一周活动预告（12.19-12.25）

汽液相变或者液固相变是自然界和工业生产中的常见现象，涉及到流场、自由界面、温度和浓度场的耦合计算，发展精确的计算方法进行数值模拟是进行物理研究的重要手段。数值算法发展的难点在于相变特性不仅取决于温度场，还取决于蒸汽或者溶质成分的浓度分布，两者在两相界面处强耦合，普通算法难以维持界面两侧的物理间断。在有限体积框架下，我们提出一种陡峭型嵌入边界方法 (Embedded Boundary Method) 来求解流场、温度场和浓度场，实现了浓度场和温度场在界面处的耦合计算，保真了陡峭界面两侧的间断物理特性，从而实现了汽液固相变精确模拟。该算法的精确性和稳定性通过一些列Benchmark算例进行了验证，并将其推广到三维情况，模拟了温度场和浓度场共同作用下的液体蒸发、三维晶枝生长等现象。最后，我们模拟了相变中非常复杂的Leidenfrost液滴蒸发以及Ni-Cu合金溶液在逆温度梯度情况下的多阶段凝固，获得和理论及实验一致的结果。

In this talk, we will report recent developments of finite element methods for some nonstationary incompressible flow problems with variable density, such as the first-order Euler and two-step BDF schemes for the Navier-Stokes equations with variable density, and the first-order Euler scheme for the Navier-Stokes equations with mass diffusion (or Kazhikhov-Smagulov model).

The optimistic gradient (OG) method has received growing attention due to its favorable performance in saddle-point optimization problems, which include two-player zero-sum games (a.k.a. matrix games) as a special case. Compared to another classical method, Extragradient, OG is more suitable for repeated games since it is a no-regret algorithm and only requires one gradient call in each iteration. Although the sublinear average-iterate convergence of OG has been provided for years, the more appealing last-iterate convergence rate has only been established recently.

In this talk, we will introduce OG as well as its several variants, and discuss the convergence rate of the last iterate in saddle-point optimization problems. Moreover, we will pay particular attention to matrix games.

The advancements in machine learning and, in particular, the recent breakthrough of artificial neural networks, has promoted novel art practices in which computers play a fundamental role to support & enhance human creativity. Alongside other arts, music and dance have also benefited from the development of machine learning and artificial intelligence to support creativity, for tasks ranging from music generation to augmented dance. This talk will present an overview of my research, focused on the intersection of artistic creation and performance, mathematical modelling, machine learning and human-computer interaction – and on their entwined co-evolution.

In this talk, we introduce the Monte Carlo methods for solving PDEs involving an integral fractional Laplacian (IFL) in high dimensions. We first construct the Feynman-Kac representation based on the Green function for the IFL on the unit ball in arbitrary dimensions. Inspired by the ``walk-on-spheres" algorithm proposed in [Kyprianou, Osojnik, and Shardlow, IMA J. Numer. Anal.(2018)], we extend our algorithm for solving fractional Poisson equations in the complex domain. Then, we can compute the expectation of a multi-dimensional random variable with a known density function to obtain the numerical solution efficiently. The proposed algorithm finds it remarkably efficient in solving fractional PDEs: it only needs to evaluate the integrals of expectation form over a series of inside ball tangent boundaries with the known Green function. Moreover, we carry out the error estimates of the proposed method for the d-dimensional unit ball. Ample numerical results are presented to demonstrate the robustness and effectiveness of the proposed method. Finally, we extended the proposed algorithm to solve space-fractional diffusion equations in high dimensions.

The finite element method was invented for stress analysis of elasticity problems in the 1960s. Since the Hellinger-Reissner variational principle takes both stress and displacement as independent variables, it has become very important in numerical simulations. Surprisingly, more than half century’s research proves that it is extremely difficult to develop stable finite element schemes that are able to preserve the stress symmetry exactly. This talk presents recent mixed finite element methods of linear elasticity in both 2D and 3D. In addition, their adaptive versions and fast solvers for the corresponding discrete system are discussed.

In this talk, we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with the L2-regularization. We derive the first order optimality conditions for the NeurODEs training problem in the form of a mean-field maximum principle and show that it admits a unique control solution, which is Lipschitz continuous in time. Some instructive numerical experiments are also provided

In this talk, I will give an introduction to multiscale finite element methods (MsFEMs). The motivations and key ideas of the MsFEMs will be first explained, followed by applications for several types of underground flow problems and heterogeneous wave equations.

The subject of the presentation is the study of splitting integrators for Poisson systems perturbed by additive noise and multiplicative noise in the Stratonovich sense. This presentation is based on a joint work with G. Vilmart and on a joint work with C-E. Bréhier and T. Jahnke.

数学物理反问题的研究是现代应用数学研究的一个热点研究方向，具有重要的实际背景和理论研究价值，研究成果有很强的应用前景。对于反问题的研究学者来讲，如何提出有意义的反问题是一个令人困惑的难题。在本报告中，我们通过逆向思维的方式给出若干例子，说明如何合理地提出具有研究价值的反问题，并指出如何将现代应用数学的成果融入反问题的研究当中，解决一些工程和实际问题的关键的难点问题。