论文标题
通过使用预测的准耐将方法，通过隐式前向操作员解决反对对流问题
Solving an inverse heat convection problem with an implicit forward operator by using a Projected Quasi-Newton method
论文作者
论文摘要
我们考虑了从嘈杂的内部焓测量值确定焓依赖的热通量的准逆热对流问题（IHCP）。此问题出现在生产由钢制成的热机械控制加工（TMCP）重板的加速冷却过程中。为了调整基础材料的复杂微观结构，必须研究热表面相对于冷却液的应用的leiden冻土行为。由于热通量取决于焓，因此仅取决于基础初始边界值问题（IBVP）的解决方案，因此只能隐含地定义逆问题的参数到解决操作员，因此逆问题的正向操作员只能被隐式定义。为了保证定义明确的运营商，我们研究了两种显示IBVP解决方案存在和独特性的方法。一种方法涉及sobolev-bochner空间中的假酮操作员和所谓的强解决方案的理论。另一种理论在Hölder空间中使用经典解决方案。尽管第一种方法在温和的假设下产生了解决方案，但与第二种方法相比，它未显示出唯一性结果。此外，我们为非线性热通量提出了一种方便的参数化方法，以使参数到解决方案关系解除，并基于投影的准Newton（PQN）方法使用迭代求解器以及盒子构造方法来解决反问题。对于数值实验，我们得出了目标功能的必要梯度信息，并将差异原理作为停止规则。数值测试表明，对于计算时间和近似精度，PQN方法的表现优于Landweber方法。
We consider the quasilinear 1D inverse heat convection problem (IHCP) of determining the enthalpy-dependent heat fluxes from noisy internal enthalpy measurements. This problem arises in the Accelerated Cooling (ACC) process of producing thermomechanically controlled processed (TMCP) heavy plates made of steel. In order to adjust the complex microstructure of the underlying material, the Leidenfrost behavior of the hot surfaces with respect to the application of the cooling fluid has to be studied. Since the heat fluxes depend on the enthalpy and hence on the solution of the underlying initial boundary value problem (IBVP), the parameter-to-solution operator, and thus the forward operator of the inverse problem, can only be defined implicitly. To guarantee well-defined operators, we study two approaches for showing existence and uniqueness of solutions of the IBVP. One approach deals with the theory of pseudomonotone operators and so-called strong solutions in Sobolev-Bochner spaces. The other theory uses classical solutions in Hölder spaces. Whereas the first approach yields a solution under milder assumptions, it fails to show the uniqueness result in contrast to the second approach. Furthermore, we propose a convenient parametrization approach for the nonlinear heat fluxes in order to decouple the parameter-to-solution relation and use an iterative solver based on a Projected Quasi-Newton (PQN) method together with box-constraints to solve the inverse problem. For numerical experiments, we derive the necessary gradient information of the objective functional and use the discrepancy principle as a stopping rule. Numerical tests show that the PQN method outperforms the Landweber method with respect to computing time and approximation accuracy.
