.. _theory: Background Theory ================= The following provides an introduction to the theoretical foundation of the solver emg3d. More specific theory is covered in the docstrings of many functions, have a look at the :doc:../api/core-section or follow the links to the corresponding functions here within the theory. If you just want to use the solver, but do not care much about the internal functionality, then the function :func:emg3d.solver.solve is the only function you will ever need. It is the main entry point, and it takes care whether multigrid is used as a solver or as a preconditioner (or not at all), while the actual multigrid solver is :func:emg3d.solver.multigrid. .. note:: This section is not an independent piece of work. Most things are taken from one of the following sources: - [Muld06]_, pages 634-639: - The *Maxwell's equations* and *Discretisation* sections correspond with some adjustemens and additions to pages 634-636. - The start of *The Multigrid Method* corresponds roughly to page 637. - Pages 638 and 639 are in parts reproduced in the code-docstrings of the corresponding functions. - [BrHM00]_: This book is an excellent introduction to multigrid methods. Particularly the *Iterative Solvers* section is taken to a big extent from the book. **Please consult these original resources for more details, and refer to them for citation purposes and not to this manual.** More in-depth information can also be found in, e.g., [Hack85]_ and [Wess91]_. Maxwell's equations ------------------- Maxwell's equations in the presence of a current source :math:\mathbf{J}_\mathrm{s} are .. math:: :label: maxwell \partial_{t} \mathbf{B}(\mathbf{x},t) + \nabla\times\mathbf{E}(\mathbf{x},t) &= 0 , \\ \nabla \times \mathbf{H}(\mathbf{x}, t) - \partial_{t} \mathbf{D}(\mathbf{x}, t) &= \mathbf{J}_{\mathrm{c}}(\mathbf{x}, t) + \mathbf{J}_\mathrm{s}(\mathbf{x}, t) , where the conduction current :math:\mathbf{J}_\mathrm{c} obeys Ohm’s law, .. math:: :label: ohmslaw \mathbf{J}_{\mathrm{c}}(\mathbf{x},t) = \sigma(\mathbf{x})\mathbf{E}(\mathbf{x},t) . Here, :math:\sigma(\mathbf{x}) is the conductivity. :math:\mathbf{E}(\mathbf{x}, t) is the electric field and :math:\mathbf{H}(\mathbf{x}, t) is the magnetic field. The electric displacement :math:\mathbf{D}(\mathbf{x}, t) = \varepsilon(\mathbf{x})\mathbf{E}(\mathbf{x}, t) and the magnetic induction :math:\mathbf{B}(\mathbf{x}, t) = \mu(\mathbf{x})\mathbf{H}(\mathbf{x}, t). The dielectric constant or permittivity :math:\varepsilon can be expressed as :math:\varepsilon = \varepsilon_r \varepsilon_0, where :math:\varepsilon_r is the relative permittivity and :math:\varepsilon_0 is the vacuum value. Similarly, the magnetic permeability :math:\mu can be written as :math:\mu = \mu_r\mu_0, where :math:\mu_r is the relative permeability and :math:\mu_0 is the vacuum value. The magnetic field can be eliminated from Equation :eq:maxwell, yielding the second-order parabolic system of equations, .. math:: :label: combmax \varepsilon \partial_{t t} \mathbf{E} + \sigma \partial_{t} \mathbf{E} + \nabla \times \mu^{-1} \nabla \times \mathbf{E} = -\partial_{t} \mathbf{J}_{\mathrm{s}} . To transform from the time domain to the frequency domain, we substitute .. math:: :label: fourierdef \mathbf{E} (\mathbf{x},t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \mathbf{\hat{E}} (\mathbf{x},\omega) e^{-\mathrm{i}\omega t}\, d\omega, and use a similar representation for :math:\mathbf{H}(\mathbf{x}, t). The resulting system of equations is .. math:: :label: fdomain \mathrm{i}\omega \mu_0(\sigma - \mathrm{i}\omega\varepsilon) \mathbf{\hat{E}} - \nabla \times \mu_r^{-1} \nabla \times \mathbf{\hat{E}} = -\mathrm{i}\omega\mu_0\mathbf{\hat{J}}_s . The multigrid method converges in the case of the diffusive approximation (with its smoothing and approximation properties), but not in the high-frequency range (at least not in the implemented form of the multigrid method in emg3d). The code emg3d assumes therefore the diffusive approximation, hence only low frequencies are considered that obey :math:|\omega\varepsilon| \ll \sigma. In this case we can set :math:\varepsilon=0, and Equation :eq:fdomain simplifies to .. math:: :label: fdomaindiff \mathrm{i}\omega \mu_0 \sigma \mathbf{\hat{E}} - \nabla \times \mu_r^{-1} \nabla \times \mathbf{\hat{E}} = -\mathrm{i}\omega\mu_0\mathbf{\hat{J}}_s , From here on, the hats are omitted. We use the perfectly electrically conducting boundary .. math:: :label: pec \mathbf{n}\times\mathbf{E} = 0 \quad \text{and} \quad \mathbf{n}\cdot\mathbf{H} = 0 , where :math:\mathbf{n} is the outward normal on the boundary of the domain. The Maxwell's equations and Ohm's law are solved in the **frequency domain**. The **time-domain** solution can be obtained by taking the inverse Fourier transform. .. note:: [Muld06]_ uses the time convention :math:e^{-\mathrm{i}\omega t}, see Equation :eq:fourierdef. However, the code emg3d uses the convention :math:e^{\mathrm{i}\omega t}. This is the same convention as used in empymod and generally in the CSEM community. This change in Fourier convention can be simply obtained in these equations by replacing :math:-\mathrm{i}\omega with :math:+\mathrm{i}\omega. Equation :eq:fdomain, for instance, becomes .. math:: -\mathrm{i}\omega \mu_0(\sigma + \mathrm{i}\omega\varepsilon) \mathbf{\hat{E}} - \nabla \times \mu_r^{-1} \nabla \times \mathbf{\hat{E}} = \mathrm{i}\omega\mu_0\mathbf{\hat{J}}_s . Laplace domain  It is also possible to solve the problem in the **Laplace domain**, by using a real value for :math:s in Equation :eq:fdomaindiff, instead of the complex value :math:\mathrm{i}\omega. This simplifies the problem from complex numbers to real numbers, which accelerates the computation. It also improves the convergence rate, as the solution is a smoother function. The solver :func:emg3d.solver.solve is agnostic to the data type of the provided source field, and can solve for real and complex problems, hence frequency and Laplace domain. See the documentation of the functions :func:emg3d.fields.get_source_field and :func:emg3d.models.Model to see how you can use emg3d for Laplace-domain computations. Discretisation -------------- Equation :eq:fdomaindiff can be discretised by the finite-integration technique ([Weil77]_, [ClWe01]_). This scheme can be viewed as a finite-volume generalization of [Yee66]_'s scheme for tensor-product Cartesian grids with variable grid spacings. An error analysis for the constant-coefficient case ([MoSu94]_) showed that both the electric and magnetic field components have second-order accuracy. Consider a tensor-product Cartesian grid with nodes at positions :math:(x_k, y_l, z_m), where :math:k=0, \dots, N_x, l=0, \dots, N_y and :math:m=0, \dots, N_z. There are :math:N_x\times N_y\times N_z cells having these nodes as vertices. The cell centres are located at .. math:: :label: cellcenters x_{k+1/2} &= {\textstyle \frac{1}{2}}\left(x_k + x_{k+1}\right) , \\ y_{l+1/2} &= {\textstyle \frac{1}{2}}\left(y_l + y_{l+1}\right) , \\ z_{m+1/2} &= {\textstyle \frac{1}{2}}\left(z_m + z_{m+1}\right) . The material properties, :math:\sigma and :math:\mu_\mathrm{r}, are assumed to be given as cell-averaged values. The electric field components are positioned at the edges of the cells, as shown in :numref:Figure %s , in a manner similar to Yee’s scheme. The first component of the electric field :math:E_{1, k+1/2, l, m} should approximate the average of :math:E_1(x, y_l, z_m) over the edge from :math:x_k to :math:x_{k+1} at given :math:y_l and :math:z_m. Here, the average is defined as the line integral divided by the length of the integration interval. The other components, :math:E_{2, k, l+1/2, m} and :math:E_{3, k, l, m+1/2}, are defined in a similar way. Note that these averages may also be interpreted as point values at the midpoint of edges: .. math:: :label: edgepoints E_{1, k+1/2, l, m} \simeq E_1\left(x_{k+1/2}, y_{l}, z_{m}\right) , \\ E_{2, k, l+1/2, m} \simeq E_2\left(x_{k}, y_{l+1/2}, z_{m}\right) , \\ E_{3, k, l, m+1/2} \simeq E_3\left(x_{k}, y_{l}, z_{m+1/2}\right) . The averages and point-values are the same within second-order accuracy. .. figure:: ../_static/Muld06_Fig1.png :scale: 100 % :align: center :alt: Staggered grid sketches. :name: Muld06_Fig1 (a) A grid cell with grid nodes and edge-averaged components of the electric field. (b) The face-averaged magnetic field components that are obtained by taking the curl of the electric field. For the discretisation of the term :math:\mathrm{i}\omega\mu_0\sigma\mathbf{E} related to Ohm's law, dual volumes related to edges are introduced. For a given edge, the dual volume is a quarter of the total volume of the four adjacent cells. An example for :math:E_1 is shown in :numref:Figure %s(b) . The vertices of the dual cell are located at the midpoints of the cell faces. .. figure:: ../_static/Muld06_Fig2.png :scale: 100 % :align: center :alt: Dual volume sketches. :name: Muld06_Fig2 The first electric field component :math:E_{1,k,l,m} is located at the intersection of the four cells shown in (a). Four faces of its dual volume are sketched in (b). The first component of the curl of the magnetic field should coincide with the edge on which :math:E_1 is located. The four vectors that contribute to this curl are shown in (a). They are defined as normals to the four faces in (a). Before computing their curl, these vectors are interpreted as tangential components at the faces of the dual volume shown in (b). The curl is evaluated by taking the path integral over a rectangle of the dual volume that is obtained for constant x and by averaging over the interval :math:[x_k,x_{k+1}]. The volume of a normal cell is defined as .. math:: :label: volnormalcell V_{k+1/2, l+1/2, m+1/2} = h_{k+1/2}^x h_{l+1/2}^y h_{m+1/2}^z , where .. math:: :label: cellwidths h_{k+1/2}^x &= x_{k+1}-x_k , \\ h_{l+1/2}^y &= y_{l+1}-y_l , \\ h_{m+1/2}^z &= z_{m+1}-z_m . For an edge parallel to the x-axis on which :math:E_{1, k+1/2, l, m} is located, the dual volume is .. math:: :label: dualvolume V_{k+1/2, l, m} = {\textstyle \frac{1}{4}} h_{k+1/2}^x \sum_{m_2=0}^1 \sum_{m_3=0}^1 h_{l-1/2+m_2}^y h_{m-1/2+m_3}^z . With the definitions, .. math:: :label: dualcellwidths d_k^x &= x_{k+1/2} - x_{k-1/2} , \\ d_l^y &= y_{l+1/2} - y_{l-1/2} , \\ d_m^z &= z_{m+1/2} - z_{m-1/2} , we obtain .. math:: :label: discdualvol V_{k+1/2, l, m} &= h_{k+1/2}^x d_l^y d_m^z , \\ V_{k, l+1/2, m} &= d_k^x h_{l+1/2}^y d_m^z , \\ V_{k, l, m+1/2} &= d_k^x d_l^y h_{m+1/2}^z . Note that Equation :eq:dualcellwidths does not define :math:d_k^x, etc., at the boundaries. We may simply take :math:d^x_0 = h^x_{1/2} at :math:k = 0, :math:d^x_{N_x} = h^x_{N_x-1/2} at :math:k = N_x and so on, or use half of these values as was done by [MoSu94]_. The discrete form of the term :math:\mathrm{i}\omega\mu_0\sigma\mathbf{E} in Equation :eq:fdomaindiff, with each component multiplied by the corresponding dual volume, becomes :math:\mathcal{S}_{k+1/2, l, m}\ E_{1, k+1/2, l, m}, :math:\mathcal{S}_{k, l+1/2, m}\ E_{2, k, l+1/2, m} and :math:\mathcal{S}_{k, l, m+1/2}\ E_{3, k, l, m+1/2} for the first, second and third components, respectively. Here :math:\mathcal{S} = -s\mu_0\sigma V is defined in terms of cell-averages. At the edges parallel to the x-axis, an averaging procedure similar to :eq:dualvolume gives .. math:: :label: sterm \mathcal{S}_{k+1/2, l, m} = &{\textstyle\frac{1}{4}}\left( \mathcal{S}_{k+1/2, l-1/2, m-1/2} + \mathcal{S}_{k+1/2, l+1/2, m-1/2} \right. \\ &+ \left. \mathcal{S}_{k+1/2, l-1/2, m+1/2} + \mathcal{S}_{k+1/2, l+1/2, m+1/2} \right) . :math:\mathcal{S}_{k, l+1/2, m} and :math:\mathcal{S}_{k, l, m+1/2} are defined in a similar way. The curl of :math:\mathbf{E} follows from path integrals around the edges that bound a face of a cell, drawn in :numref:Figure %s(a) . After division by the area of the faces, the result is a face-averaged value that can be positioned at the centre of the face, as sketched in :numref:Figure %s(b) . If this result is divided by :math:\mathrm{i}\omega\mu, the component of the magnetic field that is normal to the face is obtained. In order to find the curl of the magnetic field, the magnetic field components that are normal to faces are interpreted as tangential components at the faces of the dual volumes. For :math:E_1, this is shown in :numref:Figure %s . For the first component of Equation :eq:fdomaindiff on the edge :math:(k+1/2, l, m) connecting :math:(x_k, y_l, z_m) and :math:(x_{k+1}, y_l, z_m), the corresponding dual volume comprises the set :math:[x_k, x_{k+1}] \times [y_{l-1/2}, y_{l+1/2}] \times [z_{m-1/2}, z_{m+1/2}] having volume :math:V_{k+1/2,l,m}. The scaling by :math:\mu_r^{-1} at the face requires another averaging step because the material properties are assumed to be given as cell-averaged values. We define :math:\mathcal{M} = V\mu_r^{-1}, so .. math:: :label: mterm \mathcal{M}_{k+1/2, l+1/2, m+1/2} = h_{k+1/2}^x h_{l+1/2}^y h_{m+1/2}^z \mu_{r, k+1/2, l+1/2, m+1/2}^{-1} for a given cell :math:(k+1/2, l+1/2, m+1/2). An averaging step in, for instance, the z-direction gives .. math:: :label: mtermz \mathcal{M}_{k+1/2, l+1/2, m} = {\textstyle \frac{1}{2}} \left(\mathcal{M}_{k+1/2, l+1/2, m-1/2} + \mathcal{M}_{k+1/2, l+1/2, m+1/2} \right) at the face :math:(k+1/2, l+1/2, m) between the cells :math:(k+1/2, l+1/2, m-1/2) and :math:(k+1/2, l+1/2, m+1/2). Starting with :math:\mathbf{v}=\nabla \times \mathbf{E}, we have .. math:: :label: vterm v_{1, k, l+1/2, m+1/2} &= e_{l+1/2}^y\left(E_{3, k, l+1, m+1/2} - E_{3, k, l, m+1/2}\right) \\ &-e_{m+1/2}^z\left(E_{2, k, l+1/2, m+1} - E_{2, k, l+1/2, m}\right) , \\ v_{2, k+1/2, l, m+1/2} &= e_{m+1/2}^z\left(E_{1, k+1/2, l, m+1} - E_{1, k+1/2, l, m}\right) \\ &-e_{k+1/2}^x\left(E_{3, k+1, l, m+1/2} - E_{3, k, l, m+1/2}\right) , \\ v_{3, k+1/2, l+1/2, m} &= e_{k+1/2}^x\left(E_{2, k+1/2, l+1, m} - E_{1, k+1/2, l, m}\right) \\ &-e_{l+1/2}^y\left(E_{1, k+1/2, l+1, m} - E_{1, k+1/2, l, m}\right) . Here, .. math:: :label: eterm e_{k+1/2}^x = 1/h_{k+1/2}^x, \quad e_{l+1/2}^y = 1/h_{l+1/2}^y, \quad e_{m+1/2}^z = 1/h_{m+1/2}^z . Next, we let .. math:: :label: uterm u_{1,k,l+1/2,m+1/2} &= \mathcal{M}_{k,l+1/2,m+1/2} v_{1,k,l+1/2,m+1/2} , \\ u_{2,k+1/2,l,m+1/2} &= \mathcal{M}_{k+1/2,l,m+1/2} v_{2,k+1/2,l+1/2,m} , \\ u_{3,k+1/2,l+1/2,m} &= \mathcal{M}_{k+1/2,l+1/2,m} v_{3,k+1/2,l+1/2,m} . Note that these components are related to the magnetic field components by .. math:: :label: utermmag u_{1,k,l+1/2,m+1/2} &= \mathrm{i}\omega\mu_0 V_{k,l+1/2,m+1/2} H_{1,k+1/2,l,m+1/2} , \\ u_{2,k+1/2,l,m+1/2} &= \mathrm{i}\omega\mu_0 V_{k+1/2,l,m+1/2} H_{2,k+1/2,l,m+1/2} , \\ u_{3,k+1/2,l+1/2,m} &= \mathrm{i}\omega\mu_0 V_{k+1/2,l+1/2,m} H_{3,k+1/2,l+1/2,m} , where .. math:: :label: utermmagV V_{k,l+1/2,m+1/2} &= d_k^x h_{l+1/2}^y h_{m+1/2}^z , \\ V_{k+1/2,l,m+1/2} &= h_{k+1/2}^x d_l^y h_{m+1/2}^z , \\ V_{k+1/2,l+1/2,m} &= h_{k+1/2}^x h_{l+1/2}^y d_m^z . The discrete representation of the source term :math:\mathrm{i}\omega\mu_0\mathbf{J}_\mathrm{s}, multiplied by the appropriate dual volume, is .. math:: :label: discsource s_{1,k+1/2,l,m} &= \mathrm{i}\omega\mu_0 V_{k+1/2,l,m} J_{1,k+1/2,l,m} , \\ s_{2,k,l+1/2,m} &= \mathrm{i}\omega\mu_0 V_{k,l+1/2,m} J_{2,k,l+1/2,m} , \\ s_{3,k,l,m+1/2} &= \mathrm{i}\omega\mu_0 V_{k,l,m+1/2} J_{3,k,l,m+1/2} . Let the residual for an arbitrary electric field that is not necessarily a solution to the problem be defined as .. math:: :label: residualeq \mathbf{r} = V \left(\mathrm{i} \omega \mu_0 \mathbf{J}_\mathrm{s} + \mathrm{i}\omega\mu_0\sigma \mathbf{E} - \nabla \times \mu^{-1}_\mathrm{r} \nabla \times \mathbf{E}\right) . Its discretisation is .. math:: :label: discres r_{1,k+1/2,l,m} = ~&s_{1,k+1/2,l,m} + \mathcal{S}_{k+1/2,l,m} E_{1,k+1/2,l,m} \\ &-\left[e_{l+1/2}^y u_{3,k+1/2,l+1/2,m} - e_{l-1/2}^y u_{3,k+1/2,l-1/2,m]}\right.\\ &+\left[e_{m+1/2}^z u_{2,k+1/2,l,m+1/2} - e_{m-1/2}^z u_{2,k+1/2,l,m-1/2}\right] , \\ % r_{2,k,l+1/2,m} = ~&s_{2,k,l+1/2,m} + \mathcal{S}_{k,l+1/2,m} E_{2,k,l+1/2,m} \\ &-\left[e_{m+1/2}^z u_{1,k,l+1/2, m+1/2} - e_{m-1/2}^z u_{1,k,l+1/2,m-1/2]} \right. \\ &+\left[e_{k+1/2}^x u_{3,k+1/2,l+1/2,m} - e_{k-1/2}^x u_{3,k-1/2,l+1/2,m]}\right] , \\ % r_{3,k,l,m+1/2} = ~&s_{3,k,l,m+1/2} + \mathcal{S}_{k,l,m+1/2} E_{3,k,l,m+1/2} \\ &-\left[e_{k+1/2}^x u_{2,k+1/2,l,m+1/2} - e_{k-1/2}^x u_{2,k-1/2,m+1/2]}\right.\\ &+\left[e_{l+1/2}^y u_{1,k,l+1/2,m+1/2} - e_{l-1/2}^y u_{1,k,l-1/2,m+1/2}\right] . The weighting of the differences in :math:u_1, etc., may appear strange. The reason is that the differences have been multiplied by the local dual volume. As already mentioned, the dual volume for :math:E_{1,k,l,m} is shown in :numref:Figure %s(b) . For further details of the discretisation see [Muld06]_ or [Yee66]_. The actual meshing is done using discretize _ (part of the SimPEG _-framework). The coordinate system of discretize uses a coordinate system were positive z is upwards. The method is implemented in a matrix-free manner: the large sparse linear matrix that describes the discretised problem is never explicitly formed, only its action is evaluated on the latest estimate of the solution, thereby reducing storage requirements. Iterative Solvers ----------------- The multigrid method is an iterative (or relaxation) method and shares as such the underlying idea of iterative solvers. We want to solve the linear equation system .. math:: :label: system A \mathbf{x} = \mathbf{b} , where :math:A is the :math:n\times n system matrix and :math:x the unknown. If :math:v is an approximation to :math:x, then we can define two important measures. One is the error :math:e .. math:: :label: error \mathbf{e} = \mathbf{x} - \mathbf{v} , which magnitude can be measured by any standard vector norm, for instance the maximum norm and the Euclidean or 2-norm defined respectively, by .. math:: \|\mathbf{e}\|_\infty = \max_{1\leq j \leq n}|e_j| \quad \text{and} \quad \|\mathbf{e}\|_{2} = \sqrt{\sum_{j=1}^{n} e_{j}^{2}} . However, as the solution is not known the error cannot be computed either. The second important measure, however, is a computable measure, the residual :math:r (computed in :func:emg3d.solver.residual) .. math:: :label: residual \mathbf{r} = \mathbf{b} - A\mathbf{v} . Using Equation :eq:error we can rewrite Equation :eq:system as .. math:: A\mathbf{e} = \mathbf{b} - A\mathbf{v} , from which we obtain with Equation :eq:residual the *Residual Equation* .. math:: :label: reseq A\mathbf{e} = \mathbf{r} . The *Residual Correction* is given by .. math:: :label: rescorr \mathbf{x} = \mathbf{v}+\mathbf{e} . The Multigrid Method -------------------- .. note:: If you have never heard of multigrid methods before you might want to read through the :doc:mgwhat-section. Multigrid is a numerical technique for solving large, often sparse, systems of equations, using several grids at the same time. An elementary introduction can be found in [BrHM00]_. The motivation for this approach follows from the observation that it is fairly easy to determine the local, short-range behaviour of the solution, but more difficult to find its global, long-range components. The local behaviour is characterized by oscillatory or rough components of the solution. The slowly varying smooth components can be accurately represented on a coarser grid with fewer points. On coarser grids, some of the smooth components become oscillatory and again can be easily determined. The following constituents are required to carry out multigrid. First, a sequence of grids is needed. If the finest grid on which the solution is to be found has a constant grid spacing :math:h, then it is natural to define coarser grids with spacings of :math:2h, :math:4h, etc. Let the problem on the finest grid be defined by :math:A^h \mathbf{x}^h = \mathbf{b}^h. The residual is :math:\mathbf{r}^h = \mathbf{b}^h - A^h \mathbf{x}^h (see the corresponding function :func:emg3d.solver.residual, and for more details also the function :func:emg3d.core.amat_x). To find the oscillatory components for this problem, a smoother or relaxation scheme is applied. Such a scheme is usually based on an approximation of :math:A^h that is easy to invert. After one or more smoothing steps (see the corresponding function :func:emg3d.solver.smoothing), say :math:\nu_1 in total, convergence will slow down because it is generally difficult to find the smooth, long-range components of the solution. At this point, the problem is mapped to a coarser grid, using a restriction operator :math:\tilde{I}^{2h}_h (see the corresponding function :func:emg3d.solver.restriction, and for more details, the functions :func:emg3d.core.restrict_weights and :func:emg3d.core.restrict. On the coarse-grid, :math:\mathbf{b}^{2h} = \tilde{I}^{2h}_h\mathbf{r}^h. The problem :math:\mathbf{r}^{2h} = \mathbf{b}^{2h} - A^{2h} \mathbf{x}^{2h} = 0 is now solved for :math:\mathbf{x}^{2h}, either by a direct method if the number of points is sufficiently small or by recursively applying multigrid. The resulting approximate solution needs to be interpolated back to the fine grid and added to the solution. An interpolation operator :math:I^h_{2h}, usually called prolongation in the context of multigrid, is used to update :math:\mathbf{x}^h := \mathbf{x}^h + I^h_{2h}\mathbf{x}^{2h} (see the corresponding function :func:emg3d.solver.prolongation). Here :math:I^h_{2h}\mathbf{x}^{2h} is called the coarse-grid correction. After prolongation, :math:\nu_2 additional smoothing steps can be applied. This constitutes one multigrid iteration. So far, we have not specified the coarse-grid operator :math:A^{2h}. It can be formed by using the same discretisation scheme as that applied on the fine grid. Another popular choice, :math:A^{2h} = \tilde{I}^{2h}_h A^h I^h_{2h}, has not been considered here. Note that the tilde is used to distinguish restriction of the residual from operations on the solution, because these act on elements of different function spaces. If multigrid is applied recursively, a strategy is required for moving through the various grids. The simplest approach is the V-cycle shown in :numref:Figure %s  for the case of four grids. Here, the same number of pre- and post-smoothing steps is used on each grid, except perhaps on the coarsest. In many cases, the V-cycle does not solve the coarse-grid equations sufficiently well. The W-cycle, shown in :numref:Figure %s , will perform better in that case. In a W-cycle, the number of coarse-grid corrections is doubled on subsequent coarser grids, starting with one coarse-grid correction on the finest grid. Because of its cost, it is often replaced by the F-cycle :numref:(Figure %s) . In the F-cycle, the number of coarse-grid corrections increases by one on each subsequent coarser grid. .. figure:: ../_static/Muld06_Fig3.png :scale: 100 % :align: center :alt: V-Cycle :name: Muld06_Fig3 V-cycle with :math:\nu_1 pre-smoothing steps and :math:\nu_2 post-smoothing steps. On the coarsest grid, :math:\nu_c smoothing steps are applied or an exact solver is used. The finest grid has a grid spacing :math:h and the coarsest :math:8h. A single coarse-grid correction is computed for all grids but the coarsest. .. figure:: ../_static/Muld06_Fig4.png :scale: 100 % :align: center :alt: W-Cycle :name: Muld06_Fig4 W-cycle with :math:\nu_1 pre-smoothing steps and :math:\nu_2 post-smoothing steps. On each grid except the coarsest, the number of coarse-grid corrections is twice that of the underlying finer grid. .. figure:: ../_static/Muld06_Fig5.png :scale: 100 % :align: center :alt: F-Cycle :name: Muld06_Fig5 F-cycle with :math:\nu_1 pre-smoothing steps and :math:\nu_2 post-smoothing steps. On each grid except the coarsest, the number of coarse-grid corrections increases by one compared to the underlying finer grid. One reason why multigrid methods may fail to reach convergence is strong anisotropy in the coefficients of the governing partial differential equation or severely stretched grids (which has the same effect as anisotropy). In that case, more sophisticated smoothers or coarsening strategies may be required. Two strategies are currently implemented, *semicoarsening* and *line relaxation*, which can be used on their own or combined. Semicoarsening is when the grid is only coarsened in some directions. Line relaxation is when in some directions the whole gridlines of values are found simultaneously. If slow convergence is caused by just a few components of the solution, a Krylov subspace method can be used to remove them. In this way, multigrid is accelerated by a Krylov method. Alternatively, multigrid might be viewed as a preconditioner for a Krylov method. .. _Gauss-Seidel: Gauss-Seidel  The smoother implemented in emg3d is a Gauss-Seidel smoother. The Gauss-Seidel method solves the linear equation system :math:A \mathbf{x} = \mathbf{b} iteratively using the following method: .. math:: :label: gaussseidel \mathbf{x}^{(k+1)} = L_*^{-1} \left(\mathbf{b} - U \mathbf{x}^{(k)} \right) \ , where :math:L_* is the lower triangular component, and :math:U the strictly upper triangular component, :math:A = L_* + U. On the coarsest grid it acts as direct solver, whereas on the finer grid it acts as a smoother with only few iterations. See the function :func:emg3d.solver.smoothing, and for more details, the functions :func:emg3d.core.gauss_seidel, :func:emg3d.core.gauss_seidel_x, :func:emg3d.core.gauss_seidel_y, :func:emg3d.core.gauss_seidel_z, and also :func:emg3d.core.blocks_to_amat. Choleski factorisation  The actual solver of the system :math:A\mathbf{x}=\mathbf{b} is a non-standard Cholesky factorisation without pivoting for a symmetric, complex matrix :math:A tailored to the problem of the multigrid solver, using only the main diagonal and five lower off-diagonals of the banded matrix :math:A. The result is the same as simply using, e.g., :func:numpy.linalg.solve, but faster for the particular use-case of this code. See :func:emg3d.core.solve` for more details.