scholarly journals Entropy-regularized 2-Wasserstein distance between Gaussian measures

2021 ◽  
Author(s):  
Anton Mallasto ◽  
Augusto Gerolin ◽  
Hà Quang Minh
Keyword(s):  
Fixed Point ◽  
Closed Form ◽  
Numerical Study ◽  
Wasserstein Distance ◽  
Iteration Algorithm ◽  
Probability Measures ◽  
Gaussian Distributions ◽  
Closed Form Solutions ◽  
Special Cases

AbstractGaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.

scholarly journals The nested Sinkhorn divergence to learn the nested distance

2021 ◽  
Author(s):  
Alois Pichler ◽  
Michael Weinhardt
Keyword(s):  
Stochastic Process ◽  
Fixed Point ◽  
Stochastic Processes ◽  
Numerical Experiments ◽  
Wasserstein Distance ◽  
Iteration Algorithm ◽  
Computational Advantage ◽  
The Difference ◽  
Nested Distance

AbstractThe nested distance builds on the Wasserstein distance to quantify the difference of stochastic processes, including also the evolution of information modelled by filtrations. The Sinkhorn divergence is a relaxation of the Wasserstein distance, which can be computed considerably faster. For this reason we employ the Sinkhorn divergence and take advantage of the related (fixed point) iteration algorithm. Furthermore, we investigate the transition of the entropy throughout the stages of the stochastic process and provide an entropy-regularized nested distance formulation, including a characterization of its dual. Numerical experiments affirm the computational advantage and supremacy.

scholarly journals Optimal consumption of multiple goods in incomplete markets

2018 ◽  
Vol 55 (3) ◽  
pp. 810-822
Author(s):  
Oleksii Mostovyi
Keyword(s):  
Closed Form ◽  
Incomplete Markets ◽  
Existence And Uniqueness ◽  
Dual Problem ◽  
Optimal Consumption ◽  
Closed Form Solutions ◽  
Uniqueness Of The Solution ◽  
Incomplete Models

Abstract We consider the problem of optimal consumption of multiple goods in incomplete semimartingale markets. We formulate the dual problem and identify conditions that allow for the existence and uniqueness of the solution, and provide a characterization of the optimal consumption strategy in terms of the dual optimizer. We illustrate our results with examples in both complete and incomplete models. In particular, we construct closed-form solutions in some incomplete models.

Structure Design, Kinematics and Grasp Constraint of a Metamorphic Robotic Hand for Meat Deboning Operation

2013 ◽  
Author(s):  
Guowu Wei ◽  
Vahid Aminzadeh ◽  
Evangelos Emmanouil ◽  
Jian S. Dai
Keyword(s):  
Closed Form ◽  
Structure Design ◽  
Robotic Hand ◽  
Meat Industry ◽  
Sensor Systems ◽  
Level Control ◽  
Closed Form Solutions ◽  
Low Level ◽  
Structure And Design

A four-fingered metamorphic robotic hand with a reconfigurable palm is presented in this paper with the application in deboning operation of meat industry. This robotic hand has a reconfigurable palm that generates changeable topology and augments dexterity and versatility for the hand. Mechanical structure and design of the robotic hand are presented and based on mechanism decomposition, kinematics of the metamorphic hand is investigated with closed-form solutions leading to the workspace characterization of the robotic hand. Based on the kinematics of the four-fingered metamorphic hand, utilizing product-of-exponentials formula, grasp map and grasp constraint of the hand are then formulated revealing the grasp robustness and manipulability performed by the metamorphic hand. A prototype of the four-fingered metamorphic hand is consequently fabricated and integrated with low level control and sensor systems leading to a scenario of applying the hand in the field of meat industry for deboning operation.

Closed Form Solutions For Mixed Convection With Magnetohydrodynamic Effect in a Vertical Porous Annulus Surrounding an Electric Cable

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
A. Barletta ◽  
E. Magyari ◽  
S. Lazzari ◽  
I. Pop
Keyword(s):  
Mixed Convection ◽  
Closed Form ◽  
Heat Generation ◽  
Internal Heat ◽  
Boussinesq Approximation ◽  
Closed Form Solutions ◽  
Electric Cable ◽  
Special Cases ◽  
Magnetohydrodynamic Effect ◽  
The Magnetic Field

Mixed convection Darcy flow in a vertical porous annulus around a straight electric cable is investigated. It is assumed that the flow is fully developed and parallel. Moreover, the Boussinesq approximation is used. The magnetic field with a steady electric current in the cable is radially varying according to the Biot–Savart law. Two flow regimes are investigated. The first is mixed convection with negligible effects of internal heat generation due to Joule heating and viscous dissipation. The second is forced convection with important effects of heat generation. In these two special cases, closed form expressions of the velocity profile and of the temperature profile, as well as of the flow rate and the Nusselt number, are obtained. The main features of these solutions are discussed.

Stresses in Multilayered Ceramics Subjected to Biaxial Flexure Tests

2008 ◽  
Vol 606 ◽  
pp. 79-92 ◽  
Author(s):  
C.H. Hsueh
Keyword(s):  
Solid Oxide Fuel Cells ◽  
Closed Form ◽  
Analytical Description ◽  
Test Methods ◽  
Standard Test ◽  
Fracture Load ◽  
Closed Form Solutions ◽  
American Society ◽  
Biaxial Flexure

Although standard test methods for biaxial strength measurements of ceramics have been established and the corresponding formulas for relating the biaxial strength to the fracture load have been approved by American Society for Testing and Materials (ASTM) and International Organization for Standardization, respectively, they are limited to the case of monolayered discs. Despite the increasing applications of multilayered ceramics, characterization of their strengths using biaxial flexure tests has been difficult because the analytical description of the relation between the strength and the fracture load for multilayers subjected to biaxial flexure tests is unavailable until recently. Using ring-on-ring tests as an example, the closed-form solutions for stresses in (i) monolayered discs based on ASTM formulas, (ii) bilayered discs based on Roark’s formulas, and (iii) multilayered discs based on Hsueh et al.’s formulas are reviewed in the present study. Finite element results for ring-on-rings tests performed on (i) zirconia monolayered discs, (ii) dental crown materials of porcelain/zirconia bilayered discs, and (iii) solid oxide fuel cells trilayered discs are also presented to validate the closed-form solutions. With Hsueh et al.’s formulas, the biaxial strength of multilayered ceramics can be readily evaluated using biaxial flexure tests.

scholarly journals Influence of Flexible Foundation on Isolator Wave Effects

1996 ◽  
Vol 3 (1) ◽  
pp. 61-67 ◽  
Author(s):  
Ye-Ping Xiong ◽  
Kong-Jie Song ◽  
Xing Ai
Keyword(s):  
Closed Form ◽  
Vibration Isolation ◽  
Response Ratio ◽  
New Model ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Wave Effects ◽  
Rigid Mass ◽  
Isolation Systems ◽  
Flexible Foundation

This article deals with the interaction between wave effects in mounts and resonances of foundations inflexible vibration isolation systems. A new model is proposed that is represented as a rigid mass supported by two linear unidirectional isolators on a flexible foundation beam, whose closed-form solutions for transmissibility and response ratio are then obtained, with which the influence of wave effects coupled with the flexibility of the foundation on the effectiveness of isolation is discussed. The wave effects on flexible isolation systems are analyzed under various parametric conditions and compared with those in rigid systems. In addition, several special cases are presented to show the transition between various limiting cases. Some approaches to control wave effects are also proposed.

Stress-Free Strains in Martensitic Microstructures

2013 ◽  
Vol 738-739 ◽  
pp. 10-14
Author(s):  
Michaël Peigney
Keyword(s):  
Phase Transformation ◽  
Theoretical Prediction ◽  
Shape Memory ◽  
Closed Form ◽  
Solid Phase ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Crystallographic Structures ◽  
Solid Phase Transformation ◽  
Free Strains

The peculiar properties of shape-memory alloys are the result of a solid/solid phase transformation between different crystallographic structures (austenite and martensite). This paper is concerned with the theoretical prediction of the set of strains that minimize the effective (or macroscopic) energy. Those strains, classically refered to as recoverable strains, play a central role in shape memory effect displayed by alloys such as NiTi or CuAlNi. They correspond to macroscopic strains that can be achieved in stress-free states. Adopting the framework of nonlinear elasticity, the theoretical prediction of stress-free strains amounts to find the austenite/martensite microstructures which minimize the global energy. Closed-form solutions to that problem have been obtained only in few special cases. This paper aims at complementing existing results on that problem, essentially by deriving bounds on the set of stress-free strains.

Analysis of a Thin Elastic Ring Under Arbitrary Loading

1974 ◽  
Vol 96 (3) ◽  
pp. 870-876 ◽  
Author(s):  
J. Y. Liu ◽  
Y. P. Chiu
Keyword(s):  
General Solution ◽  
Closed Form ◽  
Elastic Ring ◽  
Closed Form Solutions ◽  
Distributed Load ◽  
Arbitrary Loading ◽  
Thin Ring ◽  
Special Cases ◽  
Loading System ◽  
Thin Elastic Ring

This paper presents a general solution for a thin ring under a self-equilibrating loading system comprising any combination of radial, tangential, and moment loads. The formulations are applicable to concentrated loads as well as to distributed load functions. Closed-form solutions are obtained for each case for engineering applications. Comparisons with recent published results for some special cases are demonstrated in some of the sample problems.

Cohomology Ring of Symplectic Quotients by Circle Actions

2004 ◽  
Vol 56 (3) ◽  
pp. 553-565 ◽  
Author(s):  
Ramin Mohammadalikhani
Keyword(s):  
Fixed Point ◽  
Cohomology Ring ◽  
Circle Action ◽  
Fixed Point Set ◽  
Circle Actions ◽  
Generating Set ◽  
Special Cases ◽  
Symplectic Quotients ◽  
Point Set

AbstractIn this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

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