Let \(\mathcal{H}\) be a Hilbert space of functions defined on an open bounded domain \(\mathcal{D}\) in \(\mathbb{R}^{d}\). The authors consider the following stochastic partial differential equation (SPDE) of parabolic type: \[ \begin{cases} dU_{t}=\mathcal{A}U_{t}dt+\sigma dW_{t}\;\text{in }\mathcal{D},\;0<t<T,\\ BU_{t}=0\;\text{on\;}\partial\mathcal{D},\;U_{0}=u_{0}, \end{cases} \] where \(\mathcal{A}\) is a linear elliptic operator in \(\mathcal{H}\), \(B\) is a boundary operator for Dirichlet or Neumann boundary conditions, \(u_{0} \in\mathcal{H}\), \(\sigma>0\), \(W\) is a Wiener process in \(\mathcal{H}\) with mean zero and spatial covariance function \(R\) given by \(\mathbb{E} (W(t,x)W(t,y))=\min\{t,s\}R(x,y)\), \(x,y\in\mathcal{D}\).
Applying the implicit Euler scheme \[ U_{t_{j}}-U_{t_{j-1}}=AU_{t_{j}}\delta t+\sigma\delta W_{j}, \] \(0=t_{0}<t_{1}<\dots<t_{n}=T\), \(\delta t_{j}:=t_{j}-t_{j-1}\), \(\delta W_{j}=W_{t_{j}}-W_{t_{j-1}}\), they become an elliptic SPDE of the form \[ Pu=f+\xi\;\text{in }\mathcal{D},\;Bu=0\;\text{on }\partial\mathcal{D},\; \] where \(P:=I-\delta t\mathcal{A}\), \(f:=U_{t_{j-1}}\), \(\xi:=\sigma\delta W_{j}\).
For solving the elliptic SPDE at each time step, the authors propose a kernel-based collocation method using the covariance function \(R\) for simulating a finite collection of collocation points.