scholarly journals An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
Simone Di Marino ◽  
Augusto Gerolin
Keyword(s):  
Optimal Transport ◽  
A Priori Estimates ◽  
Regularization Parameter ◽  
Alternative Proof ◽  
Revealed Preferences ◽  
Discussion Paper ◽  
Existence Of A Solution ◽  
Trade Offs ◽  
Bridge Problem ◽  
Schrödinger Bridge

AbstractThis paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two marginals: the main new result is the proof that the Sinkhorn algorithm converges even in the continuous multi-marginal case. This provides also an alternative proof of the convergence of the Sinkhorn algorithm in two marginals.

scholarly journals Multimarginal Optimal Transport with a Tree-Structured Cost and the Schrödinger Bridge Problem

2021 ◽  
Vol 59 (4) ◽  
pp. 2428-2453
Author(s):  
Isabel Haasler ◽  
Axel Ringh ◽  
Yongxin Chen ◽  
Johan Karlsson
Keyword(s):  
Optimal Transport ◽  
Bridge Problem ◽  
Schrödinger Bridge

scholarly journals Understanding uptake of digital health products: Methodology tutorial for a Discrete Choice Experiment using a Bayesian efficient design (Preprint)

2021 ◽  
Author(s):  
Dorothy Szinay ◽  
Rory Cameron ◽  
Felix Naughton ◽  
Jennifer A. Whitty ◽  
Jamie Brown ◽  
...  
Keyword(s):  
Discrete Choice Experiment ◽  
Discrete Choice ◽  
Choice Experiment ◽  
Stated Preference ◽  
Digital Health ◽  
Revealed Preferences ◽  
Quantitative Methodology ◽  
Efficient Design ◽  
Trade Offs ◽  
Health Products

UNSTRUCTURED Understanding the preferences of potential users of digital health products is beneficial for digital health policy and planning. Stated preference methods could help elicit individuals’ preferences in the absence of observational data. A discrete choice experiment (DCE) is a commonly used stated preference method; a quantitative methodology that argues that individuals make trade-offs when engaging in a decision by choosing an alternative of a product or service that offers the greatest utility, or benefit. This methodology is widely used in health economics in situations where revealed preferences are difficult to collect but is much less used in the field of digital health. This article outlines the stages involved in developing a discrete choice experiment. As a case study, it uses the application of a DCE for revealing preferences in targeting the uptake of smoking cessation apps. It describes the establishment of attributes, the construction of choice tasks of two or more alternatives, and the development of the experimental design. This tutorial offers a guide for researchers with no prior knowledge of this research technique.

LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC KIRCHHOFF SYSTEMS

2012 ◽  
Vol 22 (02) ◽  
pp. 1150009 ◽  
Author(s):  
GIUSEPPINA AUTUORI ◽  
FRANCESCA COLASUONNO ◽  
PATRIZIA PUCCI
Keyword(s):  
A Priori Estimates ◽  
Driving Forces ◽  
A Priori ◽  
Thin Plates ◽  
Existence Of A Solution ◽  
Dissipative Term ◽  
Maximal Solutions ◽  
Elastic Bodies ◽  
Priori Estimates ◽  
Deformation Processes

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.

A nonlinear partial differential system arising in thermoelectricity

2005 ◽  
Vol 16 (6) ◽  
pp. 683-712 ◽  
Author(s):  
A. BERMÚDEZ ◽  
R. MUÑOZ-SOLA ◽  
F. PENA
Keyword(s):  
Source Term ◽  
A Priori Estimates ◽  
A Priori ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Existence Of A Solution ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Partial Differential System ◽  
Temperature Equation

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

scholarly journals EIGENELEMENTS OF A GENERAL AGGREGATION-FRAGMENTATION MODEL

2010 ◽  
Vol 20 (05) ◽  
pp. 757-783 ◽  
Author(s):  
MARIE DOUMIC JAUFFRET ◽  
PIERRE GABRIEL
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Asymptotic Behavior Of Solutions ◽  
Fragmentation Model ◽  
Existence Of A Solution ◽  
Priori Estimates ◽  
Long Time ◽  
Weighted Norms ◽  
Fragmentation Processes ◽  
Time Asymptotic Behavior

We consider a linear integro-differential equation which arises to describe both aggregation-fragmentation processes and cell division. We prove the existence of a solution (λ, [Formula: see text], ϕ) to the related eigenproblem. Such eigenelements are useful to study the long-time asymptotic behavior of solutions as well as the steady states when the equation is coupled with an ODE. Our study concerns a non-constant transport term that can vanish at x = 0, since it seems to be relevant to describe some biological processes like proteins aggregation. Non-lower-bounded transport terms bring difficulties to find a priori estimates. All the work of this paper is to solve this problem using weighted-norms.

Effect of Price and Time on Private and Shared Transportation Network Company Trips

2021 ◽  
pp. 036119812199870
Author(s):  
Scott R. Middleton ◽  
Kyle A. Schroeckenthaler ◽  
Deepak Gopalakrishna ◽  
Allen Greenberg
Keyword(s):  
Stated Preference ◽  
Transportation Network ◽  
Relative Price ◽  
Revealed Preferences ◽  
Private Party ◽  
Trade Offs ◽  
Segmentation Analysis ◽  
Sizable Proportion ◽  
Survey Results ◽  
American Cities

Transportation network companies (TNCs) offer two types of service: private-party ridehailing and shared ridehailing. Policymakers have an interest in encouraging shared over private ridehailing to promote more efficient use of the transportation network. While transportation researchers have analyzed ridehailing behavior before, there is limited literature describing the effect of price and time on a rider’s choice between private-party and shared ridehailing. This paper fills this gap by analyzing revealed preferences for private-party and shared ridehailing trips in 15 American cities coupled with a survey of 4,365 users of a large TNC that includes stated preference questions focused on various alternative options for their most recent trip choice. This study finds that an increase in the relative price difference of $1 per mile increases an individual’s probability of sharing by over 8%, while a decrease in the relative travel time difference of 1 min per mile increases the probability of sharing by over 33%. The survey results also show that that a sizable proportion of private-party TNC trips (approximately 35%) will be difficult or even impossible to convert to shared rides through a price-based incentive. Market segmentation analysis reveals user and trip types where price- and time-based incentives have a relatively greater effect on the choice between private and shared rides. Finally, heterogeneity in user time versus money trade-offs suggests new product possibilities that would increase TNC sharing.

scholarly journals Application of transport-based metric for continuous interpolation between cryo-EM density maps

2021 ◽  
Vol 7 (1) ◽  
pp. 986-999
Author(s):  
Arthur Ecoffet ◽  
◽  
Geoffrey Woollard ◽  
Artem Kushner ◽  
Frédéric Poitevin ◽  
...  
Keyword(s):  
Optimal Transport ◽  
Regularization Parameter ◽  
Conformational Heterogeneity ◽  
The Past ◽  
Density Maps ◽  
Bioinformatic Tool ◽  
Overall Performance ◽  
Continuous Interpolation ◽  
3D Shapes ◽  
Made In

<abstract><p>Cryogenic electron microscopy (cryo-EM) has become widely used for the past few years in structural biology, to collect single images of macromolecules "frozen in time". As this technique facilitates the identification of multiple conformational states adopted by the same molecule, a direct product of it is a set of 3D volumes, also called EM maps. To gain more insights on the possible mechanisms that govern transitions between different states, and hence the mode of action of a molecule, we recently introduced a bioinformatic tool that interpolates and generates morphing trajectories joining two given EM maps. This tool is based on recent advances made in optimal transport, that allow efficient evaluation of Wasserstein barycenters of 3D shapes. As the overall performance of the method depends on various key parameters, including the sensitivity of the regularization parameter, we performed various numerical experiments to demonstrate how MorphOT can be applied in different contexts and settings. Finally, we discuss current limitations and further potential connections between other optimal transport theories and the conformational heterogeneity problem inherent with cryo-EM data.</p></abstract>

scholarly journals Dirichlet series expansions of p-adic L-functions

2021 ◽  
Author(s):  
Heiko Knospe ◽  
Lawrence C. Washington
Keyword(s):  
Dirichlet Series ◽  
Series Expansion ◽  
Regularization Parameter ◽  
Complex Case ◽  
Alternative Proof ◽  
Series Expansions ◽  
Bernoulli Distribution ◽  
Natural Manner ◽  
Limiting Behavior ◽  
Dirichlet Characters

AbstractWe study p-adic L-functions $$L_p(s,\chi )$$ L p ( s , χ ) for Dirichlet characters $$\chi $$ χ . We show that $$L_p(s,\chi )$$ L p ( s , χ ) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of $$\chi $$ χ . The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $$c=2$$ c = 2 , where we obtain a Dirichlet series expansion that is similar to the complex case.

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