance with agency human-rating requirements, and require concurrence from program Technical Authorities (TA) as well as the MPCV Safety and Mission Assurance Safety and Engineering Review Panel (MSERP). To understand and reaffirm standards applied to Apollo, Space Shuttle and Orion vehicles, Orion and Deep Space Gateway Safety and Mission Assurance (S&MA) representatives conducted accelerated research to compare unique safety and reliability criteria against ground and flight anomalies, based on information contained in post-mission reports and the Problem Reporting and Corrective Action (PRACA) database. In some cases, high-profile failures and narrow escapes have reinforced decisions to maintain or adapt safety requirements. In others, empirical trends have highlighted the need for vigilance and innovative safety guidelines. Given the inability to achieve absolute compliance with evolving safety and reliability requirements, the team conducted a targeted review of DFMR and 0FT propulsion elements within the framework of changing system design, inspection, materials and process developments to formulate conclusions on technological maturity, failure density, and net changes in safety risk. Based on the aggregate performance of high-reliability and failure-tolerant systems, the authors have attempted to establish best practices and guidelines to inform future program decisions. On a somewhat cautionary note, this study is not intended to direct a universal set of requirements for future missions based on prior lessons learned. Spacecraft safety is a multi-variable problem, and attempts to mitigate past failures will not guarantee future success. However, this assessment offers a retrospective review of policy changes, implementation and effectiveness. In the future, NASA, European Space Agency (ESA) and industry partners may benefit from a more robust correlation between requirements and performance, as space-faring nations work toward more challenging, complex and long-duration commercial and deep-space ventures.

Green, Carrie↗

Stability Analysis of the Flow over a Swept Forward-Facing Step using PIV Base Flowsin a Non-Orthogonal Coordinate System

Understanding the flow of a crossflow-vortex-dominated boundary layer over a forward-facing step excrescence is necessary in order to mitigate an early transition scenario on the wings and tail of commercial aircraft. By performing BiGlobal stability analysis on the flowfield measured with high-resolution, stereographic Particle Image Velocimetry (PIV), previous work has shown that a family of unstable disturbances exists in the direct downstream vicinity of a supercritical, forward-facing step. The goal of the present paper is to improve the previously used stability approach on two fronts: 1) the stability problem is formulated in a non-orthogonal coordinate system and 2) the contribution of the out-of-plane base-flow derivatives is accounted for locally. The resulting eigen solutions feature significantly stronger growth rates and eigenfunctions that are localized above regions of reverse flow. The changes in the problem formulation furthermore establish a significant improvement in the comparison of the stability solutions with the results from the Spectral Proper Orthogonal Decomposition (SPOD) of a time-resolved measurement of the perturbation content. A large destabilizing effect by the out-of-plane base-flow derivatives is determined to be justified, despite the assumption that these derivatives are small, because the effect can be reconstructed upon using an eigenvalue-correction formula that assumes the responsible terms are infinitesimally small. The unstable perturbation mechanisms are demonstrated to have a convective nature by assessing the relation between the frequency and the out-of-plane wavenumber, which indicates that their group speed does not approach zero. The last 2 facts, that 1) the perturbations are convective and 2) that the out-of-plane base-flow-derivative terms are small, remove any qualitative suspicions that the perturbation problem is not conducive to parabolization. This opens the path to analyzing this problem with a plane-marching approach.

Boundary- Layer Transition↗

Sub-domain decomposition methods and computational controls for multibody dynamical systems

This paper presents a concurrent methodology to simulate the dynamics of flexible multibody systems with a large number of degrees of freedom. A general class of open-loop structures is treated and a redundant coordinate formulation is adopted. A range space method is used in which the constraint forces are calculated using a preconditioned conjugate gradient method. By using a preconditioner motivated by the regular ordering of the directed graph of the structures, it is shown that the method is order N in the total number of coordinates of the system. The overall formulation has the advantage that it permits fine parallelization and does not rely on system topology to induce concurrency. It can be efficiently implemented on the present generation of parallel computers with a large number of processors. Validation of the method is presented via numerical simulations of space structures incorporating large number of flexible degrees of freedom.

Menon, R. G.↗

On Distribution Grid Optimal Power Flow Development and Integration

Due to changes in electric distribution grid operation, new operation regimes have been recommended. Distribution grid optimal power flow (DOPF) has received tremendous attention in the research community, yet it has not been fully adopted across the utility industry. Our paper recognizes this problem and suggests a development and integration procedure for DOPF. We propose development of DOPF as a three step procedure of 1) processing the grid, 2) obtaining a tractable solution, and 3) implementing multiple solution algorithms and benchmarking them to improve application reliability. For the integration of DOPF, we demonstrate how a DOPF federate may be developed that can be integrated in a co-simulation environment to mimic the real-world conditions and hence improve its practicality to be deployed in the field. To demonstrate the efficacy of the proposed methods, tests on IEEE 123 bus system are performed where the usage of tractable formulation in DOPF algorithm development and its comparison to the benchmark solution are demonstrated.

Hanif, Sarmad↗

Scalable Predictive Control and Optimization for Grid Integration of Large-Scale Distributed Energy Resources

Integrating a large number of distributed energy resources (DERs) into the power grid needs a scalable power balancing method. We formulate the power balancing problem as a look-ahead optimization problem to be solved sequentially by a power distribution system aggregator based on a model predictive control (MPC) framework. Solving large-scale look-ahead control problems requires proper configuration of the control steps. In this paper, to solve large-scale control problems, we propose a variable time granularity where control time steps nearby the current control step have finer resolutions. The aggregator objective includes maximization of power production revenue and minimization of power purchasing expense, renewable power curtailment, and mileage costs for energy storage and electric vehicle (EV) charging stations while satisfying system capacity and operational constraints. The control problem is formulated as a mixed-integer linear program (MILP) and solved using the XpressMP solver. We perform simulations considering a copper plate representation of a large distribution network consisting of 2507 devices (controllable DERs), including curtailable photovoltaics (PVs), energy storage batteries, EV charging stations, and buildings with heating, ventilation, and air conditioning units (HVACs). We show the effectiveness of the proposed approach in managing DERs interactively for maximum energy trading profit and local supply-demand power balancing. Finally, we demonstrate that the proposed method outperforms other benchmark controllers regarding computation time without compromising operational performance.

DER↗

Scalable Predictive Control and Optimization for Grid Integration of Large-Scale Distributed Energy Resources

Integrating a large number of distributed energy resources (DERs) into the power grid needs a scalable power balancing method. We formulate the power balancing problem as a look-ahead optimization problem to be solved sequentially by a power distribution system aggregator based on a model predictive control (MPC) framework. Solving large-scale look-ahead control problems requires proper configuration of the control steps. In this paper, to solve large-scale control problems, we propose a variable time granularity where control time steps nearby the current control step have finer resolutions. The aggregator objective includes maximization of power production revenue and minimization of power purchasing expense, renewable power curtailment, and mileage costs for energy storage and electric vehicle (EV) charging stations while satisfying system capacity and operational constraints. The control problem is formulated as a mixed-integer linear program (MILP) and solved using the XpressMP solver. We perform simulations considering a copper plate representation of a large distribution network consisting of 2507 devices (control-lable DERs), including curtailable photovoltaics (PVs), energy storage batteries, EV charging stations, and buildings with heating, ventilation, and air conditioning units (HVACs). We show the effectiveness of the proposed approach in managing DERs interactively for maximum energy trading profit and local supply-demand power balancing. Finally, we demonstrate that the proposed method outperforms other benchmark controllers regarding computation time without compromising operational performance.

DER↗

Scalable Predictive Control and Optimization for Grid Integration of Large-Scale Distributed Energy Resources: Preprint

Integration of a large number of distributed energy resources (DERs) into the power grid needs a scalable power balancing method. We formulate the power balancing problem as a look-ahead optimization problem to be solved sequentially by a power distribution system aggregator based on a model predictive control (MPC) framework. Solving large-scale look-ahead control problem requires proper configuration of the control steps. In this paper, to solve large-scale control problems, we propose a variable time granularity where control time steps nearby the current control step have finer resolutions. The aggregator objective includes maximization of power production revenue and minimization of power purchasing expense, renewable power curtailment, and mileage costs for energy storage and electric vehicle (EV) charging stations while satisfying system capacity and operational constraints. The control problem is formulated as a mixed-integer linear program (MILP) and solved using the XpressMP solver. We perform simulations considering a copper plate representation of a large distribution network consisting of 2507 devices (controllable DERs) including curtailable photovoltaics (PVs), energy storage batteries, EV charging stations, and buildings with heating, ventilation, and air conditioning units (HVACs). We show the effectiveness of the proposed approach in managing DERs interactively for maximum energy trading profit and local supply-demand power balancing. Finally, we demonstrate that the proposed method outperformed other benchmark controllers regarding computation time without compromising operational performance.

DER↗

A fast reduced model for a shell-and-tube based latent heat thermal energy storage heat exchanger and its application for cost optimal design by nonlinear programming

Numerical simulation of latent heat thermal energy storage (LHTES) systems plays a fundamental role in studying the physical process and guiding the engineering design. Discretization of the PDEs describing the nonlinear solidification/melting process of phase change materials (PCMs) leads to a large-scale complex dynamics system, where the system behavior depends on a set of parameters. In a design setting, repeated model evaluations are required over the set of parameters results in significant computational burden. In this paper, an explicit analytic solution was built for the propagation of the solidification front in a cylindrical coordinate. The analytic solution approach is further employed to develop a low computational reduced model (RM) as a module for a shell-and-tube based LHTES heat exchanger. The levelized Cost of Energy (LCOE) is used as a design metric and the RM model is used to apply system-level constraints in the nonlinear programming formulation that facilitates efficient global optimal design of the PCM properties, flow conditions and tube geometries. The use of LCOE as the design metric prevents over design of the heat transfer rate and also establishes a fair ground for evaluation of different thermal storage technologies and their integrated applications with other systems. Optimal results showed that a higher effectiveness results in a higher LCOE; the velocity of the HTF and the length of the channel are highly correlated with each other; both larger PCM latent energy and conductivity result in lower LCOE.

25 ENERGY STORAGE↗

A theoretical framework for convergence and continuous dependence of estimates in inverse problems for distributed parameter systems

Numerical techniques for parameter identification in distributed-parameter systems are developed analytically. A general convergence and stability framework (for continuous dependence on observations) is derived for first-order systems on the basis of (1) a weak formulation in terms of sesquilinear forms and (2) the resolvent convergence form of the Trotter-Kato approximation. The extension of this framework to second-order systems is considered.

Banks, H. T.↗

Formulation of Three-Phase State Estimation Problem Using a Virtual Reference

State Estimation (SE) is the backbone of modern Energy Management System (EMS) due its capability of processing real time measurements and provide reliable information to system operators. Since its introduction to power systems in the 70’s, SE has been widely used in transmission networks. Distribution grids on the other hand lack sufficient number of real time measurements, and for that reason SE has not been widely implemented on these systems. The recent increase in the number of renewable energy sources connected to the grid at lower voltage levels, the advent of Distribution Automation (DA) and Smart Grids necessitate closer monitoring of distribution networks and thus forcing utilities to upgrade their operations and deploy Advanced Distribution Management Sytems (ADMS). Therefore, Distribution System State Estimation (DSE) is paramount to provide real time monitoring of active distribution grids. This papers investigates the formulation of the three-phase distribution state estimation where in the most general case, none of the system buses may have balanced three phase voltages. In such a case, the choice and definition of the slack bus need to be revisited. Most of the existing work arbitrarily assign the root bus of the distribution feeder as the balanced three-phase reference which may not be realistic for today’s active distribution systems. This paper addresses this shortcoming of the existing SE formulations by introducing a virtual reference bus. Here, the proposed approach is effectively validated by simulations carried out on distribution test systems.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Nonrecursive formulations of multibody dynamics and concurrent multiprocessing

Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. Historically, much of this research can be traced to applications of low dimensionality in mechanism and vehicle dynamics. Indeed, there is little doubt that recursive order N methods are the method of choice for this class of systems. This approach has the advantage that a minimal number of coordinates are utilized, parallelism can be induced for certain system topologies, and the method is of order N computational cost for systems of N rigid bodies. Despite the fact that many authors have dismissed redundant coordinate formulations as being of order N(exp 3), and hence less attractive than recursive formulations, we present recent research that demonstrates that at least three distinct classes of redundant, nonrecursive multibody formulations consistently achieve order N computational cost for systems of rigid and/or flexible bodies. These formulations are as follows: (1) the preconditioned range space formulation; (2) penalty methods; and (3) augmented Lagrangian methods for nonlinear multibody dynamics. The first method can be traced to its foundation in equality constrained quadratic optimization, while the last two methods have been studied extensively in the context of coercive variational boundary value problems in computational mechanics. Until recently, however, they have not been investigated in the context of multibody simulation, and present theoretical questions unique to nonlinear dynamics. All of these nonrecursive methods have additional advantages with respect to recursive order N methods: (1) the formalisms retain the highly desirable order N computational cost; (2) the techniques are amenable to concurrent simulation strategies; (3) the approaches do not depend upon system topology to induce concurrency; and (4) the methods can be derived to balance the computational load automatically on concurrent multiprocessors. In addition to the presentation of the fundamental formulations, this paper presents new theoretical results regarding the rate of convergence of order N constraint stabilization schemes associated with the newly introduced class of methods.

Kurdila, Andrew J.↗

Parametric Optimization Problem Formulation for Connected Hybrid Electric Vehicles using Neural Network based Equivalent Model

The dynamics of powertrain control systems are complicated and involve both nonlinear plant model and control functionalities, albeit they are well defined and formulated using first principle approaches. This constitutes difficulties in exploring implementable optimal tuning rules for some selected control parameters using vehicle-to-vehicle (V2V) communications. This paper presents a way to use neural networks (NN) to represent the problem of parameter tuning for optimizing fuel consumption. For this purpose, physical modelling and validation have been firstly performed for the closed loop powertrain system of the concerned vehicle for some given driving cycles. This is then followed by the sensitivity analysis that selects most influential control parameters to optimize. Using the data generated from the obtained physical models, an equivalent NN formulation has finally been obtained that gives simple yet unified objectives and constraints ready to be used to solve the optimization problem that produces optimal tuning rules for the selected control parameters to minimize fuel consumption.

Parameter optimization, V2X, Powertrain modeling↗

Cyber risk assessment and investment optimization using game theory and ML-based anomaly detection and mitigation for wide-area control in smart grids

The electric power grid is increasingly becoming susceptible to cyber attacks that exploit vulnerabilities in the smart grid control, information, and physical layers. Successful cyber attacks can have catastrophic impacts on the social and economic well-being of any nation all over the globe. It has, thus, become imperative to secure the smart grid against such adversarial actions to ensure stable, secure, and reliable operation of the grid. The existing research and industry practices prove to be inadequate in terms of providing pragmatic and effective defense methodologies and measures for long-term cybersecurity planning and real-time cybersecurity for grid operation. For example, existing works lack models that incorporate uncertain behavior of cyber-attackers and pragmatic defense measures for cyber risk assessment and cybersecurity investment optimization which often provide unreliable and strictly qualitative solutions to these problems. At the same time, with the growing number of cyber incidents in the grid, there still exists a need to develop attack-resilient algorithms for wide-area monitoring, protection, and control (WAMPAC) applications like the wide-area voltage control systems (WAVCS) for Flexible AC Transmissions Systems (FACTS) that lack in scalable and feasible solutions from the cybersecurity perspective. This dissertation proposes novel models and methodologies for: (1) Cybersecurity planning, and (2) Cybersecurity for system operation. The cybersecurity planning is achieved through cyber risk assessment and cybersecurity resource investment optimization for long-term cybersecurity of the grid using game theory and attack-defense trees. Cybersecurity for system operation consists of development of cyber anomaly detection and mitigation algorithms for flexible AC transmission system (FACTS) controller-based wide-area voltage control systems (WAVCS) using machine learning (ML), and software defined networking-based moving target defense network routing for achieving real-time cyber-physical security for grid operations. This is followed by hardware-in-the-loop (HIL) implementation and evaluation of these attack prevention, detection, and mitigation algorithms and methodologies showcasing their feasibility in a close to real-world environment. For cybersecurity planning, a novel approach involving a combination of game theory and attack defense trees (ADT) for optimal cybersecurity resource allocation in the smart grid is proposed. This methodology involves modeling of the cyber-physical smart grid substations as ADTs, defining attacker costs, defense costs, and attack probabilities for attack access points. Using game theoretical formulation, optimal defense strategies for the defender of the system to invest cybersecurity resources in the grid are obtained. Additionally, a game-theoretic framework is developed for quantitative cyber-physical risk assessment of the grid under a dynamically changing cyber threat space and uncertain behavior of cyber attackers which is further used to optimize investments in the smart grid's cybersecurity resources. The attacker, defender, and the smart grid system are modeled while incorporating attacker-stochasticity and federal guidelines for smart grid cybersecurity. This allows quantification of threat, vulnerabilities, and attack impact of the grid for quantitative risk assessment. The defender's budget to invest in the security resources in the grid is optimized based on the strategies leading to minimum system risk. The evaluation of the proposed solutions highlight the feasibility for practical implementation of these methodologies and algorithms in the smart grid, while taking the federal requirements and guidelines for smart grid security into consideration. For achieving cybersecurity for system operation, attack prevention, detection, and mitigation algorithms and methodologies are developed specifically for FACTS-based WAVCS. Anomaly detection and mitigation in the WAVCS are achieved using algorithms based on machine learning which involves offline training and testing of ML models with CPS datasets incorporating physics-based features that allow accurate distinction between system faults and cyber attacks. For attack prevention, a methodology based on software defined network (SDN)-based moving target defense (MTD) network routing is proposed that enables prevention of Denial of Service (DoS) type attacks on the smart grid communication system. Subsequently, these methodologies and algorithms are implemented and evaluated on an HIL testbed that allows for real-time attack prevention, detection, and mitigation of emulated cyber attacks on the WAVCS in a close to real-world environment. The results show highly accurate and efficient performance of the implemented algorithms and methodologies with the smart grid system operating within the NERC's system operation limits even in the presence of DoS and data integrity cyber attacks. This work opens up future research opportunities in other directions such as (1) Expanding cybersecurity planning methodologies to real-time cyber contingency analysis with different game formulations; and (2) Applying the cybersecurity for system operation algorithms to broader categories of wide-area control applications.

24 POWER TRANSMISSION AND DISTRIBUTION↗

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Herein, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a coarse-grid method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the present method is displayed by numerical experiments on benchmark problems.

97 MATHEMATICS AND COMPUTING↗