======================== Mathematical Formulation ======================== .. note:: I cannot use hard mathematical terms and prove the existence of a solution on this page. Here I will focus on essential information. Please go check the publication: Behzad Azmi, Andrea Petrocchi, Stefan Volkwein, *Parameter optimization for elliptic-parabolic systems by an adaptive trust-region reduced basis method*, Advances in Applied Mechanics, Elsevier, Volume 59, 2024, Pages 109-145, https://doi.org/10.1016/bs.aams.2024.07.001. Pre-print available at https://arxiv.org/abs/2307.12723. .. important:: Academic publishing companies are thieves and do not deserve our money. If you do not have access to the paper, send me an e-mail. The geometry ------------ Let :math:`T > 0` be the finite time horizon, :math:`\Omega := (0, L)` a one-dimensional interval (called in general terms domain) and the space-time domain :math:`Q_T := (0, T) \times (0, L)`. The equations ------------- We consider the following parameter-dependent parabolic-elliptic coupled system for the two state variables :math:`y, q: Q_T \to \mathbb{R}` satisfying .. math:: y_t (t,x) - \mu_1 \frac{\partial}{\partial x} \left( \kappa_1 (x) \frac{\partial}{\partial x} y (t,x) \right) - \mu_2 f \left( y(t,x), q(t,x) \right) &= 0 \qquad \text{f.a.a. } (t,x) \in Q_T, -\mu_3 \frac{\partial}{\partial x} \left( \kappa_2 (x) \frac{\partial}{\partial x} q (t,x) \right) + \mu_4 f \left( y(t,x), q(t,x) \right) &= 0 \qquad \text{f.a.a. } (t,x) \in Q_T The functions :math:`\kappa_1` and :math:`\kappa_2` are called (space-dependent) diffusion coefficients and :math:`\boldsymbol{\mu} = [\mu_1, \mu_2, \mu_3, \mu_4]` is a control vector. The term "f.a.a." means "for almost all". This means, "for all values except those in a null set" (measure theory, fun stuff). The boundary and initial conditions ----------------------------------- For the state :math:`y` we have homogeneous Neumann boundary conditions: .. math:: y_x(t,0) = y_x(t,L) = 0\quad \text{f.a.a. }t\in(0,T). For :math:`q` we have inhomogeneous Dirichlet-Neumann mixed boundary conditions: .. math:: q(t,0) = 0\text{ f.a.a. }t\in(0,T), \quad\mu_3 \kappa_2(L) q_x(t,L) = u(t)\text{ f.a.a. }t\in(0,T). The time function :math:`u` is called *input function*, it is user-defined and can change the behaviour of the two states greatly. The initial conditions is .. math:: y(0,x) =y_\circ(x)\quad \text{f.a.a. }x\in\Omega. The assumptions --------------- The functions :math:`\kappa_1` and :math:`\kappa_2` belong to the space :math:`C^{0,1} (\bar\Omega)`. Namely, they are `Lipschitz continuous `__ in the closure of :math:`\Omega`, where we use the `Hölder notation `__ :math:`C^{0,\alpha}`. In simpler terms, the functions are continuous and their derivatives are dominated by the constant 1. The initial condition :math:`y_\circ` belongs to :math:`H^1 (\Omega)`, i.e. the Sobolev space :math:`W^{1,2} (\Omega)`. This means the function and its derivative are in :math:`L^2 (\Omega)`. .. tip:: If you do not understand what I am talking about, go read a maths book. The input function :math:`u` satisfies :math:`u_\mathsf{a} (t) \le u (t) \le u_\mathsf{b} (t)` for all :math:`t \in [0,T]` with :math:`u_\mathsf{a}, u_\mathsf{b} \in L^\infty (\Omega)`. This guarantees that `u` is always finite and does not explode in time. The nonlinearity is defined as :math:`f(\mathrm y, \mathrm q) := \sqrt{\mathrm y} \sinh(\mathrm q)` for :math:`\mathrm y\in\mathbb R_{\ge}:= \{ s \in \mathbb{R} \, \vert \, s \ge 0 \}` and :math:`\mathrm q \in \mathbb R`. The weak formulation -------------------- The weak formulation is extremely helpful to define the finite element-based solution. With :math:`V=H^1(\Omega)`, :math:`H=L^2(\Omega)`, :math:`V_\circ=\{ v \in V : v(0) = 0\}`, the weak formulation reads: .. math:: \frac{\mathrm{d}}{\mathrm{d}t} \langle y (t), \varphi \rangle_H + \mu_1 a_1 (y(t), \varphi) - \mu_2 \langle g [y(t), q(t)], \varphi \rangle_{V', V} &= 0 \qquad \forall \varphi \in V, y(0) &= y_\circ \qquad \text{in } H, \mu_3 a_2 (q(t), \psi) + \mu_4 \langle g [y(t), q(t)], \psi \rangle_{V_\circ', V_\circ} - \langle b(t), \psi \rangle_{V'_\circ, V_\circ} &= 0 \qquad \forall \psi \in V_\circ, where: - :math:`y(t)=y(t,\cdot) \in V` and :math:`q(t)=q(t,\cdot) \in V_\circ` f.a.a. :math:`t`, - :math:`\varphi \in V` and :math:`\psi \in V_\circ` are the test functions, - :math:`a_i` for :math:`i=1,2` are the diffusion operators, - :math:`g` is the coupling operators, - :math:`\langle b(t), \psi \rangle_{V_\circ', V_\circ} = u(t) \psi(L)` for :math:`\psi \in V_\circ` and f.a.a. :math:`t`. The solution space ------------------ In the paper mentioned above it is proven after 33 painful pages that there exist a solution :math:`(y, q) \in \mathcal{Y}_T \times \mathcal{Q}_T` to the weak formulation, where: .. math:: \mathcal{Y}_T := W(0, T; V, V') \cap C(\bar{Q}_T), \qquad \text{and} \qquad \mathcal{Q}_T := L^\infty (0, T; V_\circ). .. tip:: Just go read the paper. The time discretization ----------------------- Here we just mention the Implicit Euler method, but the Crank-Nicolson method is also available in this package. After time discretization, :math:`t_k = k \delta` for :math:`k=0,\dots,K`, with :math:`\delta = T / K`, and :math:`K` is the number of time steps. In the following, :math:`y^k` and :math:`q^k` are the approximated values of functions :math:`y(t_k)` and :math:`q(t_k)`, respectively. After time discretization, the solution is approximated by the following nonlinear system: .. math:: \langle y^k, \varphi \rangle_H + \langle y^{k-1}, \varphi \rangle_H + \mu_1 \delta a_1 (y^k, \varphi) - \mu_2 \delta \langle g [y^k, q^k], \varphi \rangle_{V', V} &= 0 \qquad \forall \varphi \in V, \ k=1,\dots, K, y^0 &= y_\circ \qquad \text{in } H, \mu_3 a_2 (q^k, \psi) + \mu_4 \langle g [y^k, q^k], \psi \rangle_{V_\circ', V_\circ} - \langle b^k, \psi \rangle_{V'_\circ, V_\circ} &= 0 \qquad \forall \psi \in V_\circ, \ k=0, \dots, K.