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All sourcesNASA NTRSDOE OSTI
DOE OSTI2017

Advances in Cross-Cutting Ideas for Computational Climate Science

This report presents results from the DOE-sponsored workshop titled, ``Advancing X-Cutting Ideas for Computational Climate Science Workshop,'' known as AXICCS, held on September 12--13, 2016 in Rockville, MD. The workshop brought together experts in climate science, computational climate science, computer science, and mathematics to discuss interesting but unsolved science questions regarding climate modeling and simulation, promoted collaboration among the diverse scientists in attendance, and brainstormed about possible tools and capabilities that could be developed to help address them. Emerged from discussions at the workshop were several research opportunities that the group felt could advance climate science significantly. These include (1) process-resolving models to provide insight into important processes and features of interest and inform the development of advanced physical parameterizations, (2) a community effort to develop and provide integrated model credibility, (3) including, organizing, and managing increasingly connected model components that increase model fidelity yet complexity, and (4) treating Earth system models as one interconnected organism without numerical or data based boundaries that limit interactions. The group also identified several cross-cutting advances in mathematics, computer science, and computational science that would be needed to enable one or more of these big ideas. It is critical to address the need for organized, verified, and optimized software, which enables the models to grow and continue to provide solutions in which the community can have confidence. Effectively utilizing the newest computer hardware enables simulation efficiency and the ability to handle output from increasingly complex and detailed models. This will be accomplished through hierarchical multiscale algorithms in tandem with new strategies for data handling, analysis, and storage. These big ideas and cross-cutting technologies for enabling breakthrough climate simulation advancements also need the "glue" of outreach and learning across the scientific domains to be successful. The workshop identified several strategies to allow productive, continuous engagement across those who have a broad knowledge of the various angles of the problem. Specific ideas to foster education and tools to make material progress were discussed. Examples include follow-on cross-cutting meetings that enable unstructured discussions of the types this workshop fostered. A concerted effort to recruit undergraduate and graduate students from all relevant domains and provide them experience, training, and networking across their immediate expertise is needed. This will broaden and expand their exposure to the future needs and solutions, and provide a pipeline of scientists with a diversity of knowledge and know-how. Providing real-world experience with subject matter experts from multiple angles may also motivate the students to attack these problems and even come up with the missing solutions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Advances in Cross-Cutting Ideas for Computational Climate Science

This report presents results from the DOE-sponsored workshop titled, Advancing X-Cutting Ideas for Computational Climate Science Workshop,'' known as AXICCS, held on September 12--13, 2016 in Rockville, MD. The workshop brought together experts in climate science, computational climate science, computer science, and mathematics to discuss interesting but unsolved science questions regarding climate modeling and simulation, promoted collaboration among the diverse scientists in attendance, and brainstormed about possible tools and capabilities that could be developed to help address them. Emerged from discussions at the workshop were several research opportunities that the group felt could advance climate science significantly. These include (1) process-resolving models to provide insight into important processes and features of interest and inform the development of advanced physical parameterizations, (2) a community effort to develop and provide integrated model credibility, (3) including, organizing, and managing increasingly connected model components that increase model fidelity yet complexity, and (4) treating Earth system models as one interconnected organism without numerical or data based boundaries that limit interactions. The group also identified several cross-cutting advances in mathematics, computer science, and computational science that would be needed to enable one or more of these big ideas. It is critical to address the need for organized, verified, and optimized software, which enables the models to grow and continue to provide solutions in which the community can have confidence. Effectively utilizing the newest computer hardware enables simulation efficiency and the ability to handle output from increasingly complex and detailed models. This will be accomplished through hierarchical multiscale algorithms in tandem with new strategies for data handling, analysis, and storage. These big ideas and cross-cutting technologies for enabling breakthrough climate simulation advancements also need the "glue" of outreach and learning across the scientific domains to be successful. The workshop identified several strategies to allow productive, continuous engagement across those who have a broad knowledge of the various angles of the problem. Specific ideas to foster education and tools to make material progress were discussed. Examples include follow-on cross-cutting meetings that enable unstructured discussions of the types this workshop fostered. A concerted effort to recruit undergraduate and graduate students from all relevant domains and provide them experience, training, and networking across their immediate expertise is needed. This will broaden and expand their exposure to the future needs and solutions, and provide a pipeline of scientists with a diversity of knowledge and know-how. Providing real-world experience with subject matter experts from multiple angles may also motivate the students to attack these problems and even come up with the missing solutions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Prefeasibility Assessment for Solar PV and Storage for Critical Community Facilities in Chernihiv, Ukraine [Slides]

A prefeasibility analysis is performed for integrating solar photovoltaics and battery energy storage at four critical facilities in Chernihiv, Ukraine. The facilities were identified by Chernihiv city officials and include Hospital No. 2, the Maternity Hospital, Secondary School No. 11, and Preschool No. 4. The analyses were performed using NREL's REopt decision-support software tool. The analysis identifies potential capacities for PV and battery energy storage to provide both economic and resilience benefits. The conceptual architecture and estimates of key summary financial and performance metrics are presented.

14 SOLAR ENERGY↗
DOE OSTI2015

Interactive statistical decision tools (GiSdT): An open source software integration of decision analysis and stakeholder judgment for sustainable waste management, disposal and remediation - 15650

The vast majority of remediation and waste management decisions are made without quantitative consideration of economic and socio-political factors. They are made instead based on quantitative metrics of human health risk. Sustainable decisions need to be made based on all three 'pillars of sustainability' (economics, environment and social), and require understanding and characterization of the costs and values associated with each pillar. In addition, such decisions need to conform to regulatory or other legal requirements, which often constrains the decision space of interest. Although efforts are often made to include factors across all three pillars (for example, for decisions made under CERCLA), these efforts are usually qualitative, and hence difficult to defend. They lack technical defensibility, transparency and traceability. The purpose of GiSdT (Guided Interactive Statistical Decision Tools) is to provide a quantitative framework whereby all aspects of a decision problem, such as remediation and waste management decisions, can be addressed quantitatively, and hence, defensibly, transparently and traceably. Most decision-makers do not currently have access to useful or usable methods and approaches when are presented with choices that have significant impacts across all three pillars of sustainability. The goal of GiSdT is to provide that access by identifying or developing effective and user-friendly decision methods and approaches that empower decision-makers to explicitly and routinely incorporate all aspects of sustainability into their decision-making. To identify and develop these methods and approaches, GiSdT provides the tools needed for decision-makers and stakeholders to understand and characterize their knowledge of their current decision-making processes, and what methods and approaches they need to pro-actively and quantitatively address all aspects of sustainability in their decisions. GiSdT is an open source interactive framework program that provides the tools needed to address the relevant components of a sustainability-based decision problem, including describing the decision landscape, translating that landscape into goals and objectives, valuing the measurable attributes that describe each objective, identifying decision options and addressing the uncertainty in the attributes (e.g., human health risk, institutional controls) through appropriate probabilistic models. In addition to addressing these quantitative technical components, the GiSdT program is aimed at facilitating stakeholder involvement so that the perspectives, values and objectives are all relevant stakeholders are addressed explicitly in the decision making process. The technical underpinnings of our software program, called Guided Interactive Statistical Decision Tools (GiSdT), are described. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2025

Feature review of photovoltaic modeling software utilizing blind performance assessment

While confidence in photovoltaic (PV) modeling software has always been essential, the rapid pace of new PV plant developments makes accuracy and credibility more critical than ever. Independent assessments, particularly through blind modeling comparisons, are therefore necessary to ensure unbiased benchmarking across PV modeling software. Previous studies have been limited by a narrow range of models compared, anonymized results, or system size. This study presents results from the first-ever onymous blind modeling comparison, evaluated using both lab- and utility-scale fixed-tilt, monofacial, south-facing systems at sub-hourly time intervals. Seven commercially used PV software tools were compared: 3E SynaptiQ, PlantPredict, PVsyst, RatedPower, SAM, SolarFarmer, and Solargis Evaluate. Predictions were submitted directly by software representatives, providing unique insights into each software’s implementation and resulting prediction behavior. Notable features, including plane-of-array (POA) transposition model, module temperature model, shading model, and performance model were analyzed and compared. Four summary tables compile these features of the software, serving as a resource to help users understand the methodological differences and select the most suitable software for their applications. The software tools show deviations from mean error in annual yield up to 2.5 % in the lab-scale system, increasing to 6.0 % for the utility-scale system. These differences arise from a combination of user decisions and the inherent behavior of the software, indicating the need for continuous and rigorous validation of modeling methods using these software tools against complex, real-world systems.

14 SOLAR ENERGY↗
DOE OSTI2026

EASY-SHIFT v Alpha

The software is a generic, price- and load-responsive control algorithm integrating heat pumps with thermal energy storage. The algorithm leverages simple models of the system and easily accessible data to schedule operation of heat pumps and thermal energy storage in ways that minimize the cost of operating the heating/cooling system. This tool is specifically designed to be easy to interact with, and something that industry partners are able to adopt. There are two current state of the art approaches. Industry tends to develop very simple algorithms, with predetermined schedules that are not capable of changing operation in response to changes in operating environment. For example, a control designed to avoid high-price electricity from 5-8 PM will not be able to adapt if the high-price period changes to 4-9 PM. Academia commonly develops algorithms called Model predictive control (MPC). MPC requires extensive data and highly trained staff to develop a specific type of simulation model of the building, connect the building to optimization algorithms, and leverage powerful computers. Industry, with limited time/finance budgets for any project, is resistant to adopting MPC due to the associated high complexity and cost.

Grant, Peter [Lawrence Berkeley National Laborator↗
DOE OSTI2016

Multiyear Plan for Validation of EnergyPlus Multi-Zone HVAC System Modeling using ORNL's Flexible Research Platform

This document describes the Oak Ridge National Laboratory (ORNL) multiyear experimental plan for validation and uncertainty characterization of whole-building energy simulation for a multi-zone research facility using a traditional rooftop unit (RTU) as a baseline heating, ventilating, and air conditioning (HVAC) system. The project’s overarching objective is to increase the accuracy of energy simulation tools by enabling empirical validation of key inputs and algorithms. Doing so is required to inform the design of increasingly integrated building systems and to enable accountability for performance gaps between design and operation of a building. The project will produce documented data sets that can be used to validate key functionality in different energy simulation tools and to identify errors and inadequate assumptions in simulation engines so that developers can correct them. ASHRAE Standard 140, Method of Test for the Evaluation of Building Energy Analysis Computer Programs (ASHRAE 2004), currently consists primarily of tests to compare different simulation programs with one another. This project will generate sets of measured data to enable empirical validation, incorporate these test data sets in an extended version of Standard 140, and apply these tests to the Department of Energy’s (DOE) EnergyPlus software (EnergyPlus 2016) to initiate the correction of any significant deficiencies. The fitness-for-purpose of the key algorithms in EnergyPlus will be established and demonstrated, and vendors of other simulation programs will be able to demonstrate the validity of their products. The data set will be equally applicable to validation of other simulation engines as well.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

Synthesis of Correct Digital Controller Models from Specifications by Model Transformation (21-0320)

The design of high consequence controllers (in weapons systems, autonomy, etc.) that do what they are supposed to do is a significant challenge. Testing simply does not come close to meeting the requirements for assurance. Today circuit designers at Sandia (and elsewhere) typically capture the core behavior of their components using state models in tools such as STATEFLOW. They then check that their models meet certain requirements (e.g. “The system bus must not deadlock” or “both traffic lights at an intersection must not be green at the same time”) using tools called model checkers. If the model checker returns “yes” then the property is guaranteed to be satisfied by the model. However, there are several drawbacks to this industry practice: (1) there is a lot of detail to get right, this is particularly challenging when there are multiple components requiring complex coordination (2) any errors returned by the model checker have to be traced back through the design and fixed, necessitating rework, (3) there are severe scalability problems with this approach, particularly when dealing with concurrency. All this places high demands on the designers who now face not only an accelerated schedule but also controllers of increasing complexity. This report describes a new and fundamentally different approach to the construction of safety-critical digital controllers. Instead of directly constructing a complete model and then trying to verify it, the designer can start with an initial abstract (think “sketch”) model plus the requirements, from which a correct concrete model is automatically synthesized. There is no need for post-hoc verification of required functional properties. Having tool to carry this out will significantly impact the nation’s ability to ensure the safety of high-consequence digital systems. The approach has been implemented in a prototype tool, along with a suite of examples, including ones that reflect actual problems faced by designers. Our approach operates on a variant of Statecharts developed at Sandia called Qspecs. Statecharts are a widely used formalism for developing concurrent reactive systems, supporting scalability through allowing state models containing composite states, which are the serial or parallel composition of substates which can themselves contain statecharts. Statecharts enable an incremental style of development, in which states are progressively refined to incorporate greater detail in an incremental model of software development. Our approach formulates a set of constraints from the structure of the models and the requirements and propagates these constraints to a fixpoint. The solution to the constraints is an inductive invariant along with guards on the transitions. We also show how our approach extends to implementation refinement, decomposition, composition, and elaboration. We currently handle safety requirements written in LTL (Linear Temporal Logic)

42 ENGINEERING↗
DOE OSTI2023

Zero-Order Reaction Kinetics (Zero-RK): Enabling the Use of Detailed Chemical Kinetics in Combustion Simulations (CRADA Final Report)

This was a collaborative effort between Lawrence Livermore National Security, LLC (LLNS), as manager and operator of Lawrence Livermore National Laboratory (LLNL) and Gamma Technologies, LLC (GT or Participante), to incorporate the ability to access LLNL chemical kinetics technologies while using GT-SUITE, GT’s market leading engine simulation software. At the end of the project, LLNL has released Zero-RK version 3.5 with zero- and one-dimensional (0-D and 1-D) solver functionality that interfaces with GT’s GT-SUITE v2023 and later releases. GT has tested its product to assure their customers that the interface can provide reduction in chemistry solution time for detailed chemistry simulations. The process has also positioned GT to easily benefit from future improvements of the Zero-RK suite of tools developed under the DOE Vehicle Technologies Office.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2026

Zero Order Reactioin Kinetics: Enabling the use of Detailed Chemical Kinetics in Combustion Simulations (Final CRADA Report)

This was a collaborative effort between Lawrence Livermore National Security, LLC (LLNS), as manager and operator of Lawrence Livermore National Laboratory (LLNL) and Gamma Technologies, LLC (GT or Participante), to incorporate the ability to access LLNL chemical kinetics technologies while using GT-SUITE, GT’s market leading engine simulation software. At the end of the project, LLNL has released Zero-RK version 3.5 with zero- and one-dimensional (0-D and 1-D) solver functionality that interfaces with GT’s GT-SUITE v2023 and later releases. GT has tested its product to assure their customers that the interface can provide reduction in chemistry solution time for detailed chemistry simulations. The process has also positioned GT to easily benefit from future improvements of the Zero-RK suite of tools developed under the DOE Vehicle Technologies Office.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2018

Towards a theory of economic value for modeling and simulation: incremental cost of parallel simulation (wip)

A useful perspective on the economic value of simulation in engineering comes from considering the role simulation plays in constructing a system. We propose a conceptual model that combines an embryonic concept of the value of an information bit with cost estimation tools used in software engineering. The model provides an economic justification for the common conception that models should be built to address specific questions rather than to comprehensively represent a referent system. Extending these ideas, we offer a model for the incremental value of parallel execution to explain why simulation applications that take advantage of sophisticated parallel discrete event simulation tools form a relatively narrow niche despite the ubiquity of parallel computing hardware. The proposed model enables us to bring value of information criteria into model development and to make predictions about the future of parallel simulation applications.

Nutaro, James J↗
DOE OSTI2020

Holistic Measurement Driven Resilience: Combining Operational Fault and Failure Measurements and Fault Injection for Quantifying Fault Detection, Propagation and Impact. Final report

For HPC systems to date, application resilience to faults and failures has been accomplished by the brute- force method of checkpoint/restart, which allows an application to make forward progress in the face of system and application faults, errors, and failures independent of root cause or end result. It has remained the primary resilience mechanism because we lack a way to identify faults and anticipate consequences early enough to take meaningful mitigating action. However, checkpoint/restart implementations put a tremendous burden on system resources and on the applications themselves and is becoming less feasible at scale. Because we have not yet operated at scales at which checkpoint/restart fails to provide forward progress, despite increasing costs, vendors have had little motivation to provide the instrumentation necessary for early identification of faults and failures. However, as we move from petascale to exascale, component mean time to failure (MTTF) will render the existing techniques ineffectual and/or too expensive. Furthermore, fault recovery mechanisms such as failover and/or error correction introduce performance inconsistency. Instrumentation allowing early indication of problems and tools to enable use of such information by systems, operating systems, and applications offer an alternative, more scalable and less costly solution. In the HMDR project, we built on our experience and expertise developed and accumulated over years of research on design, monitoring, measurement, and assessment of resilient computing systems. Analysis of field data on the current and past generations of extreme-scale systems revealed several challenges that, if not addressed in increasingly larger and more complex systems, may hinder the effectiveness of future exascale computing systems. Specifically, i) file systems and interconnects in current-generation large-scale systems already operate at the margins of resiliency, including consistent performance, and may not scale to larger deployments; ii) automated, software-based failover mechanisms are frequently inadequate and can introduce wider failures, such that failures during recovery may lead to system/application failures, including system-wide outages; and iii) silent data corruption represents a critical fault mode and will require efficient detection mechanisms if next-generation applications are to take full advantage of exascale hardware. To address the above challenges, we assembled a team of world-renowned experts in resilient extreme- scale computing from the University of Illinois (Electrical and Computer Engineering, Computer Science, and NCSA), SNL, LANL, NERSC, and Cray. Our team includes representatives from centers that house many of the largest HPC resources in the world, both today and over the coming years. The team has a unique track record of research in i) system and application failure characterization based on the analysis of field data, ii) data-driven design of fault/error detection mechanisms, and iii) experimental characterization of system/application resiliency. The team includes system owners/operators who provide continuous data collection and access and ensure installation of appropriate analysis tools.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

SPEL: Software tool for Porting E3SM Land Model with OpenACC in a Function Unit Test Framework

Most high-end computers adopt hybrid architecture, porting a large-scale scientific code onto accelerators is necessary. The paper presents a generic method for porting large-scale scientific code onto accelerators using compiler directives within a modularized function unit test platform. We have implemented the method and designed a software tool (SPEL) to port the E3SM Land Model (ELM) onto the GPUs in the Summit computer. SPEL automatically generates GPU-ready test modules for all ELM functions, such as CanopyFlux, SoilTemperature, and EcosystemDynamics. SPEL breaks the ELM into a collection of standalone unit test programs for easy code verification and further performance improvement. We further optimize several ELM test modules with advanced techniques, including memory reduction, reconstructed parallel loops, and asynchronous GPU kernel launch. We hope our study will inspire new toolkit developments that expedite large-scale scientific code porting with compiler directives.

Schwartz, Peter↗
DOE OSTI2023

Quantitative Scanning Transmission Electron Microscopy for Materials Science: Imaging, Diffraction, Spectroscopy, and Tomography

Scanning transmission electron microscopy (STEM) is one of the most powerful characterization tools in materials science research. Due to instrumentation developments such as highly coherent electron sources, aberration correctors, and direct electron detectors, STEM experiments can examine the structure and properties of materials at length scales of functional devices and materials down to single atoms. STEM encompasses a wide array of flexible operating modes, including imaging, diffraction, spectroscopy, and 3D tomography experiments. This review outlines many common STEM experimental methods with a focus on quantitative data analysis and simulation methods, especially those enabled by open source software. The hope is to introduce both classic and new experimental methods to materials scientists and summarize recent progress in STEM characterization. The review also discusses the strengths and weaknesses of the various STEM methodologies and briefly considers promising future directions for quantitative STEM research.

36 MATERIALS SCIENCE↗
DOE OSTI2022

mesoflow [SWR-22-56]

Mesoflow is a continuum scale simulation tool developed specifically for modeling transport and chemistry at the mesoscale. Our solver utilizes Cartesian block-structured adaptive mesh refinement to resolve complex surface morphologies (of catalysts/biomass particles among others) directly obtained from X-ray tomography data. An immersed boundary based formulation enables rapid representation of complex geometries prevalent in most mesoporous interfaces. The solver is developed on top of open-source performance portable library, AMReX, providing parallel execution capabilities on current and upcoming high-performance-computing (HPC) architectures. Our flexible software framework enables integration of complex chemical mechanisms at heterogenous interfaces and time-split algorithms for circumventing highly disparate reaction and flow time-scales. Our current studies indicate a ten-fold performance gain by using graphics-processing-units (GPU) compared to a single processor for representative problem sizes (2 million cell mesh).

Sitaraman, Hariswaran↗
DOE OSTI2018

pnnl/ZebrafishBMD

ZebrafishBMD: A software tool to perform benchmark dose calculations using toxicity screening data of high-throughput zebrafish assays. High-content imaging of larval zebrafish behavior can be used as a screening approach to rapidly evaluate the relative potential for chemicals to cause toxicity. However, most statistical methods applied to these data transform movement values to incidence-based “hits” and calculate lowest effect levels (LELs), which loses individual fish resolution of behavior and defies hazard ranking due to reliance on applied dose levels. We developed a parallelizable workflow to calculate benchmark dose (BMD) values from dynamic, high-content zebrafish behavior data that scales for high throughput chemical screening. To capture the zebrafish movement response from light to dark stimulus, we summarized time-dependent data using both area under the curve and the immediate change at the transition point into two novel metrics that characterized abnormal behavior as a function of chemical concentration. The BMD workflow was applied to calculate BMD10 values of 1,060 ToxCast chemicals for 24 zebrafish endpoints, including behavior, mortality and morphology. ZebrafishBMD implements the BMD workflow and provides Matlab scripts to read the data (stored in a specific structure) from a .mat file, to calculate the behavioral endpoint values from movement versus time data, to construct concentrationresponse curves, to perform curve-fitting, and to calculate BMD10 values.

Thomas, Dennis↗
DOE OSTI2022

$\mathrm{CROPSR}$: an automated platform for complex genome-wide $\mathrm{CRISPR}$ g$\mathrm{RNA}$ design and validation

CRISPR/Cas9 technology has become an important tool to generate targeted, highly specific genome mutations. The technology has great potential for crop improvement, as crop genomes are tailored to optimize specific traits over generations of breeding. Many crops have highly complex and polyploid genomes, particularly those used for bioenergy or bioproducts. The majority of tools currently available for designing and evaluating gRNAs for CRISPR experiments were developed based on mammalian genomes that do not share the characteristics or design criteria for crop genomes. We have developed an open source tool for genome-wide design and evaluation of gRNA sequences for CRISPR experiments, CROPSR. The genome-wide approach provides a significant decrease in the time required to design a CRISPR experiment, including validation through PCR, at the expense of an overhead compute time required once per genome, at the first run. To better cater to the needs of crop geneticists, restrictions imposed by other packages on design and evaluation of gRNA sequences were lifted. A new machine learning model was developed to provide scores while avoiding situations in which the currently available tools sometimes failed to provide guides for repetitive, A/T-rich genomic regions. We show that our gRNA scoring model provides a significant increase in prediction accuracy over existing tools, even in non-crop genomes. CROPSR provides the scientific community with new methods and a new workflow for performing CRISPR/Cas9 knockout experiments. CROPSR reduces the challenges of working in crops, and helps speed gRNA sequence design, evaluation and validation. We hope that the new software will accelerate discovery and reduce the number of failed experiments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2012

Evaluating the Efficiency of a Multi-core Aware Multi-objective Optimization Tool for Calibrating the SWAT Model

The efficiency of calibrating physically-based complex hydrologic models is a major concern in the application of those models to understand and manage natural and human activities that affect watershed systems. In this study, we developed a multi-core aware multi-objective evolutionary optimization algorithm (MAMEOA) to improve the efficiency of calibrating a worldwide used watershed model (Soil and Water Assessment Tool (SWAT)). The test results show that MAMEOA can save about 1-9%, 26-51%, and 39-56% time consumed by calibrating SWAT as compared with sequential method by using dual-core, quad-core, and eight-core machines, respectively. Potential and limitations of MAMEOA for calibrating SWAT are discussed. MAMEOA is open source software.

97 MATHEMATICS AND COMPUTING↗