Mimetic interpolation

Last updated August 19, 2023

In mathematics, mimetic interpolation is a method for interpolating differential forms. In contrast to other interpolation methods, which estimate a field at a location given its values on neighboring points, mimetic interpolation estimates the field's $k$ -form given the field's projection on neighboring grid elements. The grid elements can be grid points as well as cell edges or faces, depending on $k=0,1,2,\cdots$ .

Interpolation of integrated forms

Let $\omega ^{k}$ be a differential $k$ -form, then mimetic interpolation is the linear combination

{\bar {\omega }}^{k}=\sum _{i}\left(\int _{M_{i}}\omega ^{k}\right)\phi _{i}^{k}

where ${\bar {\omega }}^{k}$ is the interpolation of $\omega ^{k}$ , and the coefficients $\int _{M_{i}}\omega ^{k}$ represent the strengths of the field on grid element $M_{i}$ . Depending on $k$ , $M_{i}$ can be a node ( $k=0$ ), a cell edge ( $k=1$ ), a cell face ( $k=2$ ) or a cell volume ( $k=3$ ). In the above, the $\phi _{i}^{k}$ are the interpolating $k$ -forms, which are centered on $M_{i}$ and decay away from $M_{i}$ in a way similar to the tent functions. Examples of $\phi _{i}^{k}$ are the Whitney forms^[1]^[2] for simplicial meshes in $n$ dimensions.

An important advantage of mimetic interpolation over other interpolation methods is that the field strengths $\int _{M_{i}}\omega ^{k}$ are scalars and thus coordinate system invariant.

Interpolating forms

In many cases it is desirable for the interpolating forms $\phi _{i}^{k}$ to pick the field's strength on particular grid elements without interference from other $\phi _{j}^{k}$ . This allows one to assign field values to specific grid elements, which can then be interpolated in-between. A common case is linear interpolation for which the interpolating functions ( $0$ -forms) are zero on all nodes except on one, where the interpolating function is one. A similar construct can be applied to mimetic interpolation

$\int _{M_{j}}\phi _{i}^{k}=\delta _{ij}.$

That is, the integral of $\phi _{i}^{k}$ is zero on all cell elements, except on $M_{i}$ where the integral returns one. For $k=0$ this amounts to $\phi _{i}^{0}(x_{j})=\delta _{ij}$ where $x_{j}$ is a grid point. For $k=1$ the integral is over edges and hence the integral $\int _{M_{j}}\phi _{i}^{1}$ is zero expect on edge $M_{i}$ . For $k=2$ the integral is over faces and for $k=3$ over cell volumes.

Conservation properties

Mimetic interpolation respects the properties of differential forms. In particular, Stokes' theorem

\int _{M}{\overline {d\omega ^{k}}}=\int _{\partial M}{\bar {\omega }}^{k}

is satisfied with ${\overline {d\omega ^{k}}}$ denoting the interpolation of $d\omega ^{k}$ . Here, $d$ is the exterior derivative, $M$ is any manifold of dimensionality $k$ and $\partial M$ is the boundary of $M$ . This confers to mimetic interpolation conservation properties, which are not generally shared by other interpolation methods.

Commutativity between the interpolation operator and the exterior derivative

De Rham complex. Top: the spaces of differential forms in three dimensions. Bottom: the corresponding discretized versions of the differential forms obtained after interpolation. The commutativity condition ensures that the dashed and dash-dotted paths give the same result for each de Rham cell.

To be mimetic, the interpolation must satisfy

I_{k+1}(d\omega ^{k})=d(I_{k}\omega ^{k})

where $I_{k}$ is the interpolation operator of a $k$ -form, i.e. ${\bar {\omega }}^{k}=I_{k}\omega ^{k}$ . In other words, the interpolation operators and the exterior derivatives commute.^[3] Note that different interpolation methods are required for each type of form ( $k=0,1,\cdots$ ), $I_{k+1}\neq I_{k}$ . The above equation is all that is needed to satisfy Stokes' theorem for the interpolated form

\int _{M}{\overline {d\omega ^{k}}}=\int _{M}I_{k+1}(d\omega ^{k})=\int _{M}d(I_{k}\omega ^{k})=\int _{\partial M}I_{k}\omega ^{k}=\int _{\partial M}{\bar {\omega }}^{k}.

Other calculus properties derive from the commutativity between interpolation and $d$ . For instance, $d^{2}\omega =0$ ,

\int _{M}I_{k+2}(d^{2}\omega ^{k})=\int _{M}d(I_{k+1}d\omega ^{k})=\int _{\partial M}I_{k+1}d\omega ^{k}=\int _{\partial M}d(I_{k}\omega ^{k})=0.

The last step gives zero since $\int _{\partial M}d(\cdot )=0$ when integrated over the boundary $\partial M$ .

Projection

The interpolated ${\bar {\omega }}^{k}$ is often projected onto a target, $k$ -dimensional, oriented manifolds $T$ ,

\int _{T}{\bar {\omega }}^{k}=\sum _{i}\left(\int _{M_{i}}\omega ^{k}\right)\left(\int _{T}\phi _{i}^{k}\right).

For $k=0$ the target is a point, for $k=1$ it is a line, for $k=2$ an area, etc.

Applications

Many physical fields can be represented as $k$ -forms. When discretizing fields in numerical modeling, each type of $k$ -form often acquires its own staggering in accordance with numerical stability requirements, e.g. the need to prevent the checkerboard instability.^[4] This led to the development of the exterior finite element^[5] and discrete exterior calculus methods, both of which rely on a field discretization that are compatible with the field type.

The table below lists some examples of physical fields, their type, their corresponding form and interpolation interpolation method, as well as software that can be leveraged to interpolate, remap or regrid the field:


field example	field type	k-form equivalent	target	staggering	Interpolation method (example)	example of software
temperature	scalar	0-form	point	nodal	bilinear, trilinear	ESMF^[6]
electric field	vector	1-form	line	edge	edge	MINT^[7]
magnetic field	pseudo-vector	2-form	area	face	face	MINT
density	pseudo-scalar	3-form	volume	cell	area weighted, conservative	SCRIP,^[8] ESMF

Example

Consider quadrilateral cells in two dimensions with their node indexed $i=0,1,2,3$ in the counterclockwise direction. Further, let $\xi _{1}$ and $\xi _{2}$ be the parametric coordinates of each cell ( $0\leq \xi _{1},\xi _{2}\leq 1$ ). Then

{\begin{aligned}\phi _{0}^{0}&=(1-\xi _{1})(1-\xi _{2})\\[0.6ex]\phi _{1}^{0}&=\xi _{1}(1-\xi _{2})\\[0.6ex]\phi _{2}^{0}&=\xi _{1}\xi _{2}\\[0.6ex]\phi _{3}^{0}&=(1-\xi _{1})\xi _{2}.\end{aligned}}

are the bilinear interpolating forms of $I_{0}$ in the unit square ( $\xi _{1},\xi _{2}$ ). The corresponding $I_{1}$ edge interpolating forms^[9]^[10] are

{\begin{aligned}\phi _{0}^{1}&=(1-\xi _{2})d\xi _{1}\\[0.6ex]\phi _{1}^{1}&=\xi _{1}d\xi _{2}\\[0.6ex]\phi _{2}^{1}&=\xi _{2}d\xi _{1}\\[0.6ex]\phi _{3}^{1}&=(1-\xi _{1})d\xi _{2},\end{aligned}}

The four vector fields which are dual to the four edge interpolating forms attached to a quadrilateral cell. The vector fields are strongest on their supporting edge and decrease to zero towards the opposite edge. Note how the vector fields bend to enforce perpendicularity with respect to edges that are adjacent to the supporting edge. PhiEdge2D.svg — The four vector fields which are dual to the four edge interpolating forms attached to a quadrilateral cell. The vector fields are strongest on their supporting edge and decrease to zero towards the opposite edge. Note how the vector fields bend to enforce perpendicularity with respect to edges that are adjacent to the supporting edge.

were we assumed the edges to be indexed in counterclockwise direction and with the edges pointing to the east and north. At lowest order, there is only one interpolating form for $I_{2}$ ,

\phi _{0}^{2}=d\xi _{1}\wedge d\xi _{2},

where $\wedge$ is the wedge product.

We can verify that the above interpolating forms satisfy the mimetic conditions $I_{1}d\omega ^{0}=d(I_{0}\omega ^{0})$ and $d(I_{1}\omega ^{1})=I_{2}d\omega ^{1}$ . Take for instance,

{\begin{aligned}d(I_{0}\omega ^{0})&=d\left(f_{0}(1-\xi _{1})(1-\xi _{2})+f_{1}\xi _{1}(1-\xi _{2})+f_{2}\xi _{1}\xi _{2}+f_{3}(1-\xi _{1})\xi _{2}\right)\\[0.6ex]&=(f_{1}-f_{0})(1-\xi _{2})d\xi _{1}+(f_{2}-f_{1})\xi _{1}d\xi _{2}+(f_{2}-f_{3})\xi _{2}d\xi _{1}+(f_{3}-f_{0})(1-\xi _{1})d\xi _{2}\\[0.6ex]&=(f_{1}-f_{0})\phi _{0}^{1}+(f_{2}-f_{1})\phi _{1}^{1}+(f_{2}-f_{3})\phi _{2}^{1}+(f_{3}-f_{0})\phi _{3}^{1}\\[0.6ex]&=I_{1}d\omega ^{0}\end{aligned}}

where $f_{0}=\omega ^{0}(0,0)$ , $f_{1}=\omega ^{0}(1,0)$ , $f_{2}=\omega ^{0}(1,1)$ and $f_{3}=\omega ^{0}(0,1)$ are the field values evaluated at the corners of the quadrilateral in the unit square space. Likewise, we have

{\begin{aligned}d(I_{1}\omega ^{1})&=d(g_{0}(1-\xi _{2})d\xi _{1}+g_{1}\xi _{1}d\xi _{2}+g_{2}\xi _{2}d\xi _{1}+g_{3}(1-\xi _{1})d\xi _{2})\\[0.6ex]&=(g_{0}+g_{1}-g_{2}-g_{3})d\xi _{1}\wedge d\xi _{2}\\[0.6ex]&=I_{2}d\omega ^{1}\end{aligned}}

with $g_{i}=\int _{M_{i}}\omega ^{1}$ , $i\in \{0,1,2,3\}$ , being the 1-form $\omega ^{1}$ projected onto edge $i$ . Note that $g_{i}$ is also known as the pullback. If $F:\mathbb {R} \rightarrow \mathbb {R} ^{2}$ is the map that parametrizes edge $i$ , $x=F_{i}(t)$ , $0\leq t\leq 1$ , then $g_{i}=\int {F_{i}}^{*}\omega ^{1}$ where the integration is performed in $t$ space. Consider for instance edge $i=2$ , then $F_{2}(t)=x_{3}+t(x_{2}-x_{3})$ with $x_{2}$ and $x_{3}$ denoting the start and points. For a general 1-form $\omega ^{1}=a(x)d\xi _{1}+b(x)d\xi _{2}$ , one gets $g_{2}=\int _{0}^{1}\left(a(t){\frac {\partial \xi _{1}}{\partial t}}+b(t){\frac {\partial \xi _{2}}{\partial t}}\right)dt$ .

Related Research Articles

<span class="mw-page-title-main">Wave equation</span> Differential equation important in physics

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics – such as mechanical waves or electromagnetic waves. It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media.

<span class="mw-page-title-main">Fourier transform</span> Mathematical transform that expresses a function of time as a function of frequency

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into terms of the intensity of its constituent pitches.

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form:

In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.

In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

In spectroscopy, the Autler–Townes effect, is a dynamical Stark effect corresponding to the case when an oscillating electric field is tuned in resonance to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line. The AC Stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes.

An $- superprocess,, within mathematics probability theory is a stochastic process on that is usually constructed as a special limit of near-critical branching diffusions.$

In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

Filtering in the context of large eddy simulation (LES) is a mathematical operation intended to remove a range of small scales from the solution to the Navier-Stokes equations. Because the principal difficulty in simulating turbulent flows comes from the wide range of length and time scales, this operation makes turbulent flow simulation cheaper by reducing the range of scales that must be resolved. The LES filter operation is low-pass, meaning it filters out the scales associated with high frequencies.

The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of stochastic processes. Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system.

Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad $, where is a pair of real null vectors and is a pair of complex null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature$

Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Frønsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang–Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.

References

↑ Whitney, Hassler (1957). Geometric Integration Theory. Dover Books on Mathematics.
↑ Hiptmair, R (2022-06-12). "Higher Order Whitney Forms". Progress in Electromagnetics Research. 32: 271–299. doi:10.2528/PIER00080111.
↑ Pletzer, Alexander; Behrens, Erik; Little, Bill (2022-06-27). "MINT". Proceedings of the Platform for Advanced Scientific Computing Conference. Basel Switzerland: ACM. pp. 1–7. doi: 10.1145/3539781.3539786 . ISBN 978-1-4503-9410-9.
↑ Trottenberg, Ulrich; Oosterlee, Cornelius W.; Schüller, Anton (2001). Multigrid. Academic Press. p. 314.
↑ Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar (2022-06-12). "Finite element exterior calculus, homological techniques, and applications". Acta Numerica. 15: 1–155. doi:10.1017/S0962492906210018. S2CID 122763537.
↑ "Earth System Modeling Framework Regridding".{{cite web}}: CS1 maint: url-status (link)
↑ "Mimetic Interpolation on the Sphere". GitHub . 4 March 2022. Retrieved 2022-06-09.{{cite web}}: CS1 maint: url-status (link)
↑ "SCRIP". GitHub . 11 April 2022. Retrieved 2022-06-09.
↑ Pletzer, Alexander; Fillmore, David (2015-12-01). "Conservative interpolation of edge and face data on n dimensional structured grids using differential forms". Journal of Computational Physics. 302: 21–40. Bibcode:2015JCoPh.302...21P. doi:10.1016/j.jcp.2015.08.029. ISSN 0021-9991.
↑ Pletzer, Alexander; Hayek, Wolfgang (2019-01-01). "Mimetic Interpolation of Vector Fields on Arakawa C/D Grids". Monthly Weather Review. 147 (1): 3–16. Bibcode:2019MWRv..147....3P. doi:10.1175/MWR-D-18-0146.1. ISSN 0027-0644. S2CID 125214770.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Whitney, Hassler (1957). Geometric Integration Theory. Dover Books on Mathematics.

[2] Hiptmair, R (2022-06-12). "Higher Order Whitney Forms". Progress in Electromagnetics Research. 32: 271–299. doi:10.2528/PIER00080111.

[3] Pletzer, Alexander; Behrens, Erik; Little, Bill (2022-06-27). "MINT". Proceedings of the Platform for Advanced Scientific Computing Conference. Basel Switzerland: ACM. pp. 1–7. doi: 10.1145/3539781.3539786 . ISBN 978-1-4503-9410-9.

[4] Trottenberg, Ulrich; Oosterlee, Cornelius W.; Schüller, Anton (2001). Multigrid. Academic Press. p. 314.

[5] Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar (2022-06-12). "Finite element exterior calculus, homological techniques, and applications". Acta Numerica. 15: 1–155. doi:10.1017/S0962492906210018. S2CID 122763537.

[6] "Earth System Modeling Framework Regridding".{{cite web}}: CS1 maint: url-status (link)

[7] "Mimetic Interpolation on the Sphere". GitHub . 4 March 2022. Retrieved 2022-06-09.{{cite web}}: CS1 maint: url-status (link)

[8] "SCRIP". GitHub . 11 April 2022. Retrieved 2022-06-09.

[9] Pletzer, Alexander; Fillmore, David (2015-12-01). "Conservative interpolation of edge and face data on n dimensional structured grids using differential forms". Journal of Computational Physics. 302: 21–40. Bibcode:2015JCoPh.302...21P. doi:10.1016/j.jcp.2015.08.029. ISSN 0021-9991.

[10] Pletzer, Alexander; Hayek, Wolfgang (2019-01-01). "Mimetic Interpolation of Vector Fields on Arakawa C/D Grids". Monthly Weather Review. 147 (1): 3–16. Bibcode:2019MWRv..147....3P. doi:10.1175/MWR-D-18-0146.1. ISSN 0027-0644. S2CID 125214770.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]