SAND 0-50 C Effective & Efficient Handling of Ill - Conditioned Coelation atices in Kiging & adient Enhanced Kiging Emulatos hough Pivoted Cholesky Factoization Keith Dalbey, PhD Sandia National Labs, Dept 44 Optimization & Uncetainty Quantification David Day, PhD, Dept 44, Numeical Analysis ak Hoemmen, Dept 46, Scalable Algoithms Apil, 0 Sandia National Laboatoies is a multi-pogam laboatoy managed and opeated by Sandia Copoation, a wholly owned subsidiay of Lockheed atin Copoation, fo the U.S. Depatment of Enegy s National Nuclea Secuity Administation unde contact DE-AC04-94AL85000
Outline Intoduction Kiging Emulatos adient Enhanced Kiging (EK) Ways to Handle Ill-Conditioning Pivoted Cholesky fo Kiging Pivoted Cholesky fo EK Pivoted Cholesky Details esults Conclusions Cuent / Futue Wok
Intoduction Emulatos built fom small amount of build data can be used as fast suogates fo computationally epensive simulatos Kiging/P emulatos ae popula because they poduce: Best guess estimate that intepolates build data (as long as coelation mati,, is eal symmetic and positive definite) & Estimate of pediction eo away fom build points. But pooly spaced o lage amounts of build data can make ill-conditioned (ead as numeically singula) which voids the waanty so we need to handle ill-conditioned adient Enhanced Kiging (EK) is becoming popula because of new simulation techniques (e.g. Automatic Diffeentiation, yan Stakey) that poduce function value + gadient fo cost of about o fewe function values (highe info/cost atio)
Kiging Emulatos Also known as aussian Pocess Emulatos Bayes Linea ethod BLUP BLUE Diffeences among them ae mino. All have: unadjusted mean (fequently a least squaes fit) coection/adjustment to mean based on data estimated distibution about adjusted mean of possible tue sufaces
g g y Va g y E ˆ Coelation paametes,, elated to coelation lengths, L, by d =/( L d ) Kiging Emulatos d d j d d j j, ep, i j j i i i j i i j j i g,,,,, N N ˆ
Pos: oe info (equations) pe cost uch bette conditioned fo same # of equations Cons: uch wose conditioned fo same numbe of points N N N,,,,,, How?: Deive with assumptions simila to egula Kiging o eplace adient Enhanced Kiging (EK)
I J I J J I J I adient Enhanced Kiging (EK)
Some Options: Ways to Handle Ill-Conditioning Shink coelation lengths, L: but adjusted mean may degeneate into unadjusted mean too quickly Add nugget to diagonal of : but causes Kiging o EK to appoimate (smooth) instead of intepolate (ask youself if eally need to intepolate, you might not) Use mitue coelation model: weighted sum of Shot L and with Long L (weight () s too) L Use good subset of available build data: will intepolate subset you keep but how do you efficiently select good subset? New Answe: Pivoted Cholesky Factoization (checks diffeent optimal subset fo each candidate L o ) S with
Pivoted Cholesky fo Kiging Pivoted Cholesky sots equations in accoding to how much new infomation they contain. When soted, equations with moe infomation come befoe equations with less infomation Can then use LAPACK cond() estimate and bisection seach to efficiently detemine how many low info equations need to be dopped off the end Dopped equations ae ones that contain the least new info and so ae easiest to pedict Caig Lucas, LAPACK-Style Codes fo Level and 3 Pivoted Cholesky Factoizations, LAPACK Woking Note 6, 004 Lucas level 3 can (and often does) default to level and is not cost competitive with highly optimized level 3 LAPACK (non-pivoted) Cholesky fo lage matices
Pivoted Cholesky fo EK can be much lage than so Pivoted Cholesky on is vey slow (compaed to LAPACK s Cholesky) Pivoted Cholesky on also pefes deivative equations (highe infomation content) ove esponse values (bad because function values ae moe eliable than deivatives) Solution: Do pivoted Cholesky on not, then apply same odeing to whole points (a point s function value immediately followed by its deivatives) in LAPACK Cholesky on eodeed Cholesky cost by facto of (+) 3. hen do. educes Pivoted
Pivoted Cholesky Details Adaptive least squaes tend ode If using maimum likelihood to choose, need to optimize the pe equation log likelihood obj log ˆ only contains infomation about and inputs, it does not contain infomation about outputs so discontinuity in output not taken into consideation (but can compae diffeence between pedicted and actual discaded using ahalanobis distance*) det log log det N N Pivoted Cholesky can be used fo sample design * Using ahalanobis distance was suggested by ony O Hagan
EK esults fo osenbock
EK esults fo osenbock
Conclusions Pivoted Cholesky can handle ill-conditioning of Kiging s coelation mati by efficiently selecting an optimal subset of available build data points Dopped points ae ones that contain the least new infomation and so ae easiest to pedict Pivoted Cholesky can be used: to detect discontinuities in the esponse fo sample design to detemine good set of Long coelation lengths in mitue model Kiging / P
Cuent / Futue Wok aussian Pocess guided Adaptive mitue Impotance Sampling (PAIS, doesn t need to intepolate, does need all data so add nugget to handle ill-conditioning) to be pesented husday 9:30-9:55 A in S67 by Laua Swile & at Quality & Poductivity eseach Confeence June 4-7, 0 Synopsis: A tool fo managing V&V / UQ activities and data UI font end fo SVN epositoy that implements fomal best pactices pocess (including incemental eview and appoval, assessment, etc.) fo V&V and UQ P. Knupp and A. Ubina, "A Design fo a V&V and UQ Discovey Pocess," SAND0-6677, Sandia National Laboatoies, 0.
Calculating Pobability of Failue with PAIS Impotance Sampling educes onte Calo s eo vaiance by dawing moe samples fom impotant egions & appopiately down-weighting I unknown so optimal impotance density is unknown aussian Pocess Adaptive Impotance Sampling uses seies of impoving P appoimations of, in a mitue appoimation of, itue impotance sampling is not much wose than impotance sampling fom the best of the mitue components [Owen & Zhou 000] j-th component P appoimation is eal valued Epected Indicato is the point-wise potion of P s aussian CDF past the failue theshold
Hebie Function
PAIS On Hebie Function
PAIS On Hebie Function
Synopsis Sceen Shot