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Relevant bibliographies by topics / High-dimensional PDEs

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Academic literature on the topic 'High-dimensional PDEs'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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Journal articles on the topic "High-dimensional PDEs"

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Cohen, Albert, and Ronald DeVore. "Approximation of high-dimensional parametric PDEs." Acta Numerica 24 (April 27, 2015): 1–159. http://dx.doi.org/10.1017/s0962492915000033.

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Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, and uncertainty quantification. In most of these applications, the number of parameters is large or perhaps even infinite. Thus, the development of numerical methods for these parametric problems is faced with the possible curse of dimensionality. This article is directed at (i) identifying and understanding which properties of parametric equations allow one to avoid this curse and (ii) developing and analysing effective numerical methods which fully exploit these proper

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Schwab, Christoph, and Claude Jeffrey Gittelson. "Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs." Acta Numerica 20 (April 28, 2011): 291–467. http://dx.doi.org/10.1017/s0962492911000055.

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Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series are established. Recent results on the

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Han, Jiequn, Arnulf Jentzen, and Weinan E. "Solving high-dimensional partial differential equations using deep learning." Proceedings of the National Academy of Sciences 115, no. 34 (2018): 8505–10. http://dx.doi.org/10.1073/pnas.1718942115.

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Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy funct

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Huré, Côme, Huyên Pham, and Xavier Warin. "Deep backward schemes for high-dimensional nonlinear PDEs." Mathematics of Computation 89, no. 324 (2020): 1547–79. http://dx.doi.org/10.1090/mcom/3514.

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Boelens, Arnout M. P., Daniele Venturi, and Daniel M. Tartakovsky. "Parallel tensor methods for high-dimensional linear PDEs." Journal of Computational Physics 375 (December 2018): 519–39. http://dx.doi.org/10.1016/j.jcp.2018.08.057.

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Dektor, Alec, and Daniele Venturi. "Dynamic tensor approximation of high-dimensional nonlinear PDEs." Journal of Computational Physics 437 (July 2021): 110295. http://dx.doi.org/10.1016/j.jcp.2021.110295.

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Guo, Wenjie, Jianfeng Zhang, and Jia Zhuo. "A monotone scheme for high-dimensional fully nonlinear PDEs." Annals of Applied Probability 25, no. 3 (2015): 1540–80. http://dx.doi.org/10.1214/14-aap1030.

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Eigel, Martin, Johannes Neumann, Reinhold Schneider, and Sebastian Wolf. "Non-intrusive Tensor Reconstruction for High-Dimensional Random PDEs." Computational Methods in Applied Mathematics 19, no. 1 (2019): 39–53. http://dx.doi.org/10.1515/cmam-2018-0028.

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AbstractThis paper examines a completely non-intrusive, sample-based method for the computation of functional low-rank solutions of high-dimensional parametric random PDEs, which have become an area of intensive research in Uncertainty Quantification (UQ). In order to obtain a generalized polynomial chaos representation of the approximate stochastic solution, a novel black-box rank-adapted tensor reconstruction procedure is proposed. The performance of the described approach is illustrated with several numerical examples and compared to (Quasi-)Monte Carlo sampling.

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Warin, Xavier. "Nesting Monte Carlo for high-dimensional non-linear PDEs." Monte Carlo Methods and Applications 24, no. 4 (2018): 225–47. http://dx.doi.org/10.1515/mcma-2018-2020.

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Abstract A new method based on nesting Monte Carlo is developed to solve high-dimensional semi-linear PDEs. Depending on the type of non-linearity, different schemes are proposed and theoretically studied: variance error are given and it is shown that the bias of the schemes can be controlled. The limitation of the method is that the maturity or the Lipschitz constants of the non-linearity should not be too high in order to avoid an explosion of the computational time. Many numerical results are given in high dimension for cases where analytical solutions are available or where some solutions

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Dektor, Alec, and Daniele Venturi. "Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs." Journal of Computational Physics 404 (March 2020): 109125. http://dx.doi.org/10.1016/j.jcp.2019.109125.

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Dissertations / Theses on the topic "High-dimensional PDEs"

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Wissmann, Rasmus. "Expansion methods for high-dimensional PDEs in finance." Thesis, University of Oxford, 2015. http://ora.ox.ac.uk/objects/uuid:c791d5e9-dfa3-4bd1-86ec-82e29839aea9.

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We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principa

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Rossini, Eugenio. "Deep Learning and Nonlinear PDEs in High-Dimensional Spaces." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/16859/.

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Questa tesi è incentrata sull'analisi di un algoritmo che permette di approssimare una soluzione per PDE ad alta dimensionalità. Tale algoritmo utilizza le nuove tecniche sviluppate in ambito della teoria delle Reti Neurali (Deep Learning) per affrontare un problema di difficile risoluzione. Il problema principale che affligge i modelli governati da PDE semilineari paraboliche in dimensione 100 prende il nome di "maledizione della dimensionalità". Questo non permette di utilizzare algoritmi deterministici come Galerkin o Elementi Finiti, poichè il costo computazionale cresce esponenzialmente r

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Heene, Mario [Verfasser], and Dirk [Akademischer Betreuer] Pflüger. "A massively parallel combination technique for the solution of high-dimensional PDEs / Mario Heene ; Betreuer: Dirk Pflüger." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2018. http://d-nb.info/1162497254/34.

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Grandin, Magnus. "Adaptive Solvers for High-Dimensional PDE Problems on Clusters of Multicore Processors." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-234984.

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Accurate numerical solution of time-dependent, high-dimensional partial differential equations (PDEs) usually requires efficient numerical techniques and massive-scale parallel computing. In this thesis, we implement and evaluate discretization schemes suited for PDEs of higher dimensionality, focusing on high order of accuracy and low computational cost. Spatial discretization is particularly challenging in higher dimensions. The memory requirements for uniform grids quickly grow out of reach even on large-scale parallel computers. We utilize high-order discretization schemes and implement ad

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Gustafsson, Magnus. "Towards an adaptive solver for high-dimensional PDE problems on clusters of multicore processors." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-169259.

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Accurate numerical simulation of time-dependent phenomena in many spatial dimensions is a challenging computational task apparent in a vast range of application areas, for instance quantum dynamics, financial mathematics, systems biology and plasma physics. Particularly problematic is that the number of unknowns in the governing equations (the number of grid points) grows exponentially with the number of spatial dimensions introduced, often referred to as the curse of dimensionality. This limits the range of problems that we can solve, since the computational effort and requirements on memory

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Figueroa, Leonardo E. "Deterministic simulation of multi-beaded models of dilute polymer solutions." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:4c3414ba-415a-4109-8e98-6c4fa24f9cdc.

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We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method wit

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Books on the topic "High-dimensional PDEs"

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Sabelfeld, Karl K., and Nikolai A. Simonov. Stochastic Methods for Boundary Value Problems: Numerics for High-Dimensional PDEs and Applications. De Gruyter, Inc., 2016.

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Book chapters on the topic "High-dimensional PDEs"

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Schwab, Christoph. "Methods for High-Dimensional Parametric and Stochastic Elliptic PDEs." In Encyclopedia of Applied and Computational Mathematics. Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_526.

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Schneider, Reinhold, Thorsten Rohwedder, and Örs Legeza. "Numerical Approaches for High-Dimensional PDEs for Quantum Chemistry." In Encyclopedia of Applied and Computational Mathematics. Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_245.

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Heene, Mario, Alfredo Parra Hinojosa, Hans-Joachim Bungartz, and Dirk Pflüger. "A Massively-Parallel, Fault-Tolerant Solver for High-Dimensional PDEs." In Euro-Par 2016: Parallel Processing Workshops. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58943-5_51.

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Reisinger, Christoph, and Rasmus Wissmann. "Finite Difference Methods for Medium- and High-Dimensional Derivative Pricing PDEs." In High-Performance Computing in Finance. Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781315372006-6.

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Hinojosa, Alfredo Parra, Hans-Joachim Bungartz, and Dirk Pflüger. "Scalable Algorithmic Detection of Silent Data Corruption for High-Dimensional PDEs." In Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-75426-0_5.

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Gustafsson, Magnus, and Sverker Holmgren. "An Implementation Framework for Solving High-Dimensional PDEs on Massively Parallel Computers." In Numerical Mathematics and Advanced Applications 2009. Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11795-4_44.

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Kircheis, Robert, and Stefan Körkel. "Parameter Estimation for High-Dimensional PDE Models Using a Reduced Approach." In Contributions in Mathematical and Computational Sciences. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-23321-5_5.

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"Solving One-Dimensional Partial Differential Equations." In Advances in Systems Analysis, Software Engineering, and High Performance Computing. IGI Global, 2021. http://dx.doi.org/10.4018/978-1-7998-7078-4.ch007.

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This chapter describes the pdepe command, which is used to solve spatially one-dimensional partial differential equations (PDEs). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the pdepe solver uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics are presented in the final part of the chapter. They illustrate how to solve: heat transfer PDE with temperature dependent material properties, startup velocities of the fluid flow in a pipe, Burger's PDE, and coupled FitzHugh-Nagumo PDE.

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"Solving Two-Dimensional Partial Differential Equations." In Advances in Systems Analysis, Software Engineering, and High Performance Computing. IGI Global, 2021. http://dx.doi.org/10.4018/978-1-7998-7078-4.ch008.

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This chapter describes the PDE Modeler tool, which is used to solve spatially two-dimensional partial differential equations (PDE). It begins with a description of the standard forms of PDEs and its initial and boundary conditions that the tool uses. It is shown how various PDEs and boundary conditions can be represented in standard forms. Applications to the mechanics and tribology are presented in the final part of the chapter. They illustrate the use of PDE Modeler to solve the Reynolds equation describing the hydrodynamic lubrication, to implement the mechanical stress modeler application for a plate with an elliptical hole, to solve the transient heat equation with temperature-dependent material properties, and to study vibration of a rectangular membrane.

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Chen, J., S. Yuan, and J. Li. "A PDEM-based dimension-reduction of FPK equation for high-dimensional stochastic dynamics." In Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures. CRC Press, 2014. http://dx.doi.org/10.1201/b16387-143.

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Conference papers on the topic "High-dimensional PDEs"

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Alexanderian, Alen, William Reese, Ralph C. Smith, and Meilin Yu. "Efficient Uncertainty Quantification for Biotransport in Tumors With Uncertain Material Properties." In ASME 2018 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/imece2018-86216.

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We consider modeling of single phase fluid flow in heterogeneous porous media governed by elliptic partial differential equations (PDEs) with random field coefficients. Our target application is biotransport in tumors with uncertain heterogeneous material properties. We numerically explore dimension reduction of the input parameter and model output. In the present work, the permeability field is modeled as a log-Gaussian random field, and its covariance function is specified. Uncertainties in permeability are then propagated into the pressure field through the elliptic PDE governing porous med

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Michopoulos, John G., Athanasios P. Iliopoulos, John C. Steuben, and Andrew J. Birnbaum. "On the Multiphysics Modeling of the Sliding Wear Between Deformable Heat Conducting Bodies." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-86077.

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An approach exploiting the relevant conservation laws associated with the wear due to sliding between deformable heat conducting bodies is presented in this work. The proposed methodology considers a pair of wearing objects in contact where their wear behaviors are encapsulated by semantically reduced one-dimensional, time-dependent ordinary differential equations (ODEs) as a replacement to the full mass conservation PDEs governing mass loss due to the various mechanisms present at the interface. At the same time, the conservation of energy and momentum are still expressed by the full form of

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Hajimirza, Shima. "A Dimensionality-Reduced First Order Method for Industrial Optimization: A Case Study in Renewable Energy Technologies." In ASME 2015 9th International Conference on Energy Sustainability collocated with the ASME 2015 Power Conference, the ASME 2015 13th International Conference on Fuel Cell Science, Engineering and Technology, and the ASME 2015 Nuclear Forum. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/es2015-49058.

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We propose and study a dimensionality-reduced first order method for solving complex optimization problems of high-dimensional search space. We demonstrate that the proposed method is very efficient in design problems where the computational bottleneck is mostly due to the time-consuming nature of the forward problem in contrast to the complexity of the function behavior in the search space or other computational overheads. Many industrial problems are of this nature including design problems based on back testing or simulation of an evolutionary equation or a dynamic system in time, frequency

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Nara, Y., H. Sugino, T. Arakawa, Y. Shirasaki, T. Funatsu, and S. Shoji. "High Through-Put Parallel Biomolecules Sorting Microsystem with Three Dimensional PDMS Stack." In TRANSDUCERS 2007 - 2007 International Solid-State Sensors, Actuators and Microsystems Conference. IEEE, 2007. http://dx.doi.org/10.1109/sensor.2007.4300118.

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Zhou, Xiaoyu, Hongxia Li, and Yi Huang. "Modified Probability Density Evolution Method for the Solution of Multi-Degree-of-Freedom Nonlinear Stochastic Dynamical Systems." In ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/omae2017-62382.

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A probability density evolution based exponential polynomial regression (PDEM-EPR) method to calculate the probability density function (PDF) of the high dimensional nonlinear stochastic dynamical systems is presented in this paper. Several typical examples, such as linear oscillator and Duffing oscillator are solved by PDEM-EPR method, and the results fit well with the analytical solutions. An engineering practice problem of ship nonlinear random roll in the beam waves is involved in this paper. The results obtained by PDEM-EPR is compared with those obtained by path integral method. The late

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Ren, Beibei, Paul Frihauf, Miroslav Krstic, and Robert J. Rafac. "Laser Pulse Shaping via Iterative Learning Control and High-Dimensional Extremum Seeking." In ASME 2011 Dynamic Systems and Control Conference and Bath/ASME Symposium on Fluid Power and Motion Control. ASMEDC, 2011. http://dx.doi.org/10.1115/dscc2011-6009.

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We investigate pulse shaping and optimization for a laser amplifier. Due to the complex character of the nonlinear PDE dynamics involved in the laser model, it is of interest to consider non-model based methods for pulse shaping. We determine input pulse shapes for an unknown laser dynamics model using iterative learning control (ILC) and high-dimensional extremum seeking (ES), which is a real-time optimization strategy. We utilize ILC to obtain the input pulse shape that generates a desired output pulse shape and ES to find the input pulse that maximizes the energy amplifier gain. Both single

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Raghupathy, Arun Prakash, Urmila Ghia, and Karman Ghia. "Variations in the Flow Field of a Pulse Detonation Engine for Different Operating Conditions." In ASME 2006 2nd Joint U.S.-European Fluids Engineering Summer Meeting Collocated With the 14th International Conference on Nuclear Engineering. ASMEDC, 2006. http://dx.doi.org/10.1115/fedsm2006-98129.

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Pulse Detonation Engine (PDE) is considered to be the propulsion system of future air and space vehicles because of its low cost, light weight, and high performance. Hybrid PDE is a relatively new concept where a turbine is integrated with a PDE. This hybrid system is expected to operate under fuel-rich conditions during take-off (stoichiometric), and fuel-lean (φ = 0.44) conditions during cruise. Hence, the objective of the present study is to simulate the external flow field of a stand alone PDE system and study its variation during the above mentioned operating conditions. In order to study

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Gonzalez-Domenzain, Walter, and Ashwin A. Seshia. "Rapid Prototyping of PDMS Devices With Applications to Protein Crystallization." In ASME 2007 5th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2007. http://dx.doi.org/10.1115/icnmm2007-30046.

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This paper describes a microfabrication process for constructing three-dimensional microfluidic structures in polydimethylsiloxane (PDMS). Rapid prototyping of microfluidic devices is possible starting from ink-jet printed masks and by utilising replica molding to create fluidic structures in PDMS from SU-8 and SPR-220 masters pre-patterned on a silicon or glass substrate. Multi-layer bonded and stacked alignment of up to 13 different functional polymer microfluidic layers with through-layer fluidic interconnects has been demonstrated. Pneumatically actuated valves have also been demonstrated

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Jiang, Huicong, and Hua Tan. "One Dimensional Simulation of Droplet Ejection of Drop-on-Demand Inkjet." In ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/imece2017-71190.

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In this study, we present a 1D method to predict the droplet ejection of a drop-on-demand (DoD) inkjet which includes the drop breakup, coalescence, and the meniscus movement at nozzle orifice. A simplified 1D slender-jet analysis based on the lubrication approximation is used to study the drop breakup. In this model, the free-surface (liquid-air interface) is represented by a shape function so that the full Navier-Stokes (NS) equations can be linearized into a set of simple partial differential equations (PDEs) which are solved by method of lines (MOL). The shape-preserving piecewise cubic in

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Jung, Yung J., Laila Jaber-Ansari, Xugang Xiong, et al. "Highly Organized Carbon Nanotube-PDMS Hybrid System for Multifunctional Flexible Devices." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35442.

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We will present a method to fabricate a new class of hybrid composite structures based on highly organized multiwalled carbon nanotube (MWNT) and singlewalled carbon nanotube (SWNT) network architectures and a polydimethylsiloxane (PDMS) matrix for the prototype high performance flexible systems which could be used for many daily-use applications. To build 1–3 dimensional highly organized network architectures with carbon nanotubes (both MWNT and SWNT) in macro/micro/nanoscale we used various nanotube assembly processes such as selective growth of carbon nanotubes using chemical vapor depositi

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Reports on the topic "High-dimensional PDEs"

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Zabaras, Nicolas J. Modeling and simulation of high dimensional stochastic multiscale PDE systems at the exascale. Office of Scientific and Technical Information (OSTI), 2016. http://dx.doi.org/10.2172/1331205.

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Kevrekidis, Ioannis. Modeling and Simulation of High Dimensional Stochastic Multiscale PDE Systems at the Exascale. Office of Scientific and Technical Information (OSTI), 2017. http://dx.doi.org/10.2172/1347551.

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