Adaptive algorithms benchmarking - CMMSE2013

Introduction Motivation hp-adaptivity with Rothe’s method Benchmarking Conclusions

Computational Comparison of Various FEM Adaptivity Approaches

Lukas Korous, Pavel Solın, Pavel Karban, Frantisek Mach, Pavel Kus,

Lukas Korous, Ivo Dolezel, et al

Department of Theory of Electrical Engineering

Faculty of Electrical Engineering

University of West Bohemia

Czech Republic

May 22, 2013

1/55 L. Korous, P. Solin, P. Karban, F. Mach, P. Kus, I. Dolezel: Computational Comparison of Various FEM Adaptivity Approaches


Introduction

Context



Introduction

Context

General purpose (hp-)FEM (& (hp-)DG) software

non-linear, transient, coupled, real/complex, ... problemsmultitude of settings of automatic adaptivity

1 understanding effects of adaptivity settings on the whole computation

2 automatic self-tuning of adaptivity algorithm for arbitrary problem

Automatic solution of general engineering-level problems

1 accurately (error estimation - adaptivity)

2 fast (effective implementation, optimization, parallelization)



Introduction

Goals

very good error information (important on its own) ... OK

small requirements on the input mesh, on user experience ... OK

only do automatic (local) refinements when beneficial ...∼ OK

use the ”best” adaptivity type (”best” - speed, problem size, ...), with ”best” settings ...∼ OK

Mid-goal: Adaptivity benchmarking

measure all kinds of performance aspects (speed, memory, parallelization)

improve performance / tune implementation according to data

implement robust heuristics to automatically self-optimize the implemented adaptive algorithms

use gained experience with implementing and optimizing various hp-adaptive aspects in 2D and evolve the

3D code



Motivation

Theory⇐⇒ reality

Why do adaptivity in the first place?

While achieving comparable resolution as without adaptivity, it (generally) leads to:

reduced algebraic problems sizes (DOFs)

reduced mesh size (Elements * Poly order)

The previous is related to:

reduced algebraic structures memory demands (nnz, ...) ?

reduced algebraic solver memory consumption ?

reduced mesh size (bytes) ?

reduced size of utility data structures ?

CPU time ?

Aim: inspect / test the relationship.



hp-adaptivity with Rothe’s method

A-posteriori mesh(space)-adaptivity with Rothe’s method

Solving Lu = f on a mesh Th1.

Do the following until an error condition is satisfied: {1 solve Lu = f on a mesh Thn

2 inspect the error condition

if True=⇒ End

3 determine the refinement of Thn

mapping e → True/False

(optional) determine how to refine the identified elements

4 refine the appropriate elements and obtain Thn+1

}




Determination where to refine the mesh

Error calculation

mapping e → True/False in our implementation:

1 error estimation based on coarse and reference solution difference

2 error quantity (L2

error norm, H1

error norm, L∞

error norm, ..) is calculated element-wise

3 elements of the mesh sorted descending by the error quantity

4 a Stopping Criterion Threshold T% is prescribed:

refine ONLY elements with error larger than T% of the maximum element error.

In the calculated examples, 7 different levels (Lowest - 5%, ..., Highest - 95%) were tested.




Determination of refinement type - illustration

GAMM channel - use of directional polynomial orders




Determination of refinement type

I. Basic refinement types.

1. noSelectionH

standard isotropic h-refinement

2. noSelectionHP

refine both in h and in p

II. Selection based on scoring refinement candidates.

- local projection of the reference solution to space created by a particular candidate.

- measure the local error in the appropriate quantity.

II.i Use the error itself

3. hXORpError

refine either in h or in p, select the one with the smaller error.




Determination of refinement type

II.ii Calculate a score for each refinement candidate:

candidate.score = log

(original.error

candidate.error

)1

(candidate.dofs − original.dofs)

and select the one with highest score.

4. hORpDOFs - 3 candidates:

isotropic h-refinement

isotropic p-refinement

both isotropic h- and isotropic p- refinement

5. isoHPDOFs - adds the following candidates:

isotropic h-refinement with order permutations ranging from (original p - 1) to (original p + 1)

6. anisoHPDOFs - adds the following candidates:

anisotropic h-refinement with order permutations ranging from (original p - 1) to (original p + 1)

anisotropic p-refinement



Benchmarking

Attributes inspected

Steps the adaptive algorithm must make in order to achieve the error threshold.

Number of unknowns in the last adaptivity step (where the error threshold is met).

Cumulative number of unknowns of all systems solved in the process.

Capabilities of the algorithm to perform faster by caching of a sort, in this sense, how many local stiffness

matrices and rhs vectors can be reused.

Cumulative direct solver factorization size, used memory and flops.

Measurement how well a particular adaptive strategy followed the prescribed error threshold (by not

dropping unnecessarily much below it, etc.).

Error estimate and exact error.

CPU time (implementation-specific, for illustration)



Benchmarking

Illustration - 1



Benchmarking

Illustration - 2 - improvement



Benchmarking

Benchmark examples

Thanks to: W. MITCHELL, A Collection of 2D Elliptic Problems for Testing Adaptive Algorithms

10 different ”difficulty” settings

5 resolution(error) thresholds



Benchmarking

Benchmark 1 - NIST-04 - exponential peak

Solution of various difficulty settings:

Error thresholds: {0.1%, 0.025%, 0.00625%, 0.0015625%, 0.000390625%}



Benchmarking


Number of unknowns reached by the adaptive algorithm

total per all difficulties combined

total per all error thresholds combined



Benchmarking




error level: 0.1%



Benchmarking




error level: 0.000390625%



Benchmarking


Cumulative number of unknowns of all systems solved





Benchmarking




error level: 0.1% error threshold



Benchmarking







Benchmarking


, percentage of total cache searches





Benchmarking






Benchmarking

Benchmark 1 - NIST-04 - summary

h-adaptivity is competitive down to 0.1% error threshold

winners: (3.simple h- XOR p- refinement), (2.simplest hp-refinement)

(3.simple h- XOR p- refinement) leads to better matrix sparsity pattern

in both cases, best stopping criterion: 30− 40%

losers: (4.h- OR p- refinement) - too few candidates (unnecessary DOFs)



Benchmarking

Benchmark 2 - NIST-06 - boundary layer





Benchmarking




3 error levels, with the lowest one= 0.00625%



Benchmarking







Benchmarking



highest difficulty

highest error level: = 0.000390625%



Benchmarking




all error levels



Benchmarking


Local stiffness matrices - new calculation, percentage of total cache searches





Benchmarking







Benchmarking


Local stiffness matrices - forced recalculation, percentage of total cache searches





Benchmarking







Benchmarking


Summary of benchmark data

even though boundary layers should favor (6. anisotropic refinements), it does so only for very high

accuracies

for higher error levels (0.0015625% and up), anisotropic refinement candidates bring only small

improvement

winners: (6. anisotropic refinements), (5.isotropic refinements), (2.simplest hp-refinement)

roughly same numbers of adaptivity steps, (6. anisotropic refinements) always slightly less DOFs

in all cases, best stopping criterion: 5% (due to large error difference boundary layer⇔ rest of the domain)



Benchmarking

Benchmark 3 - NIST-08 - oscillatory





Benchmarking




all error levels



Benchmarking


High difficulty test example



Benchmarking




all error levels



Benchmarking

Benchmark 3 - NIST-08 - high difficulty






Benchmarking




all error levels



Benchmarking


Average ratio of achieved error and prescribed error threshold


all error levels



Benchmarking



all error levels



Benchmarking


high difficulty example: (4.h- OR p- refinement) - achieved less DOFs than strategies with more candidates

(why?)

winners: (6. anisotropic refinements), (5.isotropic refinements), (2.simplest hp-refinement)

(2.simplest hp-refinement) less adaptivity steps (comparable DOFs of steps)

in all cases, best stopping criterion: 20%

losers: (4.h- OR p- refinement) for less difficult versions - too few candidates (unnecessary DOFs) + too

many steps



Benchmarking

Benchmark 4 - NIST-12 - multiple difficulties


Error thresholds: {5.0%, 2.5%, 1.25%, 0.625%, 0.3125%, 0.05%}



Benchmarking




error level: 0.625%



Benchmarking







Benchmarking




error level: 0.05%



Benchmarking


hp-adaptivity comparably efficient only below error level of 0.05%

above this threshold, h-adaptivity (reference solution only refined in h!) performs (much) better

number of adaptive steps comparable

⇒ important to include h-adaptivity (non-p-) to any general heuristics



Conclusions

More benchmarking needed (verification, phenomena variety)

Already observable

It pays off to spend time tuning the parameters

Selections based on scoring refinement candidates may have troubles showing results paying for their

implementation costs

Anisotropic refinements do not exhibit improvements in general⇒ other adaptivity features will have higher

implementation priority

Depending on the application, a fine-tuned h-adaptivity might be generally sufficient for engineering-level

(low) accuracy demands

Heuristics automatically tuning adaptivity algorithms must take into account:

available resources (memory, no. of cores, etc.)

”problem characteristics”

error threshold, other user-defined settings

previous (up-to-now) adaptivity algorithm behavior on the current problem

even the stopping criteria and refinement selections can (from the data it is obvious that should) be chosen

in an adaptive way

???



Conclusions

Thank you for your attention.


Adaptive algorithms benchmarking - CMMSE2013

Software

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