Problem Steady single-phase flow can be described by: a p = f - PowerPoint PPT Presentation

M ULTISCALE B ASIS O PTIMIZATION FOR D ARCY F LOW James M. Rath work with Todd Arbogast in the Center for Subsurface Modeling / ICES at the University of Texas at Austin

Problem Steady single-phase flow can be described by: −∇ · a ∇ p = f This PDE can be approximated in a number of ways. In this talk, we will focus on applying 2D piecewise linear finite elements on triangles.

Challenges Steady single-phase flow can be described by: −∇ · a ∇ p = f The coefficient a depends on the permeability. The permeability is often geostatistically generated at high resolution. It can be very heterogeneous. Together these conditions make for an ill-conditioned and computationally expensive problem.

Goal Calculate the approximation at the full resolution of the problem capturing all the details of the flow.

Method Calculate the approximation at the full resolution of the problem capturing all the details of the flow. We propose a new iterative method for solving the problem. The principle per iteration costs are only a coarse problem solve and a fine-scale residual evaluation. The principle start-up costs are a static condensation of subgrid DOFs into the coarse problem, and a coarse solve.

Teaser Calculate the approximation at the full resolution of the problem capturing all the details of the flow. We propose a new iterative method for solving the problem. The principle per iteration costs are only a coarse problem solve and a fine-scale residual evaluation. The principle start-up costs are a static condensation of subgrid DOFs into the coarse problem, and a coarse solve. The method has a number of unusual features: • number of iterations appears independent of the fine-scale resolution • number of iterations appears independent of the heterogeneity of the coefficient a (i.e., the permeability) • global, monotone, superlinear convergence

Fine-scale degrees of freedom Let V be piecewise linear functions on a fine mesh. Degrees of freedom (DOFs) are shown above.

Goal Solve Ap = f with pressure p ∈ V , data f ∈ V ′ , and matrix A : V → V ′ .

Multiscale degrees of freedom From V , take out coarse edge DOFs to get V H .

Multiscale degrees of freedom: corner shape From V , take out coarse edge DOFs to get V H . Fix shapes for corner DOFs using the usual multiscale basis shapes.

Multiscale problem Solve A H p H = f H for p H ∈ V H where • I H : V H → V is the natural inclusion, • A H = I T H AI H is the coarsened matrix, • and f H = I T H f is the coarsened data. Note that these follow from the Galerkin procedure applied to V H ⊂ V .

Multiscale solution quality Solve A H p H = f H for p H ∈ V H where • I H : V H → V is the natural inclusion, • A H = I T H AI H is the coarsened matrix, • and f H = I T H f is the coarsened data. Note that these follow from the Galerkin procedure applied to V H ⊂ V . The multiscale solution p H is a pretty good approximation for p : we use almost all the same DOFs and just take out a few. (And multiscale problems are just as easy to solve as coarse ones.)

Oops Solve A H p H = f H for p H ∈ V H where • I H : V H → V is the natural inclusion, • A H = I T H AI H is the coarsened matrix, • and f H = I T H f is the coarsened data. Note that these follow from the Galerkin procedure applied to V H ⊂ V . The multiscale solution p H is a pretty good approximation for p : we use almost all the same DOFs and just take out a few. (And multiscale problems are just as easy to solve as coarse ones.) But almost always p � = p H , and — even worse — p / ∈ V H . That is, we couldn’t possibly get p as the result of a multiscale problem no matter how hard we try; we’re missing some degrees of freedom.

Supplemented multiscale degrees of freedom From V H , add back in some edge shapes to form V β .

Supplemented multiscale degrees of freedom: edge shape From V H , add back in some edge shapes to form V β . Fix shapes along each coarse edge.

Supplemented multiscale degrees of freedom: another edge shape From V H , add back in some edge shapes to form V β . Fix shapes along each coarse edge. Pick any shape you like ...

Supplemented multiscale degrees of freedom: another edge shape From V H , add back in some edge shapes to form V β . Fix shapes along each coarse edge. Pick any shape you like ...

Supplemented multiscale degrees of freedom: another edge shape From V H , add back in some edge shapes to form V β . Fix shapes along each coarse edge. Pick any shape you like ... But just pick one (for each coarse edge) for any given computation.

Supplemented multiscale degrees of freedom: parameterized family From V H , add back in some edge shapes to form V β Fix shapes along each coarse edge. Pick any shape you like ... Record the heights (of the shapes along coarse edges) in a list β .

Supplemented multiscale problem As before, solve A β p β = f β for p β ∈ V β with • I β : V β → V as the natural inclusion, • A β = I T β AI β as the coarsened matrix, • and f β = I T β f as the coarsened data.

A light at the end of the tunnel? As before, solve A β p β = f β for p β ∈ V β with • I β : V β → V as the natural inclusion, • A β = I T β AI β as the coarsened matrix, • and f β = I T β f as the coarsened data. Again, usually p � = p β and p / ∈ V β . That is, for any particular β we’re still missing some degrees of freedom. But at least now V = � V β . β That is, by adjusting β we can find a V β that accomodates p .

Motivation As before, solve A β p β = f β for p β ∈ V β with • I β : V β → V as the natural inclusion, • A β = I T β AI β as the coarsened matrix, • and f β = I T β f as the coarsened data. Again, usually p � = p β and p / ∈ V β . That is, for any particular β we’re still missing some degrees of freedom. But at least now V = � V β . β That is, by adjusting β we can find a V β that accomodates p . We have traded solving a linear problem for solving a non-linear one. But this is sensible because we trade a large linear system for a small non-linear one.

An algorithm • Pick a β and solve the multiscale problem: A β p β = f β .

An algorithm • Pick a β and solve the multiscale problem: A β p β = f β . • Calculate the fine-scale residual: r β = f − AI β p β .

An algorithm: goal • Pick a β and solve the multiscale problem: A β p β = f β . • Calculate the fine-scale residual: r β = f − AI β p β . • Find β so that r β = 0 .

An algorithm • Pick a β and solve the multiscale problem: A β p β = f β . • Calculate the fine-scale residual: r β = f − AI β p β . • Find β so that r β = 0 . • Use Newton’s method!

An algorithm • Pick a β and solve the multiscale problem: A β p β = f β . • Calculate the fine-scale residual: r β = f − AI β p β . • Compute step: � † r β . r ′ � δβ = − β • Update: β ← β + δβ . • Repeat as needed.

Jacobian computation is too much • Pick a β and solve the multiscale problem: A β p β = f β . • Calculate the fine-scale residual: r β = f − AI β p β . • Compute step: � † r β . r ′ � δβ = − β • Update: β ← β + δβ . • Repeat as needed. Explicitly computing r ′ β is expensive! (So approximate it instead.)

Quick and easy example We apply this method to solving the flow problem on the unit square with constant permeability. There are sources at the bottom-left and top-right of the domain (a quarter five-spot) and no gravity. A 2 × 2 coarse grid is used with a 6 × 6 subgrid (corresponding to a 12 × 12 fine grid). 1 2 3 4 -2.5 log 10 � residual � -5 -7.5 -10 -12.5 -15 -17.5 iteration number Note the axis scales: we get quadratic convergence — on a linear problem!

Problem size insensitivity: Fix H and let h → 0 10 # of Newton iterations 8 6 4 2 10 15 20 30 50 1 /h If we solve the flow problem with the same coarse grid (fixing H ), but let the subgrid get finer and finer (letting h → 0 ), then we only need a constant number of Newton iterations regardless of h .

Problem size insensitivity: Fix H/h and let h → 0 10 # of Newton iterations 8 6 4 2 10 15 20 30 1 /h If we solve the flow problem with the same subgrid (fixing H/h ), but let the overall grid get finer and finer (let H → 0 and h → 0 ), then we only need a constant number of Newton iterations regardless of h .

Kick it up a notch: a more heterogeneous permeability The above graphic plots the variation from a statistically generated perme- ability field. Darker areas indicate low permeability; lighter areas indicate high permeability. The permeabilities span about five orders of magnitude ( 10 5 ).

Problem size insensitivity: Fix H and let h → 0 25 20 # of Newton iterations # of Newton 15 iterations: maximum 10 median minimum 5 10 15 20 30 50 1 /h Again, fix H but let h → 0 . At each h , grab several statistical subsamples of the (very-fine) heterogeneous permeability field. Roughly a constant number of Newton iterations is needed regardless of h .

Heterogeneity insensitivity?: κ max /κ min → + ∞ 30 25 # of Newton iterations 20 1 /h = 8 1 /h = 12 15 1 /h = 16 1 /h = 20 10 5 2 4 6 8 10 log 10 ( κ max /κ min ) Take a subsample of the heterogeneous permeability field and rescale it so that κ max /κ min gets large.