Authorship：Lead author  

In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the Wasserstein space by using the Neumann heat semigroup and prove the existence of minimizers of relaxed problems. Furthermore, we introduce the filtered Wasserstein gradient flow and derive the error estimate between the original Wasserstein gradient flow and the filtered one in terms of the Wasserstein distance. We also construct a candidate for the optimal mass distribution for a given fixed total mass and simultaneously obtain the shape of the material by the numerical calculation of filtered Wasserstein gradient flows.

arXiv

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Other Link： https://arxiv.org/pdf/2601.14332v1

Authorship：Lead author  

In this paper, we apply the framework of optimal transport to the formulation
of optimal design problems. By considering the Wasserstein space as a set of
design variables, we associate each probability measure with a shape
configuration of a material in some ways. In particular, we focus on
connections between differentials on the Wasserstein space and sensitivities in
the standard setting of shape and topology optimization in order to regard the
optimization procedure of those problems as gradient flows on the Wasserstein
space.

arXiv

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Other Link： http://arxiv.org/pdf/2411.12234v1

In this article, we deal with stochastic horizontal lifts and
anti-developments of semimartingales with jumps on complete and connected
Riemannian manifolds without any assumption for their curvatures. We prove two
one-to-one correspondences among some classes of discontinuous semimartingales
on Riemannian manifolds, orthonormal frame bundles and Euclidean spaces by
using the stochastic differential geometry with jumps introduced by Cohen
(1996). Both of these two results are extension of the one shown in
Pontier-Estrade (1992). The first result is the correspondence in the case
where jumps of semimartingales are regarded as initial velocities of geodesics
which are not necessarily minimal. In the second result, we also established
the correspondence in the situation where jumps of semimartingales are given by
connection rules, but we impose the condition that the jumps of semimartingales
are small. The latter result enables us to construct martingales for a given
connection rule with small jumps on any compact manifold from local martingales
on a Euclidean space through horizontal semimartingales on orthonormal
semimartingales.

arXiv

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Other Link： http://arxiv.org/pdf/2105.04824v2

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Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

Abstract

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.

DOI： 10.1007/s11118-024-10129-5

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Other Link： https://link.springer.com/article/10.1007/s11118-024-10129-5/fulltext.html

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