Artykuły w czasopismach: „Entropic optimal transport”

1

Altschuler, Jason M., Jonathan Niles-Weed i Austin J. Stromme. "Asymptotics for Semidiscrete Entropic Optimal Transport". SIAM Journal on Mathematical Analysis 54, nr 2 (14.03.2022): 1718–41. http://dx.doi.org/10.1137/21m1440165.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

2

Keriven, Nicolas. "Entropic Optimal Transport on Random Graphs". SIAM Journal on Mathematics of Data Science 5, nr 4 (29.11.2023): 1028–50. http://dx.doi.org/10.1137/22m1518281.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

3

Amari, Shun-ichi, Ryo Karakida, Masafumi Oizumi i Marco Cuturi. "Information Geometry for Regularized Optimal Transport and Barycenters of Patterns". Neural Computation 31, nr 5 (maj 2019): 827–48. http://dx.doi.org/10.1162/neco_a_01178.

Pełny tekst źródła

Streszczenie:

We propose a new divergence on the manifold of probability distributions, building on the entropic regularization of optimal transportation problems. As Cuturi ( 2013 ) showed, regularizing the optimal transport problem with an entropic term is known to bring several computational benefits. However, because of that regularization, the resulting approximation of the optimal transport cost does not define a proper distance or divergence between probability distributions. We recently tried to introduce a family of divergences connecting the Wasserstein distance and the Kullback-Leibler divergence from an information geometry point of view (see Amari, Karakida, & Oizumi, 2018 ). However, that proposal was not able to retain key intuitive aspects of the Wasserstein geometry, such as translation invariance, which plays a key role when used in the more general problem of computing optimal transport barycenters. The divergence we propose in this work is able to retain such properties and admits an intuitive interpretation.

Style APA, Harvard, Vancouver, ISO itp.

4

Rigollet, Philippe, i Jonathan Weed. "Entropic optimal transport is maximum-likelihood deconvolution". Comptes Rendus Mathematique 356, nr 11-12 (listopad 2018): 1228–35. http://dx.doi.org/10.1016/j.crma.2018.10.010.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

5

Clason, Christian, Dirk A. Lorenz, Hinrich Mahler i Benedikt Wirth. "Entropic regularization of continuous optimal transport problems". Journal of Mathematical Analysis and Applications 494, nr 1 (luty 2021): 124432. http://dx.doi.org/10.1016/j.jmaa.2020.124432.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

6

Junge, Oliver, Daniel Matthes i Bernhard Schmitzer. "Entropic transfer operators". Nonlinearity 37, nr 6 (16.04.2024): 065004. http://dx.doi.org/10.1088/1361-6544/ad247a.

Pełny tekst źródła

Streszczenie:

Abstract We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

Style APA, Harvard, Vancouver, ISO itp.

7

Amid, Ehsan, Frank Nielsen, Richard Nock i Manfred K. Warmuth. "Optimal Transport with Tempered Exponential Measures". Proceedings of the AAAI Conference on Artificial Intelligence 38, nr 10 (24.03.2024): 10838–46. http://dx.doi.org/10.1609/aaai.v38i10.28957.

Pełny tekst źródła

Streszczenie:

In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, ``a-la-Kantorovich'', which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, ``a-la-Sinkhorn-Cuturi'', which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.

Style APA, Harvard, Vancouver, ISO itp.

8

PEYRÉ, GABRIEL, LÉNAÏC CHIZAT, FRANÇOIS-XAVIER VIALARD i JUSTIN SOLOMON. "Quantum entropic regularization of matrix-valued optimal transport". European Journal of Applied Mathematics 30, nr 6 (28.09.2017): 1079–102. http://dx.doi.org/10.1017/s0956792517000274.

Pełny tekst źródła

Streszczenie:

This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This “quantum” formulation of optimal transport (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycentres within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis.

Style APA, Harvard, Vancouver, ISO itp.

9

Ito, Kaito, i Kenji Kashima. "Entropic model predictive optimal transport over dynamical systems". Automatica 152 (czerwiec 2023): 110980. http://dx.doi.org/10.1016/j.automatica.2023.110980.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

10

Lo, Chi-Jen (Roger), Mahesh K. Marina, Nishanth Sastry, Kai Xu, Saeed Fadaei i Yong Li. "Shrinking VOD Traffic via Rényi-Entropic Optimal Transport". Proceedings of the ACM on Measurement and Analysis of Computing Systems 8, nr 1 (16.02.2024): 1–34. http://dx.doi.org/10.1145/3639033.

Pełny tekst źródła

Streszczenie:

In response to the exponential surge in Internet Video on Demand (VOD) traffic, numerous research endeavors have concentrated on optimizing and enhancing infrastructure efficiency. In contrast, this paper explores whether users' demand patterns can be shaped to reduce the pressure on infrastructure. Our main idea is to design a mechanism that alters the distribution of user requests to another distribution which is much more cache-efficient, but still remains 'close enough' (in the sense of cost) to fulfil each individual user's preference. To quantify the cache footprint of VOD traffic, we propose a novel application of Rényi entropy as its proxy, capturing the 'richness' (the number of distinct videos or cache size) and the 'evenness' (the relative popularity of video accesses) of the on-demand video distribution. We then demonstrate how to decrease this metric by formulating a problem drawing on the mathematical theory of optimal transport (OT). Additionally, we establish a key equivalence theorem: minimizing Rényi entropy corresponds to maximizing soft cache hit ratio (SCHR) --- a variant of cache hit ratio allowing similarity-based video substitutions. Evaluation on a real-world, city-scale video viewing dataset reveals a remarkable 83% reduction in cache size (associated with VOD caching traffic). Crucially, in alignment with the above-mentioned equivalence theorem, our approach yields a significant uplift to SCHR, achieving close to 100%.

Style APA, Harvard, Vancouver, ISO itp.

11

Lo, Chi-Jen (Roger), Mahesh K. Marina, Nishanth Sastry, Kai Xu, Saeed Fadaei i Yong Li. "Shrinking VOD Traffic via Rényi-Entropic Optimal Transport". ACM SIGMETRICS Performance Evaluation Review 52, nr 1 (11.06.2024): 75–76. http://dx.doi.org/10.1145/3673660.3655081.

Pełny tekst źródła

Streszczenie:

In response to the exponential surge in Video on Demand (VOD) traffic, numerous research endeavors have concentrated on optimizing and enhancing infrastructure efficiency. In contrast, this paper explores whether users' demand patterns can be shaped to reduce the pressure on infrastructure. Our main idea is to design a mechanism that alters the distribution of user requests to another distribution which is much more cache-efficient, but still remains 'close enough' (in terms of cost) to fulfil individual user's preference.

Style APA, Harvard, Vancouver, ISO itp.

12

Dupuy, Arnaud, Alfred Galichon i Yifei Sun. "Estimating matching affinity matrices under low-rank constraints". Information and Inference: A Journal of the IMA 8, nr 4 (23.08.2019): 677–89. http://dx.doi.org/10.1093/imaiai/iaz015.

Pełny tekst źródła

Streszczenie:

Abstract In this paper, we address the problem of estimating transport surplus (a.k.a. matching affinity) in high-dimensional optimal transport problems. Classical optimal transport theory specifies the matching affinity and determines the optimal joint distribution. In contrast, we study the inverse problem of estimating matching affinity based on the observation of the joint distribution, using an entropic regularization of the problem. To accommodate high dimensionality of the data, we propose a novel method that incorporates a nuclear norm regularization that effectively enforces a rank constraint on the affinity matrix. The low-rank matrix estimated in this way reveals the main factors that are relevant for matching.

Style APA, Harvard, Vancouver, ISO itp.

13

Tenetov, Evgeny, Gershon Wolansky i Ron Kimmel. "Fast Entropic Regularized Optimal Transport Using Semidiscrete Cost Approximation". SIAM Journal on Scientific Computing 40, nr 5 (styczeń 2018): A3400—A3422. http://dx.doi.org/10.1137/17m1162925.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

14

Benamou, Jean-David, Wilbert L. Ijzerman i Giorgi Rukhaia. "An entropic optimal transport numerical approach to the reflector problem". Methods and Applications of Analysis 27, nr 4 (2020): 311–40. http://dx.doi.org/10.4310/maa.2020.v27.n4.a1.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

15

Carlier, Guillaume, Vincent Duval, Gabriel Peyré i Bernhard Schmitzer. "Convergence of Entropic Schemes for Optimal Transport and Gradient Flows". SIAM Journal on Mathematical Analysis 49, nr 2 (styczeń 2017): 1385–418. http://dx.doi.org/10.1137/15m1050264.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

16

Wang, Shuchan, Photios A. Stavrou i Mikael Skoglund. "Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality". Entropy 24, nr 2 (21.02.2022): 306. http://dx.doi.org/10.3390/e24020306.

Pełny tekst źródła

Streszczenie:

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies.

Style APA, Harvard, Vancouver, ISO itp.

17

Shi, Liangliang, Zhaoqi Shen i Junchi Yan. "Double-Bounded Optimal Transport for Advanced Clustering and Classification". Proceedings of the AAAI Conference on Artificial Intelligence 38, nr 13 (24.03.2024): 14982–90. http://dx.doi.org/10.1609/aaai.v38i13.29419.

Pełny tekst źródła

Streszczenie:

Optimal transport (OT) is attracting increasing attention in machine learning. It aims to transport a source distribution to a target one at minimal cost. In its vanilla form, the source and target distributions are predetermined, which contracts to the real-world case involving undetermined targets. In this paper, we propose Doubly Bounded Optimal Transport (DB-OT), which assumes that the target distribution is restricted within two boundaries instead of a fixed one, thus giving more freedom for the transport to find solutions. Based on the entropic regularization of DB-OT, three scaling-based algorithms are devised for calculating the optimal solution. We also show that our DB-OT is helpful for barycenter-based clustering, which can avoid the excessive concentration of samples in a single cluster. Then we further develop DB-OT techniques for long-tailed classification which is an emerging and open problem. We first propose a connection between OT and classification, that is, in the classification task, training involves optimizing the Inverse OT to learn the representations, while testing involves optimizing the OT for predictions. with this OT perspective, we first apply DB-OT to improve the loss, and the Balanced Softmax is shown as a special case. Then we apply DB-OT for inference in the testing process. Even with vanilla Softmax trained features, our experiments show that our method can achieve good results with our improved inference scheme in the testing stage.

Style APA, Harvard, Vancouver, ISO itp.

18

Di Marino, Simone, i Lénaïc Chizat. "A tumor growth model of Hele-Shaw type as a gradient flow". ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 103. http://dx.doi.org/10.1051/cocv/2020019.

Pełny tekst źródła

Streszczenie:

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

Style APA, Harvard, Vancouver, ISO itp.

19

Guex, Guillaume, Ilkka Kivimäki i Marco Saerens. "Randomized optimal transport on a graph: framework and new distance measures". Network Science 7, nr 1 (marzec 2019): 88–122. http://dx.doi.org/10.1017/nws.2018.29.

Pełny tekst źródła

Streszczenie:

AbstractThe recently developed bag-of-paths (BoP) framework consists in setting a Gibbs–Boltzmann distribution on all feasible paths of a graph. This probability distribution favors short paths over long ones, with a free parameter (the temperatureT) controlling the entropic level of the distribution. This formalism enables the computation of new distances or dissimilarities, interpolating between the shortest-path and the resistance distance, which have been shown to perform well in clustering and classification tasks. In this work, the bag-of-paths formalism is extended by adding two independent equality constraints fixing starting and ending nodes distributions of paths (margins).When the temperature is low, this formalism is shown to be equivalent to a relaxation of the optimal transport problem on a network where paths carry a flow between two discrete distributions on nodes. The randomization is achieved by considering free energy minimization instead of traditional cost minimization. Algorithms computing the optimal free energy solution are developed for two types of paths: hitting (or absorbing) paths and non-hitting, regular, paths and require the inversion of ann×nmatrix withnbeing the number of nodes. Interestingly, for regular paths on an undirected graph, the resulting optimal policy interpolates between the deterministic optimal transport policy (T→ 0+) and the solution to the corresponding electrical circuit (T→ ∞). Two distance measures between nodes and a dissimilarity between groups of nodes, both integrating weights on nodes, are derived from this framework.

Style APA, Harvard, Vancouver, ISO itp.

20

Huizing, Geert-Jan, Gabriel Peyré i Laura Cantini. "Optimal transport improves cell–cell similarity inference in single-cell omics data". Bioinformatics 38, nr 8 (14.02.2022): 2169–77. http://dx.doi.org/10.1093/bioinformatics/btac084.

Pełny tekst źródła

Streszczenie:

Abstract Motivation High-throughput single-cell molecular profiling is revolutionizing biology and medicine by unveiling the diversity of cell types and states contributing to development and disease. The identification and characterization of cellular heterogeneity are typically achieved through unsupervised clustering, which crucially relies on a similarity metric. Results We here propose the use of Optimal Transport (OT) as a cell–cell similarity metric for single-cell omics data. OT defines distances to compare high-dimensional data represented as probability distributions. To speed up computations and cope with the high dimensionality of single-cell data, we consider the entropic regularization of the classical OT distance. We then extensively benchmark OT against state-of-the-art metrics over 13 independent datasets, including simulated, scRNA-seq, scATAC-seq and single-cell DNA methylation data. First, we test the ability of the metrics to detect the similarity between cells belonging to the same groups (e.g. cell types, cell lines of origin). Then, we apply unsupervised clustering and test the quality of the resulting clusters. OT is found to improve cell–cell similarity inference and cell clustering in all simulated and real scRNA-seq data, as well as in scATAC-seq and single-cell DNA methylation data. Availability and implementation All our analyses are reproducible through the OT-scOmics Jupyter notebook available at https://github.com/ComputationalSystemsBiology/OT-scOmics. Supplementary information Supplementary data are available at Bioinformatics online.

Style APA, Harvard, Vancouver, ISO itp.

21

Damodaran, Bharath Bhushan, Rémi Flamary, Vivien Seguy i Nicolas Courty. "An Entropic Optimal Transport loss for learning deep neural networks under label noise in remote sensing images". Computer Vision and Image Understanding 191 (luty 2020): 102863. http://dx.doi.org/10.1016/j.cviu.2019.102863.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

22

Muratore-Ginanneschi, Paolo, i Luca Peliti. "Classical uncertainty relations and entropy production in non-equilibrium statistical mechanics". Journal of Statistical Mechanics: Theory and Experiment 2023, nr 8 (1.08.2023): 083202. http://dx.doi.org/10.1088/1742-5468/ace3b3.

Pełny tekst źródła

Streszczenie:

Abstract We analyze Fürth’s 1933 classical uncertainty relations in the modern language of stochastic differential equations. Our interest is motivated by their application to non-equilibrium classical statistical mechanics. We show that Fürth’s uncertainty relations are a property inherent in martingales within the framework of a diffusion process. This result implies a lower bound on the fluctuations in current velocities of entropic quantifiers associated with transitions in stochastic thermodynamics. In cases of particular interest, we recover a well-known inequality for optimal mass transport relating the mean kinetic energy of the current velocity and the squared quadratic Wasserstein distance between the probability distributions of the entropy. We take advantage in particular of an unpublished suggestion by Krzysztof Gawȩdzki to derive a lower bound to the entropy production by a transition described by a Langevin–Kramers process in terms of the squared quadratic Wasserstein distance between the initial and final states of the transition. Finally, we illustrate how Fürth’s relations admit a straightforward extension to piecewise deterministic processes. We show that the results presented in this paper pertain to the characteristics exhibited by general Markov processes.

Style APA, Harvard, Vancouver, ISO itp.

23

Vashisht, Sagar, Dibakar Rakshit, Satyam Panchal, Michael Fowler i Roydon Fraser. "Quantifying the Effects of Temperature and Depth of Discharge on Li-Ion Battery Heat Generation: An Assessment of Resistance Models for Accurate Thermal Behavior Prediction". ECS Meeting Abstracts MA2023-02, nr 3 (22.12.2023): 445. http://dx.doi.org/10.1149/ma2023-023445mtgabs.

Pełny tekst źródła

Streszczenie:

Li-ion batteries (LiBs) are widely adopted in electric vehicles (EVs) owing to their superior properties, such as high energy density, low discharge rate, long lifespan, and lightweight construction. Since the battery pack is the sole energy source for an EV, its performance is critical for optimal vehicle operation. However, the battery's calendar life, cycle life, and overall performance are significantly affected by temperature variations. The Li-ion batteries used in EVs may encounter challenging working conditions, leading to thermal problems such as significant capacity and power loss. In contrast, thermal runaways can occur at temperatures above a specific threshold, leading to severe health deterioration and sometimes catastrophic safety hazards such as fires and explosions. As the temperature significantly impacts Li-ion batteries, a battery thermal management system that can efficiently dissipate heat is crucial to ensure the battery's optimal performance and longevity. Hence, it is crucial to develop accurate algorithms for battery thermal management systems to precisely and dynamically estimate the temperature dynamics of the batteries integrated within the battery pack. While experimental data can be used to estimate battery temperatures, the dynamic and diverse operating conditions of electric vehicles (EVs) present a significant challenge. Therefore, accurately predicting thermal response within batteries is critical. Various thermal models have been developed to predict the thermal behavior of batteries and quantify the amount of heat generated. The simplified thermal model only considers joule heating and reversible entropic heating. However, more accurate physics-based models consider reversible heat caused by the side reactions, heat generated by mass transport loss, and even mixing-induced heat. The amount of heat generated inside a Li-ion battery is determined by its equivalent internal resistance, open circuit voltage, and entropy change, which are in turn influenced by temperature and depth of discharge (DoD). To the best of the authors' knowledge, previous research on the heat generation of Li-ion batteries has been limited in some respects. Specifically, there has been little investigation into the combined impact of temperature and depth of discharge (DoD) across a wide temperature range. Most studies have been conducted under ambient temperature conditions, and only a few have focused on high temperatures within a narrow range with low discharge rates. Thus, this study aims to address the research gap regarding the impact of temperature and depth of discharge (DoD) on heat generation in Li-ion batteries by analyzing these parameters using a transient battery thermal model. The research intends to improve the accuracy and precision of battery thermal behavior prediction, which has broad implications for battery-powered applications. This study aims to evaluate the impact of different resistance models on heat generation in Li-ion batteries, explicitly comparing a constant resistance model with a model that considers resistance as a function of temperature and depth of discharge (DoD). Investigating the interdependent impact of battery temperature and DoD on heat generation is crucial to create an accurate battery thermal model with high fidelity. The current study uses a two-dimensional battery thermal model to comprehensively analyze thermal behavior of a LiFePO4-20Ah Li-ion pouch cell. In this research study, heat generation in a Li-ion battery is evaluated by estimating the internal resistance and entropic change obtained from experimentation. The energy equation is then solved using the finite difference method in MATLAB to obtain the transient thermal response of the battery. The developed transient electrothermal model is validated against experimental data under varying C rates to assess the accuracy and precision of the proposed model. The simulation results show that the thermal response obtained considering the effect of temperature and DoD on heat generation shows more accurate results than the constant resistance values. The thermal behavior of a LiFePO4 pouch cell, considering constant values for heat generation, has a maximum relative error of roughly 19.99% compared to experimental data at a 4C discharge rate. While this maximum relative error was reduced to 6.29% when considering the effect of temperature and DoD on heat generation. In the constant resistance model, more significant errors can be attributed to the fact that the resistance of a Li-ion battery varies with the depth of discharge (DoD). While the initial discharge phase of the battery exhibits minimal changes in resistance values, a substantial increase in resistance occurs during the final stages of discharge. This contrasts with the actual behavior of Li-ion batteries, which demonstrate significant variations in resistance values throughout the discharge process. Thus, coupling the effects of DoD and temperature on heat generation is necessary to accurately predict the thermal behavior of Li-ion battery. Figure 1

Style APA, Harvard, Vancouver, ISO itp.

24

Wu, Liming. "Entropical Optimal Transport, Schrödinger’s System and Algorithms". Acta Mathematica Scientia 41, nr 6 (listopad 2021): 2183–97. http://dx.doi.org/10.1007/s10473-021-0623-1.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

25

Bao, Han, i Shinsaku Sakaue. "Sparse Regularized Optimal Transport with Deformed q-Entropy". Entropy 24, nr 11 (10.11.2022): 1634. http://dx.doi.org/10.3390/e24111634.

Pełny tekst źródła

Streszczenie:

Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy (q=1) and the squared 2-norm (q=0), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed.

Style APA, Harvard, Vancouver, ISO itp.

26

Bonafini, Mauro, i Bernhard Schmitzer. "Domain decomposition for entropy regularized optimal transport". Numerische Mathematik 149, nr 4 (19.11.2021): 819–70. http://dx.doi.org/10.1007/s00211-021-01245-0.

Pełny tekst źródła

Streszczenie:

AbstractWe study Benamou’s domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback–Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient implementation, such as adaptive sparsity, a coarse-to-fine scheme and parallelization, paving the way to numerically solving large-scale optimal transport problems. We demonstrate efficient numerical performance for computing the Wasserstein-2 distance between 2D images and observe that, even without parallelization, domain decomposition compares favorably to applying a single efficient implementation of the Sinkhorn algorithm in terms of runtime, memory and solution quality.

Style APA, Harvard, Vancouver, ISO itp.

27

Tong, Qijun, i Kei Kobayashi. "Entropy-Regularized Optimal Transport on Multivariate Normal and q-normal Distributions". Entropy 23, nr 3 (3.03.2021): 302. http://dx.doi.org/10.3390/e23030302.

Pełny tekst źródła

Streszczenie:

The distance and divergence of the probability measures play a central role in statistics, machine learning, and many other related fields. The Wasserstein distance has received much attention in recent years because of its distinctions from other distances or divergences. Although computing the Wasserstein distance is costly, entropy-regularized optimal transport was proposed to computationally efficiently approximate the Wasserstein distance. The purpose of this study is to understand the theoretical aspect of entropy-regularized optimal transport. In this paper, we focus on entropy-regularized optimal transport on multivariate normal distributions and q-normal distributions. We obtain the explicit form of the entropy-regularized optimal transport cost on multivariate normal and q-normal distributions; this provides a perspective to understand the effect of entropy regularization, which was previously known only experimentally. Furthermore, we obtain the entropy-regularized Kantorovich estimator for the probability measure that satisfies certain conditions. We also demonstrate how the Wasserstein distance, optimal coupling, geometric structure, and statistical efficiency are affected by entropy regularization in some experiments. In particular, our results about the explicit form of the optimal coupling of the Tsallis entropy-regularized optimal transport on multivariate q-normal distributions and the entropy-regularized Kantorovich estimator are novel and will become the first step towards the understanding of a more general setting.

Style APA, Harvard, Vancouver, ISO itp.

28

Venerus, David C., David Nieto Simavilla i Jay D. Schieber. "THERMAL TRANSPORT IN CROSS-LINKED ELASTOMERS SUBJECTED TO ELONGATIONAL DEFORMATIONS". Rubber Chemistry and Technology 92, nr 4 (1.10.2019): 639–52. http://dx.doi.org/10.5254/rct.19.80382.

Pełny tekst źródła

Streszczenie:

ABSTRACT Investigations on thermal transport in cross-linked elastomers subjected to elongational deformations are reviewed and discussed. The focus is on experimental research, in which the deformation-induced anisotropy of the thermal conductivity tensor in several common elastomeric materials is measured using novel optical techniques developed in our laboratory. These sensitive and noninvasive techniques allow for the reliable measurement of thermal conductivity (diffusivity) tensor components on samples in a deformed state. When combined with measurements of the stress in deformed samples, we are able to examine the validity of the stress–thermal rule, which predicts a linear relationship between the thermal conductivity and stress tensor in deformed polymeric materials. These results are used to shed light on possible underlying mechanisms for anisotropic thermal transport in elastomers. We also present results from a novel experimental technique that show evidence of a deformation dependence of the heat capacity, which implies that, in addition to the usual entropic contribution, there is an energetic contribution to the stress in deformed elastomers.

Style APA, Harvard, Vancouver, ISO itp.

29

Santambrogio, Filippo. "Dealing with moment measures via entropy and optimal transport". Journal of Functional Analysis 271, nr 2 (lipiec 2016): 418–36. http://dx.doi.org/10.1016/j.jfa.2016.04.009.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

30

Gentil, Ivan, Christian Léonard i Luigia Ripani. "About the analogy between optimal transport and minimal entropy". Annales de la faculté des sciences de Toulouse Mathématiques 26, nr 3 (2017): 569–600. http://dx.doi.org/10.5802/afst.1546.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

31

Dolbeault, Jean, i Xingyu Li. "φ-Entropies: convexity, coercivity and hypocoercivity for Fokker–Planck and kinetic Fokker–Planck equations". Mathematical Models and Methods in Applied Sciences 28, nr 13 (6.12.2018): 2637–66. http://dx.doi.org/10.1142/s0218202518500574.

Pełny tekst źródła

Streszczenie:

This paper is devoted to [Formula: see text]-entropies applied to Fokker–Planck and kinetic Fokker–Planck equations in the whole space, with confinement. The so-called [Formula: see text]-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and [Formula: see text] estimates. We review some of their properties in the case of diffusion equations of Fokker–Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker–Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operator plays an essential role in the relaxation process. Here we adopt the [Formula: see text] point of view and establish a sharp decay rate. Rather than giving general but quantitatively vague estimates, our goal here is to consider simple cases, benchmark available methods and obtain sharp estimates on a key example. Some [Formula: see text]-entropies give rise to improved entropy–entropy production inequalities and, as a consequence, to faster decay rates for entropy estimates of solutions to non-degenerate diffusion equations. We prove that faster entropy decay also holds at kinetic level away from equilibrium and that optimal decay rates are achieved only in asymptotic regimes.

Style APA, Harvard, Vancouver, ISO itp.

32

Mihelich, M., D. Faranda, B. Dubrulle i D. Paillard. "Statistical optimization for passive scalar transport: maximum entropy production versus maximum Kolmogorov–Sinai entropy". Nonlinear Processes in Geophysics 22, nr 2 (25.03.2015): 187–96. http://dx.doi.org/10.5194/npg-22-187-2015.

Pełny tekst źródła

Streszczenie:

Abstract. We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov–Sinai entropy for a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov–Sinai entropy, seen as functions of a parameter f connected to the jump probability, admit a unique maximum denoted fmaxEP and fmaxKS. The behaviour of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this paper is that fmaxEP and fmaxKS have the same Taylor expansion at first order in the deviation from equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP(N) tends towards a non-zero value, while fmaxKS(N) tends to 0 when N goes to infinity. For values of N typical of those adopted by Paltridge and climatologists working on maximum entropy production (N ≈ 10–100), we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N* such that fmaxEP and fmaxKS coincide, at least up to a second-order parameter proportional to the non-equilibrium fluxes imposed to the boundaries. We find that the optimal resolution N* depends on the non-equilibrium fluxes, so that deeper convection should be represented on finer grids. This result points to the inadequacy of using a single grid for representing convection in climate and weather models. Moreover, the application of this principle to passive scalar transport parametrization is therefore expected to provide both the value of the optimal flux, and of the optimal number of degrees of freedom (resolution) to describe the system.

Style APA, Harvard, Vancouver, ISO itp.

33

Mihelich, M., D. Faranda, B. Dubrulle i D. Paillard. "Statistical optimization for passive scalar transport: maximum entropy production vs. maximum Kolmogorov–Sinay entropy". Nonlinear Processes in Geophysics Discussions 1, nr 2 (18.11.2014): 1691–713. http://dx.doi.org/10.5194/npgd-1-1691-2014.

Pełny tekst źródła

Streszczenie:

Abstract. We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov–Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov–Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at first order in the deviation of equilibrium. We find that fmaxEP hardly depends on N whereas fmaxKS depends strongly on N. In particular, for a fixed difference of potential between the reservoirs, fmaxEP(N) tends towards a non-zero value, while fmaxKS(N) tends to 0 when N goes to infinity. For values of N typical of that adopted by Paltridge and climatologists (N ≈ 10 ~ 100), we show that fmaxEP and fmaxKS coincide even far from equilibrium. Finally, we show that one can find an optimal resolution N* such that fmaxEP and fmaxKS coincide, at least up to a second order parameter proportional to the non-equilibrium fluxes imposed to the boundaries. We find that the optimal resolution N* depends on the non equilibrium fluxes, so that deeper convection should be represented on finer grids. This result points to the inadequacy of using a single grid for representing convection in climate and weather models. Moreover, the application of this principle to passive scalar transport parametrization is therefore expected to provide both the value of the optimal flux, and of the optimal number of degrees of freedom (resolution) to describe the system.

Style APA, Harvard, Vancouver, ISO itp.

34

Bazaluk, Oleg, Sergiy Kotenko i Vitalii Nitsenko. "Entropy as an Objective Function of Optimization Multimodal Transportations". Entropy 23, nr 8 (24.07.2021): 946. http://dx.doi.org/10.3390/e23080946.

Pełny tekst źródła

Streszczenie:

This article considers the use of the entropy method in the optimization and forecasting of multimodal transport under conditions of risks that can be determined simultaneously by deterministic, stochastic and fuzzy quantities. This will allow to change the route of transportation in real time in an optimal way with an unacceptable increase in the risk at one of its next stages and predict the redistribution of the load of transport nodes. The aim of this study is to develop a mathematical model for the optimal choice of an alternative route, the best for one or more objective functions in real time. In addition, it is proposed to use this mathematical model to estimate the dynamic change in turnover through intermediate transport nodes, forecasting their loading over time under different conditions that also include long-term risks which are significant in magnitude. To substantiate the feasibility of the proposed mathematical model, the analysis and forecast of cargo turnover through the seaports of Ukraine are presented, taking into account and analysing the existing risks.

Style APA, Harvard, Vancouver, ISO itp.

35

Islas, Carlos, Pablo Padilla i Marco Antonio Prado. "Information Processing in the Brain as Optimal Entropy Transport: A Theoretical Approach". Entropy 22, nr 11 (29.10.2020): 1231. http://dx.doi.org/10.3390/e22111231.

Pełny tekst źródła

Streszczenie:

We consider brain activity from an information theoretic perspective. We analyze the information processing in the brain, considering the optimality of Shannon entropy transport using the Monge–Kantorovich framework. It is proposed that some of these processes satisfy an optimal transport of informational entropy condition. This optimality condition allows us to derive an equation of the Monge–Ampère type for the information flow that accounts for the branching structure of neurons via the linearization of this equation. Based on this fact, we discuss a version of Murray’s law in this context.

Style APA, Harvard, Vancouver, ISO itp.

36

Datta, Nilanjana, i Cambyse Rouzé. "Relating Relative Entropy, Optimal Transport and Fisher Information: A Quantum HWI Inequality". Annales Henri Poincaré 21, nr 7 (5.02.2020): 2115–50. http://dx.doi.org/10.1007/s00023-020-00891-8.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

37

Bhatia, Rajendra, Tanvi Jain i Yongdo Lim. "Strong convexity of sandwiched entropies and related optimization problems". Reviews in Mathematical Physics 30, nr 09 (25.09.2018): 1850014. http://dx.doi.org/10.1142/s0129055x18500149.

Pełny tekst źródła

Streszczenie:

We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrized version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimization problems for these functions. The case of the classical fidelity is of special interest for the multimarginal optimal transport problem (the [Formula: see text]-coupling problem) for Gaussian measures.

Style APA, Harvard, Vancouver, ISO itp.

38

Liero, Matthias, Alexander Mielke i Giuseppe Savaré. "Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures". Inventiones mathematicae 211, nr 3 (14.12.2017): 969–1117. http://dx.doi.org/10.1007/s00222-017-0759-8.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

39

Bigot, Jérémie, Elsa Cazelles i Nicolas Papadakis. "Central limit theorems for entropy-regularized optimal transport on finite spaces and statistical applications". Electronic Journal of Statistics 13, nr 2 (2019): 5120–50. http://dx.doi.org/10.1214/19-ejs1637.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

40

Li, Haifeng, Jun Liu, Li Cui, Haiyang Huang i Xue-Cheng Tai. "Volume preserving image segmentation with entropy regularized optimal transport and its applications in deep learning". Journal of Visual Communication and Image Representation 71 (sierpień 2020): 102845. http://dx.doi.org/10.1016/j.jvcir.2020.102845.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

41

Feng, Qi, i Wuchen Li. "Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold". Entropy 25, nr 5 (11.05.2023): 786. http://dx.doi.org/10.3390/e25050786.

Pełny tekst źródła

Streszczenie:

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.

Style APA, Harvard, Vancouver, ISO itp.

42

Christen, Thomas, i Frank Kassubek. "Entropy production moment closures and effective transport coefficients". Journal of Physics D: Applied Physics 47, nr 36 (21.08.2014): 363001. http://dx.doi.org/10.1088/0022-3727/47/36/363001.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

43

Hobbs, Bruce E., i Alison Ord. "The mechanics of granitoid systems and maximum entropy production rates". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, nr 1910 (13.01.2010): 53–93. http://dx.doi.org/10.1098/rsta.2009.0202.

Pełny tekst źródła

Streszczenie:

A model for the formation of granitoid systems is developed involving melt production spatially below a rising isotherm that defines melt initiation. Production of the melt volumes necessary to form granitoid complexes within 10 4 –10 7 years demands control of the isotherm velocity by melt advection. This velocity is one control on the melt flux generated spatially just above the melt isotherm, which is the control valve for the behaviour of the complete granitoid system. Melt transport occurs in conduits initiated as sheets or tubes comprising melt inclusions arising from Gurson–Tvergaard constitutive behaviour. Such conduits appear as leucosomes parallel to lineations and foliations, and ductile and brittle dykes. The melt flux generated at the melt isotherm controls the position of the melt solidus isotherm and hence the physical height of the Transport/Emplacement Zone. A conduit width-selection process, driven by changes in melt viscosity and constitutive behaviour, operates within the Transport Zone to progressively increase the width of apertures upwards. Melt can also be driven horizontally by gradients in topography; these horizontal fluxes can be similar in magnitude to vertical fluxes. Fluxes induced by deformation can compete with both buoyancy and topographic-driven flow over all length scales and results locally in transient ‘ponds’ of melt. Pluton emplacement is controlled by the transition in constitutive behaviour of the melt/magma from elastic–viscous at high temperatures to elastic–plastic–viscous approaching the melt solidus enabling finite thickness plutons to develop. The system involves coupled feedback processes that grow at the expense of heat supplied to the system and compete with melt advection. The result is that limits are placed on the size and time scale of the system. Optimal characteristics of the system coincide with a state of maximum entropy production rate.

Style APA, Harvard, Vancouver, ISO itp.

44

Le, Xuan Hoang Khoa, Hakan F. Oztop, Fatih Selimefendigil i Mikhail A. Sheremet. "Entropy Analysis of the Thermal Convection of Nanosuspension within a Chamber with a Heat-Conducting Solid Fin". Entropy 24, nr 4 (7.04.2022): 523. http://dx.doi.org/10.3390/e24040523.

Pełny tekst źródła

Streszczenie:

Heat transport augmentation in closed chambers can be achieved using nanofluids and extended heat transfer surfaces. This research is devoted to the computational analysis of natural convection energy transport and entropy emission within a closed region, with isothermal vertical borders and a heat-conducting solid fin placed on the hot border. Horizontal walls were assumed to be adiabatic. Control relations written using non-primitive variables with experimentally based correlations for nanofluid properties were computed by the finite difference technique. The impacts of the fin size, fin position, and nanoadditive concentration on energy transfer performance and entropy production were studied. It was found that location of the long fin near the bottom wall allowed for the intensification of convective heat transfer within the chamber. Moreover, this position was characterized by high entropy generation. Therefore, the minimization of the entropy generation can define the optimal location of the heat-conducting fin using the obtained results. An addition of nanoparticles reduced the heat transfer strength and minimized the entropy generation.

Style APA, Harvard, Vancouver, ISO itp.

45

Huebener, R. P., i H. C. Ri. "Vortex transport entropy in cuprate superconductors and Boltzmann constant". Physica C: Superconductivity and its Applications 591 (grudzień 2021): 1353975. http://dx.doi.org/10.1016/j.physc.2021.1353975.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

46

Cheng, Jiaxin, Yue Wu, Ayush Jaiswal, Xu Zhang, Pradeep Natarajan i Prem Natarajan. "User-Controllable Arbitrary Style Transfer via Entropy Regularization". Proceedings of the AAAI Conference on Artificial Intelligence 37, nr 1 (26.06.2023): 433–41. http://dx.doi.org/10.1609/aaai.v37i1.25117.

Pełny tekst źródła

Streszczenie:

Ensuring the overall end-user experience is a challenging task in arbitrary style transfer (AST) due to the subjective nature of style transfer quality. A good practice is to provide users many instead of one AST result. However, existing approaches require to run multiple AST models or inference a diversified AST (DAST) solution multiple times, and thus they are either slow in speed or limited in diversity. In this paper, we propose a novel solution ensuring both efficiency and diversity for generating multiple user-controllable AST results by systematically modulating AST behavior at run-time. We begin with reformulating three prominent AST methods into a unified assign-and-mix problem and discover that the entropies of their assignment matrices exhibit a large variance. We then solve the unified problem in an optimal transport framework using the Sinkhorn-Knopp algorithm with a user input ε to control the said entropy and thus modulate stylization. Empirical results demonstrate the superiority of the proposed solution, with speed and stylization quality comparable to or better than existing AST and significantly more diverse than previous DAST works. Code is available at https://github.com/cplusx/eps-Assign-and-Mix.

Style APA, Harvard, Vancouver, ISO itp.

47

Zheng, Yunpeng, Mingchu Zou, Wenyu Zhang, Di Yi, Jinle Lan, Ce-Wen Nan i Yuan-Hua Lin. "Electrical and thermal transport behaviours of high-entropy perovskite thermoelectric oxides". Journal of Advanced Ceramics 10, nr 2 (29.01.2021): 377–84. http://dx.doi.org/10.1007/s40145-021-0462-5.

Pełny tekst źródła

Streszczenie:

AbstractOxide-based ceramics could be promising thermoelectric materials because of their thermal and chemical stability at high temperature. However, their mediocre electrical conductivity or high thermal conductivity is still a challenge for the use in commercial devices. Here, we report significantly suppressed thermal conductivity in SrTiO3-based thermoelectric ceramics via high-entropy strategy for the first time, and optimized electrical conductivity by defect engineering. In high-entropy (Ca0.2Sr0.2Ba0.2Pb0.2La0.2)TiO3 bulks, the minimum thermal conductivity can be 1.17 W/(m·K) at 923 K, which should be ascribed to the large lattice distortion and the huge mass fluctuation effect. The power factor can reach about 295 μW/(m·K2) by inducing oxygen vacancies. Finally, the ZT value of 0.2 can be realized at 873 K in this bulk sample. This approach proposed a new concept of high entropy into thermoelectric oxides, which could be generalized for designing high-performance thermoelectric oxides with low thermal conductivity.

Style APA, Harvard, Vancouver, ISO itp.

48

Dechant, Andreas. "Minimum entropy production, detailed balance and Wasserstein distance for continuous-time Markov processes". Journal of Physics A: Mathematical and Theoretical 55, nr 9 (3.02.2022): 094001. http://dx.doi.org/10.1088/1751-8121/ac4ac0.

Pełny tekst źródła

Streszczenie:

Abstract We investigate the problem of minimizing the entropy production for a physical process that can be described in terms of a Markov jump dynamics. We show that, without any further constraints, a given time-evolution may be realized at arbitrarily small entropy production, yet at the expense of diverging activity. For a fixed activity, we find that the dynamics that minimizes the entropy production is driven by conservative forces. The value of the minimum entropy production is expressed in terms of the graph-distance based Wasserstein distance between the initial and final configuration. This yields a new kind of speed limit relating dissipation, the average number of transitions and the Wasserstein distance. It also allows us to formulate the optimal transport problem on a graph via continuous-time interpolating dynamics, in complete analogy to the continuous space setting. We demonstrate our findings for simple state networks, a time-dependent pump and for spin flips in the Ising model.

Style APA, Harvard, Vancouver, ISO itp.

49

Muscato, O., i V. Romano. "Simulation of Submicron Silicon Diodes with a Non-Parabolic Hydrodynamical Model Based on the Maximum Entropy Principle". VLSI Design 13, nr 1-4 (1.01.2001): 273–79. http://dx.doi.org/10.1155/2001/52981.

Pełny tekst źródła

Streszczenie:

A hydrodynamical model for electron transport in silicon semiconductors, free of any fitting parameters, has been formulated in [1,2] on the basis of the maximum entropy principle, by considering the energy band described by the Kane dispersion relation and by including electron-non polar optical phonon and electron-acoustic phonon scattering.In [3] the validity of this model has been checked in the bulk case. Here the consistence is investigated by comparing with Monte Carlo data the results of the simulation of a submicron n+–n–n+ silicon diode for different length of the channel, bias voltage and doping profile.The results show that the model is sufficiently accurate for CAD purposes.

Style APA, Harvard, Vancouver, ISO itp.

50

Bourgade, J. P., P. Degond, N. Mauser i C. Ringhofer. "Quantum corrections to semiclassical transport in nanoscale devices using entropy principles". Journal of Computational Electronics 6, nr 1-3 (9.12.2006): 117–20. http://dx.doi.org/10.1007/s10825-006-0062-1.

Pełny tekst źródła

Style APA, Harvard, Vancouver, ISO itp.

Artykuły w czasopismach na temat „Entropic optimal transport”

Utwórz poprawne odniesienie w stylach APA, MLA, Chicago, Harvard i wielu innych