Convex variational problems arise in many fields ranging from image processing to fluid and solid mechanics communities. Interesting applications usually involve non-smooth terms, which require well-designed optimization algorithms for their resolution. The present manuscript presents the Python package called fenics_optim built on top of the FEniCS finite element software, which enables one to automate the formulation and resolution of various convex variational problems. Formulating such a problem relies on FEniCS domain-specific language and the representation of convex functions, in particular, non-smooth ones, in the conic programming framework. The discrete formulation of the corresponding optimization problems hinges on the finite element discretization capabilities offered by FEniCS, while their numerical resolution is carried out by the interior-point solver Mosek. Through various illustrative examples, we show that convex optimization problems can be formulated using only a few lines of code, discretized in a very simple manner, and solved extremely efficiently.

中文翻译：

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自动求解凸变分问题

凸变分问题出现在从图像处理到流体和固体力学社区的许多领域。有趣的应用程序通常涉及非平滑项，这需要精心设计的优化算法来解决它们。本手稿介绍了一个名为 fenics_optim 的 Python 包，它建立在 FEniCS 有限元软件之上，它使人们能够自动制定和解决各种凸变分问题。制定这样的问题依赖于 FEniCS 领域特定的语言和凸函数的表示，特别是圆锥编程框架中的非光滑函数。相应优化问题的离散公式取决于 FEniCS 提供的有限元离散化能力，而它们的数值解析是由内点求解器 Mosek 执行的。通过各种说明性示例，我们表明凸优化问题可以仅使用几行代码来制定，以非常简单的方式离散化，并且非常有效地解决。