NonlinearOperator

A nonlinear operator automatically assembles all necessary terms for the Newton method. Other linearisations of a nonlinear operator can be constructed with special constructors for BilinearOperator or LinearOperator.

Constructor

To describe a NonlinearOperator we have to specify a kernel function. These functions are 'flat' in the sense that the input and output vector contain the components of the test-function values and derivatives as specified by oa_test and oa_args respectively. The assembly of the local matrix will be done internally by multiplying the subvectors of result with its test-function counterparts. For a more detailed explanation of this see the following

function NonlinearOperator(
	[kernel!::Function],
	oa_test::Array{<:Tuple{Union{Unknown,Int}, DataType},1},
	oa_args::Array{<:Tuple{Union{Unknown,Int}, DataType},1} = oa_test;
	jacobian = nothing,
	kwargs...)

kernel!(result, input, qpinfo)

jacobian!(jac, input_args, params)

Example - NSE convection operator

For the Navier–Stokes equations, we need a kernel function for the nonlinear convection term

\[\begin{equation} (v,u\cdot\nabla u) = (v,\nabla u^T u) \end{equation}\]

In 2D the input (as specified below) will contain the two components of $u=(u_1,u_2)'$ and the four components of the gradient $\nabla u = \begin{pmatrix} u_{11} & u_{12} \\ u_{21} & u_{22}\end{pmatrix}$ in order, i.e. $(u_1,u_2,u_{11},u_{12},u_{21},u_{22})$. As the convection term is tested with $v$, the output vector $o$ only has to contain what should be tested with each component of $v$, i.e.

\[\begin{equation} A_\text{local} = (v_1,v_2)^T(o_1,o_2) = \begin{pmatrix} v_1o_1 & v_1o_2\\ v_2o_1 & v_2o_2 \end{pmatrix}. \end{equation}\]

To construct the kernel there are two options, component-wise and based on tensor_view. For the first we have to write the convection term as individual components

\[\begin{equation} o = \begin{pmatrix} u_1\cdot u_{11}+u_2\cdot u_{12}\\ u_1\cdot u_{21}+u_2\cdot u_{22}\\ \end{pmatrix} = \begin{pmatrix} u\cdot (u_11,u_12)^T\\ u\cdot (u_21,u_22)^T \end{pmatrix}. \end{equation}\]

To make our lives a bit easier we will extract the subcompontents of input as views, such that ∇u[3] actually accesses input[5], which corresponds to the third entry $u_{21}$ of $\nabla u$.

function kernel!(result, input, qpinfo)
    u, ∇u = view(input, 1:2), view(input,3:6)
    result[1] = dot(u, view(∇u,1:2))
    result[2] = dot(u, view(∇u,3:4))
    return nothing
end

To improve readability of the kernels and to make them easier to understand, we provide the function tensor_view which constructs a view and reshapes it into an object matching the given TensorDescription. See the table to see which tensor size is needed for which derivative of a scalar, vector or matrix-valued variable.

function kernel!(result, input, qpinfo)
    u = tensor_view(input,1,TDVector(2))
    v = tensor_view(result,1,TDVector(2))
    ∇u = tensor_view(input,3,TDMatrix(2))
    tmul!(v,∇u,u)
    return nothing
end

The coressponding NonlinearOperator constructor call is the same in both cases and reads

u = Unknown("u"; name = "velocity")
NonlinearOperator(kernel!, [id(u)], [id(u),grad(u)])

The second argument triggers that the evaluation of the Identity and Gradient operator of the current velocity iterate at each quadrature point go (in that order) into the input vector (of length 6) of the kernel, while the third argument triggers that the result vector of the kernel is multiplied with the Identity evaluation of the velocity test function.

u = Unknown("u"; name = "velocity")
LinearOperator(kernel!, [id(u)], [id(u),grad(u)])

function kernel_picard!(result, input_ansatz, input_args, qpinfo)
    a, ∇u = view(input_args, 1:2), view(input_ansatz,1:4)
    result[1] = dot(a, view(∇u,1:2))
    result[2] = dot(a, view(∇u,3:4))
end
u = Unknown("u"; name = "velocity")
BilinearOperator(kernel_picard!, [id(u)], [grad(u)], [id(u)])

function kernel!(result, input, qpinfo)
    u = tensor_view(input,1,TDVector(dim))
    v = tensor_view(result,1,TDVector(dim))
    ∇u = tensor_view(input,1+dim,TDMatrix(dim))
    tmul!(v,∇u,u)
    return nothing
end

Newton by local jacobians of kernel

To demonstrate the general approach consider a model problem with a nonlinear operator that has the weak formulation that seeks some function $u(x) \in X$ in some finite-dimensional space $X$ with $N := \mathrm{dim} X$, i.e., some coefficient vector $x \in \mathbb{R}^N$, such that

\[\begin{aligned} F(x) := \int_\Omega A(L_1u(x)(y)) \cdot L_2v(y) \,\textit{dy} & = 0 \quad \text{for all } v \in X \end{aligned}\]

for some given nonlinear kernel function $A : \mathbb{R}^m \rightarrow \mathbb{R}^n$ where $m$ is the dimension of the input $L_1 u(x)(y) \in \mathbb{R}^m$ and $n$ is the dimension of the result $L_2 v(y) \in \mathbb{R}^n$. Here, $L_1$ and $L_2$ are linear operators, e.g. primitive differential operator evaluations of $u$ or $v$.

Let us consider the Newton scheme to find a root of the residual function $F : \mathbb{R}^N \rightarrow \mathbb{R}^N$, which iterates

\[\begin{aligned} x_{n+1} = x_{n} - D_xF(x_n)^{-1} F(x_n) \end{aligned}\]

or, equivalently, solves

\[\begin{aligned} D_xF(x_n) \left(x_{n+1} - x_{n}\right) = -F(x_n) \end{aligned}\]

To compute the jacobian of $F$, observe that its discretisation on a mesh $\mathcal{T}$ and some quadrature rule $(x_{qp}, w_{qp})$ leads to

\[\begin{aligned} F(x) = \sum_{T \in \mathcal{T}} \lvert T \rvert \sum_{x_{qp}} A(L_1u_h(x)(x_{qp})) \cdot L_2v_h(x_{qp}) w_{qp} & = 0 \quad \text{in } \Omega \end{aligned}\]

Now, by linearity of everything involved other than $A$, we can evaluate the jacobian by

\[\begin{aligned} D_xF(x) = \sum_{T \in \mathcal{T}} \lvert T \rvert \sum_{x_{qp}} DA(L_1 u_h(x)(x_{qp})) \cdot L_2 v_h(x_{qp}) w_{qp} & = 0 \quad \text{in } \Omega \end{aligned}\]

Hence, assembly only requires to evaluate the low-dimensional jacobians $DA \in \mathbb{R}^{m \times n}$ of $A$ at $L_1 u_h(x)(x_{qp})$. These jacobians are computed by automatic differentiation via ForwardDiff.jl (or via the user-given jacobian function). If $m$ and $n$ are a little larger, e.g. when more operator evaluations $L_1$ and $L_2$ or more unknowns are involved, there is the option to use sparse_jacobians (using the sparsity detection of Symbolics.jl).