High-contrast homogenization: Analytic.DoubledMinimizerSelection #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Measurable selection of the doubled variational minimizer on a general cell #
The doubled variational problem defining the coarse block of s.introduction
minimizes the doubled energy over the affine space P + \Lpoto(U) × \Lsolo(U),
whose correction space does not depend on the coefficient field. The minimizer is
therefore a genuine function of the field, and this file proves that it is a
measurable one, on an arbitrary bounded open cell of positive volume.
The argument is a Galerkin selection. The correction space is the closure of a
separable submodule of pointwise potential/solenoidal corrections, so along a
fixed dense sequence of generators the variational quantity is the infimum of
countably many competitor energies, each a measurable function of the field.
Choosing, for each tolerance, the first generator whose energy is within that
tolerance of the infimum gives a measurable countably valued approximation, and
the energy gap controls the Hilbert distance to the minimizer, so the
approximations converge to it pointwise. A pointwise limit of strongly
measurable maps into the ambient L² space is strongly measurable, and so is
every fixed continuous readout of it.
Everything here is stated on one quantitative ellipticity slice of the cell and for an arbitrary measurable parameter space whose localized smooth entry tests are measurable; the passage to the coefficient space is a separate step.
The doubled problem of a general cell on one ellipticity slice #
The doubled operator system of a cell of positive volume, on one quantitative ellipticity slice, with the canonical potential/solenoidal correction space.
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- One or more equations did not get rendered due to their size.
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The doubled variational problem of a cell of positive volume, realized in
the ambient L² space on one quantitative ellipticity slice.
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The minimizer of the cell's doubled problem is the affine Hilbert minimizer of its energy over the canonical correction space.
The affine competitor attached to a correction generator.
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The competitor attached to a generator is the ambient image of its affine block state.
The competitor attached to a generator is admissible for the cell's doubled problem.
The quadratic energy of a generator competitor is the block energy average of its pointwise representative.
The variational quantity of the cell is the minimized quadratic energy of its doubled problem.
The variational quantity of the cell is the quadratic energy of the minimizer of its doubled problem.
The Galerkin selection over a measurable parameter space #
The slice datum attached to a parameter.
Equations
- HCPolySupport.HighContrast.Selection.sliceOf hSlice w = ⟨(A w).toFun, ⋯⟩
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The L² realization of the block coefficient field is measurable in the
parameter whenever the localized smooth entry tests are.
The competitor energies along a fixed generator sequence are measurable in the parameter.
For each tolerance and each parameter some competitor along a dense generator sequence is within that tolerance of the variational quantity.
The selected competitors converge, at every parameter, to the minimizer of the cell's doubled problem.
The minimizer of the cell's doubled variational problem is a strongly measurable function of the parameter. The Galerkin selection along a fixed dense generator sequence produces countably valued measurable approximations, and the energy gap forces them to converge to the minimizer.