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At least 1,657 records · Page 92

Transforming Energy Through Sustainable Mobility: Expanding Low-Carbon Transportation R&D Solutions

As the nation's premier laboratory for cutting-edge transportation decarbonization research and development solutions, the National Renewable Energy Laboratory (NREL) pioneers the creation and deployment of sustainable mobility technologies and strategies, with a focus on slashing transportation sector greenhouse gas emissions and combatting climate change. Trucks, planes, cargo ships, and other difficult-to-decarbonize vehicles are part of this essential transition for the transportation sector, which is currently the nation's largest source of the greenhouse gas emissions. NREL provides the scientific building blocks needed to spur innovation through multifaceted analysis, research, and engineering. This work acts as a catalyst to help industry bring affordable, high-performance, energy-efficient, and low-emission modes of transport and related infrastructure to market sooner. Our researchers collaborate closely with academic, government, and industry partners to design better batteries, drivetrains, and engines. They develop technologies for high-power charging, thermal management, energy storage, and power electronics. They are also reimagining fuels and combustion while creating sustainable lightweight materials. Unbiased expert guidance - backed by credible data and analysis, tools, and scientific rigor - empowers partners to make informed decisions about sustainable transportation. NREL recognizes that communities with limited mobility options face reduced access to employment opportunities, health care, and education, lowering overall quality of life. Alongside partners, NREL experts are creating transportation solutions that meet community-identified needs and increase mobility equity in historically underserved and overburdened communities. Rather than providing a one-size-fits-all solution, we take an interdisciplinary approach to mobility equity that considers the needs and challenges of diverse groups, maximizing benefits at the individual, community, and societal levels.

ADVANCED PROPULSION SYSTEMS↗

Disentangling Ferroelectric Wall Dynamics and Identification of Pinning Mechanisms via Deep Learning

Field-induced domain-wall dynamics in ferroelectric materials underpins multiple applications ranging from actuators to information technology devices and necessitates a quantitative description of the associated mechanisms including giant electromechanical couplings, controlled nonlinearities, or low coercive voltages. While the advances in dynamic piezoresponse force microscopy measurements over the last two decades have rendered visualization of polarization dynamics relatively straightforward, the associated insights into the local mechanisms have been elusive. This work explores the domain dynamics in model polycrystalline materials using a workflow combining deep-learning-based segmentation of the domain structures with nonlinear dimensionality reduction using multilayer rotationally invariant autoencoders (rVAE). The former allows unambiguous identification and classification of the ferroelectric and ferroelastic domain walls. Here, the rVAE discovers the latent representations of the domain wall geometries and their dynamics, consequently providing insight into the intrinsic mechanisms of polarization switching, that can further be compared to simple physical models. The rVAE disentangles the factors affecting the pinning efficiency of ferroelectric walls, offering insights into the correlation of ferroelastic wall distribution and ferroelectric wall pinning.

36 MATERIALS SCIENCE↗

Simulated performance of biomass gasification based combined power and refrigeration plant for community scale application

Thermal performance analysis and sizing of a biomass gasification based combined power and refrigeration plant (CPR) is reported in this study. The plant is capable of producing 100 kWe of electrical output while simultaneously producing a refrigeration effect, varying from 28-68 ton of refrigeration (TR). The topping gas turbine cycle is an indirectly heated all-air cycle. A combustor heat exchanger duplex (CHX) unit burns producer gas and transfer heat to air. This arrangement avoids complex gas cleaning requirements for the biomass-derived producer gas. The exhaust air of the topping GT is utilized to run a bottoming ammonia absorption refrigeration (AAR) cycle via a heat recovery steam generator (HRSG), steam produced in the HRSG supplying heat to the generator of the refrigeration cycle. Effects of major operating parameters like topping cycle pressure ratio (r{sub p}) and turbine inlet temperature (TIT) on the energetic performance of the plant are studied. Energetic performance of the plant is evaluated via energy efficiency, required biomass consumption and fuel energy savings ratio (FESR). The FESR calculation method is significant for indicating the savings in fuel of a combined power and process heat plant instead of separate plants for power and process heat. The study reveals that, topping cycle attains maximum power efficiency of 30%in pressure ratio range of 8-10. Up to a certain value of pressure ratio the required air flow rate through the GT unit decreases with increase in pressure ratio and then increases with further increase in pressure ratio. The capacity of refrigeration of the AAR unit initially decreases up to a certain value of topping GT cycle pressure ratio and then increases with further increase in pressure ratio. The FESR is found to be maximized at a pressure ratio of 9 (when TIT=1100°C), the maximum value being 53%. The FESR is higher for higher TIT. The heat exchanger sizing is also influenced by the topping cycle pressure ratio and GT-TIT.

09 BIOMASS FUELS↗

Scaling Irregular Applications through Data Aggregation and Software Multithreading

Bioinformatics, data analytics, semantic databases, knowledge discovery are emerging high performance application areas that exploit dynamic, linked data structures such as graphs, unbalanced trees or unstructured grids. These data structures usually are very large, requiring significantly more memory than available on single shared memory systems. Additionally, these data structures are difficult to partition on distributed memory systems. They also present poor spatial and temporal locality, thus generating unpredictable memory and network accesses. The Partitioned Global Address Space (PGAS) programming model seems suitable for these applications, because it allows using a shared memory abstraction across distributed-memory clusters. However, current PGAS languages and libraries are built to target regular remote data accesses and block transfers. Furthermore, they usually rely on the Single Program Multiple Data (SPMD) parallel control model, which is not well suited to the fine grained, dynamic and unbalanced parallelism of irregular applications. In this paper we present {\bf GMT} (Global Memory and Threading library), a custom runtime library that enables efficient execution of irregular applications on commodity clusters. GMT integrates a PGAS data substrate with simple fork/join parallelism and provides automatic load balancing on a per node basis. It implements multi-level aggregation and lightweight multithreading to maximize memory and network bandwidth with fine-grained data accesses and tolerate long data access latencies. A key innovation in the GMT runtime is its thread specialization (workers, helpers and communication threads) that realize the overall functionality. We compare our approach with other PGAS models, such as UPC running using GASNet, and hand-optimized MPI code on a set of typical large-scale irregular applications, demonstrating speedups of an order of magnitude.

Aggregation↗

PANTHER. Pattern ANalytics To support High-performance Exploitation and Reasoning.

Sandia has approached the analysis of big datasets with an integrated methodology that uses computer science, image processing, and human factors to exploit critical patterns and relationships in large datasets despite the variety and rapidity of information. The work is part of a three-year LDRD Grand Challenge called PANTHER (Pattern ANalytics To support High-performance Exploitation and Reasoning). To maximize data analysis capability, Sandia pursued scientific advances across three key technical domains: (1) geospatial-temporal feature extraction via image segmentation and classification; (2) geospatial-temporal analysis capabilities tailored to identify and process new signatures more efficiently; and (3) domain- relevant models of human perception and cognition informing the design of analytic systems. Our integrated results include advances in geographical information systems (GIS) in which we discover activity patterns in noisy, spatial-temporal datasets using geospatial-temporal semantic graphs. We employed computational geometry and machine learning to allow us to extract and predict spatial-temporal patterns and outliers from large aircraft and maritime trajectory datasets. We automatically extracted static and ephemeral features from real, noisy synthetic aperture radar imagery for ingestion into a geospatial-temporal semantic graph. We worked with analysts and investigated analytic workflows to (1) determine how experiential knowledge evolves and is deployed in high-demand, high-throughput visual search workflows, and (2) better understand visual search performance and attention. Through PANTHER, Sandia's fundamental rethinking of key aspects of geospatial data analysis permits the extraction of much richer information from large amounts of data. The project results enable analysts to examine mountains of historical and current data that would otherwise go untouched, while also gaining meaningful, measurable, and defensible insights into overlooked relationships and patterns. The capability is directly relevant to the nation's nonproliferation remote-sensing activities and has broad national security applications for military and intelligence- gathering organizations.

58 GEOSCIENCES↗

Intelligent Partitioning based Fully Parallel AC Security-Constrained Optimal Power Flow

Today’s power grid is becoming more diverse and integrated with high-level distributed energy resources and smart control technologies that is creating a new set of grid management challenges in terms of large-scale, nonlinear, and non-convex problem modeling, complex and time-consuming computation, as well as difficult uncertainty handling. This project focused on solving a challenging multi-period security-constrained generation scheduling problem, which is of great importance for maximizing the social welfare of real-time dispatch, day-ahead market, as well as weekly planning of power systems. Our developed software explored parallel optimization algorithms for complex and realistic power system models, and develop fast, efficient, and robust grid optimization solutions on the high-performance computing platform that will enable increased grid economics, flexibility, resilience, as well as energy security in the United States.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Progress Towards Microcalorimeter Gamma Spectroscopy Analysis for Nuclear Fuel Separations

The High Efficiency and Resolution Microcalorimeter Spectrometer, 400 pixels (HERMES-400) is an ultra-high resolution gamma-ray spectrometer at Idaho National Laboratory’s Materials and Fuels Complex. HERMES-400 is intended to enable precise nondestructive analysis of advanced fuel cycle materials. Key upgrades and repairs of the instrument were performed this year to improve performance. First, the fourth and final microcalorimeter detector module was installed, meeting the milestone for its integration into the system. Operation with all four modules enables higher detector efficiency than any previous microcalorimeter gamma-ray system. Second, a thinner magnetic shield was installed. This shields the sensitive microcalorimeter pixels from stray magnetic fields while allowing transmission of lower-energy gamma rays, which are important to isotopic analysis. Finally, the cryostat, which provides cooling of the modules to their operating temperature, was found to have a helium coolant leak, which was repaired. HERMES-400 is in a stable state to perform repeated, long-term gamma-ray measurements with good detector efficiency and energy resolution. In FY25, irradiated TRISO fuel and salts from advanced electrochemical recycling techniques are candidates for microcalorimeter measurements. A final upgrade of the magnetic shielding and vibration damping of the system is also planned to maximize the system performance.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

In-Space Engine (ISE-100) Development - Design Verification Test

In the past decade, NASA has formulated science mission concepts with an anticipation of landing spacecraft on the lunar surface, meteoroids, and other planets. Advancing thruster technology for spacecraft propulsion systems has been considered for maximizing science payload. Starting in 2010, development of In-Space Engine (designated as ISE-100) has been carried out. ISE-100 thruster is designed based on heritage Missile Defense Agency (MDA) technology aimed for a lightweight and efficient system in terms volume and packaging. It runs with a hypergolic bi-propellant system: MON-25 (nitrogen tetroxide, N2O4, with 25% of nitric oxide, NO) and MMH (monomethylhydrazine, CH6N2) for NASA spacecraft applications. The utilization of this propellant system will provide a propulsion system capable of operating at wide range of temperatures, from 50 C (122 F) down to -30 C (-22 F) to drastically reduce heater power. The thruster is designed to deliver 100 lb(sub f) of thrust with the capability of a pulse mode operation for a wide range of mission duty cycles (MDCs). Two thrusters were fabricated. As part of the engine development, this test campaign is dedicated for the design verification of the thruster. This presentation will report the efforts of the design verification hot-fire test program of the ISE-100 thruster in collaboration between NASA Marshall Space Flight Center (MSFC) and Aerojet Rocketdyne (AR) test teams. The hot-fire tests were conducted at Advance Mobile Propulsion Test (AMPT) facility in Durango, Colorado, from May 13 to June 10, 2016. This presentation will also provide a summary of key points from the test results.

Trinh, Huu P.↗

Integration of Structural Analysis and Manufacturing Process Planning for Global Optimization with Automated Fiber Placement

Design of mass-efficient composite structures intended for Automated Fiber Placement (AFP) requires close interaction between structural analysis and manufacturing process planning. Tools exist for each of these disciplines, but software interplay has been insufficient for rapid and efficient design iteration. Within the NASA Advanced Composites Consortium (ACC), the Design for Manufacturing (DFM) task has made significant progress towards linking these disciplines and respective software – HyperX (design), CAPP (process planning), and VCP (tool path generation). The initial focus in previous work was on data exchange between disciplines. The ability to both export and consume composite design and manufacturing data to and from each tool. This paper focuses on the effort to automate and streamline the connection between the tools listed above, with the goal of being able to automatically generate a composite AFP design that is mass-efficient and manufacturable. The optimization method being pursued is a bi-level approach, where each tool performs optimization within its discipline. The optimization in HyperX is focused on mass and laminate strength, while CAPP is focused on maximizing manufacturability. VCP is used to generate fiber paths for each design iteration. These sub-processes are wrapped with a global level optimization, driven by HyperX, used to converge the design. This paper describes the current state of this effort, which is a completed HyperX-VCP iteration loop and initial work on the HyperX-CAPP iteration loop. Additionally, example results are shown for a wind blade structure with double curvature.

Automated Fiber Placement↗

Life Cycle Assessment of Pasture-based Agrivoltaic Systems: Emissions and Energy Use of Integrated Rabbit Production

Agrivoltaic systems, which deliberately maximize the utility of a single parcel of land for both solar photovoltaic (PV) electricity production and agriculture, have been demonstrated as a viable technology that can ameliorate competing land uses and meet growing energy and food demands efficiently. The goal of this study is to assess the environmental impacts of a novel pasture-based agrivoltaic concept: co-farming rabbits and solar PV. A life cycle assessment (LCA) quantified the impacts of 1) the integrated agrivoltaic concept in comparison to conventional practices including 2) separate rabbit farming and PV production and 3) separate rabbit farming and conventional electricity production. The impact assessment methods employed to determine the environmental impacts were IPCC 2013 global warming potential 100a V1.03 and fossil energy demand V1.11. The results indicate that the pasture-based agrivoltaic system produces the least amount of greenhouse gas emissions (3.8 million kg CO₂ equivalent) and demands the least amount of fossil energy (46 million MJ) per functional unit of cumulative MWh output of electricity and cumulative kg of meat over 30 years in comparison to the two other scenarios under study. The pasture-based agrivoltaic system features a dual synergy that consequently produces 69.3% less emissions and demands 82.9% less fossil energy compared to non-integrated production. The potential for agrivoltaic systems to significantly reduce environmental impacts revealed by this LCA, which was limited to emissions and fossil energy demand, demonstrates that integrated solar and pasture-based agricultural systems are superior to conventional practices in terms of their comparatively lower emission and energy intensity. These findings provide empirical support for increased agrivoltaic system development more broadly.

14 SOLAR ENERGY↗

A Study on Reconstruction of Intra-Pin Information in Pin-Wise Reactor Analysis

Along with nodal calculation, accurate pin-wise calculations with parallel computational platforms have been recently developed as alternative tools to nodal methods to maximize core performance and minimize uncertainties used in reactor safety analysis. Therefore, it is necessary to have a method for accurately reconstructing intra-pin information comparable to whole-core transport calculation while being significantly more computationally efficient. In this study, a method to reconstruct intra-pin information (intra-pin reconstruction) is proposed. This method is based on combining the local intra-pin form function with the global homogeneous distribution. The form functions are generated from a single pin calculation with reflective boundary conditions using Serpent 2, a Monte Carlo neutron transport code. The global homogeneous flux was reconstructed using the analytical solution of the diffusion equation due to its superior accuracy and speed compared to other methods. The study here is limited to a two-dimensional two-group problem of rectangular geometric fuel pin element. The pin-wise boundary constraints (BCs) and reference intra-pin power used for reconstruction and comparison were obtained from the Serpent 2 reference calculation. From our study, it is indicated that maximum relative error (MRE) between reconstructed and reference values is within 5%. In addition, a sensitivity study was also performed to assess the feasibility of the method when the proposed method is coupled with a pin-wise calculation. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Exchange Coupling Determines Metal-Dependent Efficiency for Iron- and Cobalt-Catalyzed Photochemical CO 2 Reduction

Catalysts promoting multielectron charge delocalization offer selectivity for the CO 2 reduction reaction (CO 2 RR) over the competing hydrogen evolution reaction. Here, we show metal–ligand exchange coupling as an example of charge delocalization that can determine the efficiency for photocatalytic CO 2 RR. A comparative evaluation of iron and cobalt complexes supported by the redox-active ligand tpyPY2Me establishes that the two-electron reduction of [Co(tpyPY2Me)] 2+ ([Co] 2+ ) occurs at potentials 770 mV more negative than the [Fe(tpyPY2Me)] 2+ ([Fe] 2+ ) analogue by maximizing the exchange coupling in the latter compound. The positive shift in the reduction potential promoted by metal–ligand exchange coupling drives [Fe] 2+ to be among the most active and selective molecular catalysts for photochemical CO 2 RR reported to date, maintaining up to 99% CO product selectivity with total turnover numbers (TONs) and initial turnover frequencies exceeding 30,000 and 900 min –1 , respectively. In contrast, [Co] 2+ shows much lower CO 2 RR activity, reaching only ca. 600 TON at 83% CO product selectivity under similar conditions accompanied by rapid catalyst decomposition. Further, the spin density plots of the two-electron reduced [Co] 0 complex implicate a paramagnetic open-shell doublet ground state compared to the diamagnetic open-shell singlet ground state of reduced [Fe] 0 , rationalizing the observed negative shift in two-electron reduction potentials from the [M] 2+ species and lowered CO 2 RR efficiency for the cobalt complex relative to its iron congener. This work emphasizes the contributions of multielectron metal–ligand exchange coupling in promoting effective CO 2 RR and provides a starting point for the broader incorporation of this strategy in catalyst design.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Increasing markets and decreasing package weight for high-specific-power photovoltaics

We report that thin-film and emerging technologies in photovoltaics (PV) offer advantages for lightweight, flexible power over the rigid silicon panels that dominate the present market. One important advantage is high specific power (the power-to-weight ratio). Here we consider niche market size, price points and value propositions that can provide a path for new PV market entrants. Examining the cost-production experience curves of Si, CdTe and CIGS PV suggests that a minimum market size of US$0.2-1 billion is required to incubate a new market entrant. Several markets requiring high specific power meet this threshold. We assess the critical role of the substrate, packaging and interconnects and provide a quantitative assessment of pathways to maximize specific power. With all requisite components included, along with requirements for safety and reliability, we estimate a lower bound for a durable lightweight module at about 300-500 g m -2 . Lastly, pairing this bound with a 15%-efficiency thin-film or 35%-efficiency III-V module would yield specific powers up to 500 W kg -1 or 1,167 W kg -1 , respectively.

14 SOLAR ENERGY↗

Radiative thermal rectification using superconducting materials

Thermal rectification can be defined as an asymmetry in the heat flux when the temperature difference between two interacting thermal reservoirs is reversed. In this Letter, we present a far-field radiative thermal rectifier based on high-temperature superconducting materials with a rectification ratio up to 80%. This value is among the highest reported in literature. Two configurations are examined: a superconductor (Tl{sub 2}Ba{sub 2}CaCu{sub 2}O{sub 8}) exchanging heat with (1) a black body and (2) another superconductor, YBa{sub 2}Cu{sub 3}O{sub 7} in this case. The first configuration shows a higher maximal rectification ratio. Besides, we show that the two-superconductor rectifier exhibits different rectification regimes depending on the choice of the reference temperature, i.e., the temperature of the thermostat. Presented results might be useful for energy conversion devices, efficient cryogenic radiative insulators engineering, and thermal logical circuits’ development.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Delocalized Excitation Transfer in Open Quantum Systems with Long-Range Interactions

The interplay between coherence and system-environment interactions is at the basis of a wide range of phenomena, from quantum information processing to charge and energy transfer in molecular systems, biomolecules, and photochemical materials. In this work, we use a Frenkel exciton model with long-range interacting qubits coupled to a damped collective bosonic mode to investigate vibrationally assisted transfer processes in donor-acceptor systems featuring internal substructures analogous to light-harvesting complexes. We find that certain delocalized excitonic states maximize the transfer rate and that the entanglement is preserved during the dissipative transfer over a wide range of parameters. We investigate the reduction in transfer caused by static disorder, white noise, and finite temperature and study how transfer efficiency scales as a function of the number of dimerized monomers and the component number of each monomer, finding which excitonic states lead to optimal transfer. Finally, we provide a realistic experimental setting to realize this model in analog trapped-ion quantum simulators. Analog quantum simulation of systems comprising many and increasingly complex monomers could offer valuable insights into the design of light-harvesting materials, particularly in the nonperturbative intermediate parameter regime examined in this study, where classical simulation methods are resource intensive.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Time Series Classification for Locating Forced Oscillation Sources

Here, this article presents a machine learning based time-series classification method for using synchrophasor measurements to locate the source of forced oscillation (FO) for fast disturbance removal. First, multivariate time series (MTS) matrices are constructed by the most informative measurements selected by sequential feature selection from each power plant. Then, the Mahalanobis matrix is trained such that the Mahalanobis distance between the MTSs from the same class (i.e., with the same FO source location) are minimized and from different classes (i.e., with different FO source locations) are maximized. This allows MTSs to be classified by classifiers with class membership corresponding to the location of each FO source. To meet the runtime requirements of online matching, class templates are constructed to reduce data size and improve matching efficiency. To account for uncertainty in identifying the exact beginning of an FO event, dynamic time warping is used to align the out-of-sync MTSs. IEEE 39bus and WECC 179bus systems are used for algorithm development and validation. Simulation results demonstrate that the algorithm meets online operation runtime requirement with high accuracy using misaligned data sets.

42 ENGINEERING↗

Redundancy in Aromatic O -Demethylation and Ring-Opening Reactions in Novosphingobium aromaticivorans and Their Impact in the Metabolism of Plant-Derived Phenolics

ABSTRACT Lignin is a plant heteropolymer composed of phenolic subunits. Because of its heterogeneity and recalcitrance, the development of efficient methods for its valorization still remains an open challenge. One approach to utilize lignin is its chemical deconstruction into mixtures of monomeric phenolic compounds, followed by biological funneling into a single product. Novosphingobium aromaticivorans DSM 12444 has been previously engineered to produce 2-pyrone-4,6-dicarboxylic acid (PDC) from depolymerized lignin by simultaneously metabolizing multiple aromatics through convergent routes involving the intermediates 3-methoxygallic acid (3-MGA) and protocatechuic acid (PCA). We investigated enzymes predicted to be responsible for O -demethylation and oxidative aromatic ring opening, two critical reactions involved in the metabolism of phenolic compounds by N. aromaticivorans . The results showed the involvement of DesA in O -demethylation of syringic and vanillic acids, LigM in O- demethylation of vanillic acid and 3-MGA, and a new O- demethylase, DmtS, in the conversion of 3-MGA into gallic acid (GA). In addition, we found that LigAB was the main aromatic ring-opening dioxygenase involved in 3-MGA, PCA, and GA metabolism and that a previously uncharacterized dioxygenase, LigAB2, had high activity with GA. Our results indicate a metabolic route not previously identified in N. aromaticivorans that involves O -demethylation of 3-MGA to GA. We predict that this pathway channels ∼15% of the carbon flow from syringic acid, with the rest following ring opening of 3-MGA. The new knowledge obtained in this study allowed for the creation of an improved engineered strain for the funneling of aromatic compounds that exhibits stoichiometric conversion of syringic acid into PDC. IMPORTANCE For lignocellulosic biorefineries to effectively contribute to reduction of fossil fuel use, they need to become efficient at producing chemicals from all major components of plant biomass. Making products from lignin will require engineering microorganisms to funnel multiple phenolic compounds to the chemicals of interest, and N. aromaticivorans is a promising chassis for this technology. The ability of N. aromaticivorans to efficiently and simultaneously degrade many phenolic compounds may be linked to having functionally redundant aromatic degradation pathways and enzymes with broad substrate specificity. A detailed knowledge of aromatic degradation pathways is thus essential to identify genetic engineering targets to maximize product yields. Furthermore, knowledge of enzyme substrate specificity is critical to redirect flow of carbon to desired pathways. This study described an uncharacterized pathway in N. aromaticivorans and the enzymes that participate in this pathway, allowing the engineering of an improved strain for production of PDC from lignin.

Biotechnology & Applied Microbiology↗

Numerical methods for fractional Fokker–Planck equation with multiplicative Marcus Lévy noises

The Fokker–Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location.

Mathematics↗