Matrix evolution equations occur in many applications, such as dynamical Lyapunov/Sylvester systems or Riccati equations in optimization and stochastic control, machine learning or data assimilation. In many cases, their tightest stability condition is coming from a linear term. Exponential time differencing (ETD) is known to produce highly stable numerical schemes by treating the linear term in an exact fashion. In particular, for stiff problems, ETD methods are a method of choice. We propose an extension of the class of ETD algorithms to matrix-valued dynamical equations. This allows us to produce highly efficient and stable integration schemes. We show their efficiency and applicability for a variety of real-world problems, from geophysical applications to dynamical problems in machine learning.

翻译：矩阵演化方程广泛存在于诸多应用领域，例如动力李雅普诺夫/西尔维斯特系统、优化与随机控制中的里卡蒂方程、机器学习或数据同化等。在许多情况下，其最严格的稳定性条件源于线性项。已知指数时间差分（ETD）方法通过对线性项进行精确处理，能够构造出高稳定性的数值格式。尤其对于刚性问题，ETD方法是一种优选方案。本文提出将ETD算法类扩展至矩阵值动力方程，从而能够构建高效且稳定的积分格式。我们通过从地球物理应用到机器学习中的动力问题等一系列实际案例，验证了所提方法的高效性与适用性。