We will consider the discretization of the stochastic differential equation$$X_t=X_0+W_t+\int_0^t b\left(s, X_s\right) d s, t \in[0, T]$$where the drift coefficient $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is measurable and satisfies the integrability condition : $\|b\|_{L^q\left([0, T], L^\rho\left(\mathbb{R}^d\right)\right)}<\infty$ for some $\rho, q \in(0,+\infty]$ such that$$\rho \geq 2 \text { and } \frac{d}{\rho}+\frac{2}{q}<1 .$$Krylov and Röckner [3] established strong existence and uniqueness under this condition.Let $n \in \mathbb{N}^*, h=\frac{T}{n}$ and $t_k=k h$ for $k \in \left [ \left [0,n \right ] \right ]$. Since there is no smoothing effect in the time variable, we introduce a sequence $\left(U_k\right)_{k \in \left [ \left [0,n-1 \right ] \right ]}$ independent from $\left(X_0,\left(W_t\right)_{t \geq 0}\right)$ of independent random variables which are respectively distributed according to the uniform law on $[k h,(k+1) h]$. The resulting scheme Euler is initialized by $X_0^h=X_0$ and evolves inductively on the regular time-grid $\left(t_k=k h\right)_{k \in \left [ \left [0,n \right ] \right ]}$ by:$$X_{t_{k+1}}^h=X_{t_k}^h+W_{t_{k+1}}-W_{t_k}+b_h\left(U_k, X_{t_k}^h\right) h$$where $b_h$ is some truncation of the drift function $b$. When $b$ is bounded, one of course chooses $b_h=b$. Then the order of weak convergence in total variation distance is $1 / 2$, as proved in [1]. It improves to 1 up to some logarithmic correction under some additional uniform in time bound on the spatial divergence of the drift coefficient. In the general case (1), we will see that for suitable truncations $b_h$, the difference between the transition densities of the stochastic differential equation and its Euler scheme is bounded from above by $C h^{\frac{1}{2}\left(1-\left(\frac{d}{\rho}+\frac{2}{q}\right)\right)}$ multiplied by some centered Gaussian density, as proved in [2].[-]

An elegant theorem by J.L. Lions establishes well-posedness of non-autonomous evolutionary problems in Hilbert spaces which are defined by a non-autonomous form. However a regularity problem remained open for many years. We give a survey on positive and negative (partially very recent) results. One of the positive results can be applied to an evolutionary network which has been studied by Dominik Dier and Marjeta Kramar jointly with the speaker. It is governed by non-autonomous Kirchhoff conditions at the vertices of the graph. Also the diffusion coefficients may depend on time. Besides existence and uniqueness long-time behaviour can be described. When conductivity and diffusion coefficients match (so that mass is conserved) the solutions converge to an equilibrium.[-]

We consider the numerical evaluation of integrals with respect to self-similar measures supported on fractal sets, with a weakly singular integrand of loga-rithmic or algebraic type. We show that, in many cases, the self-similarity of the measures, combined with the homogeneity properties of the integrand, can be exploited to express the singular integral exactly in terms of regular inte-grals, which can be readily approximated numerically using e.g. a composite barycentre rule. Our approach applies to measures supported on many well-known fractals including Cantor sets and dusts, the Sierpinski triangle, carpet and tetrahedron, the Vicsek fractal, and the Koch snowflake. We illustrate our approach via numerical examples computed using our IFSIntegrals.jl Julia code. This is joint work with Andrew Gibbs, Botond Major and Andrea Moiola.[-]

Motivated by the task of sampling measures in high dimensions we will discuss a number of gradient flows in the spaces of measures, including the Wasserstein gradient flows of Maximum Mean Discrepancy and Hellinger gradient flows of relative entropy, the Stein Variational Gradient Descent and a new projected dynamic gradient flows. For all the flows we will consider their deterministic interacting-particle approximations. The talk is highlight some of the properties of the flows and indicate their differences. In particular we will discuss how well can the interacting particles approximate the target measures.The talk is based on joint works wit Anna Korba, Lantian Xu, Sangmin Park, Yulong Lu, and Lihan Wang.[-]

One of the goals of shape analysis is to model and characterise shape evolution. We focus on methods where this evolution is modeled by the action of a time-dependent diffeomorphism, which is characterised by its time-derivatives: vector fields. Reconstructing the evolution of a shape from observations then amounts to determining an optimal path of vector fields whose flow of diffeomorphisms deforms the initial shape in accordance with the observations. However, if the space of considered vector fields is not constrained, optimal paths may be inaccurate from a modeling point of view. To overcome this problem, the notion of deformation module allows to incorporate prior information from the data into the set of considered deformations and the associated metric. I will present this generic framework as well as the Python library IMODAL, which allows to perform registration using such structured deformations. More specifically, I will focus on a recent implicit formulation where the prior can be expressed as a property that the generated vector field should satisfy. This imposed property can be of different categories that can be adapted to many use cases, such as constraining a growth pattern or imposing divergence-free fields.[-]

The aim of this talk is to present a new variation formulation of the time-dependent many-body electronic Schrödinger equation with Coulombic singularities. More precisely, its solution can actually be expressed as the solution of a global space-time quadratic minimization problem that proves to be useful for several tasks:
1) first, it is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation
2) it enables to yield a new variational principle for the construction dynamical low-rank approximations, that is different from the classical Dirac-Frenkel variational principle
3) it enables to obtain fully certified a posteriori error estimators between the exact solution and approximate solutions.
The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.[-]