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Cosmic Calibration Katrin Heitmann Statistical Challenges for Large-Scale Structure in the Era of LSST Oxford , April 18, 2018 Cosmological Constraints Large Scale Simulations Cosmic Emulation Cosmic Emulators Large Synoptic Survey Telescope

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SLIDE 1

Katrin Heitmann Statistical Challenges for Large-Scale Structure in the Era of LSST Oxford , April 18, 2018

Cosmic Calibration

Cosmological Constraints Cosmic Emulation

The Q Continuum Simulation

Large Scale Simulations

Large Synoptic Survey Telescope

Cosmic Emulators

http://www.hep.anl.gov/cosmology/CosmicEmu/emu.html

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SLIDE 2

Thanks to many collaborators!

  • The Beginnings -- Proof of Concept

(Heitmann et al. 2006, Habib et al. 2007)

  • The Coyote Universe + Extension

(Heitmann et al. 2009, 2010, 2013, Lawrence et al. 2010)

  • Emulators beyond P(k)

(Kwan et al. 2013a,b)

  • The Mira-Titan Universe

(Heitmann et al. 2015, Lawrence et al. 2017, Kwan et al., Bocquet et al in prep.)

+ HACC team

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SLIDE 3

Thanks to many collaborators!

  • The Beginnings -- Proof of Concept

(Heitmann et al. 2006, Habib et al. 2007)

  • The Coyote Universe + Extension

(Heitmann et al. 2009, 2010, 2013, Lawrence et al. 2010)

  • Emulators beyond P(k)

(Kwan et al. 2013a,b)

  • The Mira-Titan Universe

(Heitmann et al. 2015, Lawrence et al. 2017, Kwan et al., Bocquet et al in prep.)

+ HACC team

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SLIDE 4

Thanks to many collaborators!

  • The Beginnings -- Proof of Concept

(Heitmann et al. 2006, Habib et al. 2007)

  • The Coyote Universe + Extension

(Heitmann et al. 2009, 2010, 2013, Lawrence et al. 2010)

  • Emulators beyond P(k)

(Kwan et al. 2013a,b)

  • The Mira-Titan Universe

(Heitmann et al. 2015, Lawrence et al. 2017, Kwan et al., Bocquet et al in prep.)

+ HACC team

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SLIDE 5

Cosmic Calibration: Solving the Inverse Problem

  • Challenge: To extract cosmological constraints from observations

in nonlinear regime, need to run Marko Chain Monte Carlo code; input: 10,000 - 100,000 different models

  • Direct simulations: Cost estimate
  • HACC on Titan (fastest system in the US, 5th in the world)
  • 10 simulations fit on full machine, 24 hours per simulation
  • For 100,000 simulations this translates to ~30 years
  • Current strategy: Fitting functions for e.g. P(k), accurate at 10%

level, this is not good enough!

  • Our alternative: Emulators, fast prediction schemes built from a

manageable set of simulations

  • “Ingredients”: Optimal sampling methods to decide which models

to simulate, efficient representation of simulation outcome, powerful interpolation scheme

  • Example here: Power spectrum emulator
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SLIDE 6

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Cosmic Calibration Framework

Run suite of simulations (40,100,...) with chosen parameter values Design optimal simulation campaign over (~20) parameter range Statistics Package (Gaussian Process Modeling, MCMC) Response surface; emulator Calibration Distribution Observation input Predictive Distribution Model inadequacy, self calibration

Modeling/Sims; Observations; Observations+Modeling

  • Step 1: Design simulation

campaign, rule of thumb: O(10) models for each parameter

  • Step 2: Carry out simulation

campaign and extract quantity of interest, in our case, power spectrum

  • Step 3: Choose suitable

interpolation scheme to interpolate between models, here Gaussian Processes

  • Step 4: Build emulator
  • Step 5: Use emulator to analyze

data, determine model inadequacy, refine simulation and modeling strategy...

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SLIDE 7

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

  • Observational considerations
  • CMB provides very accurate

measurements of “vanilla parameters”

  • In particular, ω , ω , n known

at the 2-3% level

  • w, σ less well known
  • For good emulator performance

from very small number of runs

  • Not too broad priors
  • Not too many parameters

The Coyote Simulation Design for wCDM Cosmologies

The (original) Coyote Universe

0.020 ≤ ω ≤ 0.025 0.11 ≤ ω ≤ 0.15 0.85 ≤ n ≤ 1.05

  • 1.3 ≤ w ≤ -0.7

0.6 ≤σ ≤ 0.9

b m

Priors:

8

Design Parameters Derived Parameters

s b m s 8

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SLIDE 8

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Coyote Universe

  • 37 model runs + ΛCDM
  • 16 low resolution realizations

(green)

  • 4 medium resolution realizations

(red)

  • 1 high resolution realization (blue)
  • 11 outputs per run between z = 0 - 3
  • Restricted priors to minimize

necessary number of runs

  • 1.3 Gpc boxes, mp ~10¹¹M
  • ~1000 simulations, 60TB °

.

Background Visualization with ParaView by J. Woodring

Priors:

0.020 ≤ ω ≤ 0.025 0.11 ≤ ω ≤ 0.15 0.85 ≤ n ≤ 1.05

  • 1.3 ≤ w ≤ -0.7

0.6 ≤ σ ≤ 0.9

8 s m b

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SLIDE 9

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Next step: Smooth Power Spectrum

Baryon wiggles

∆2(k) = k3P(k) 2π2 ; P( ~ k) = hδ2( ~ k)i

Gadget PM, 2048³ PM, 1024³

  • Each simulation represents one

possible realization of the Universe in a finite volume

  • Need smooth prediction for

building the emulator for each model

  • Major challenge: Make sure that

baryon features are not washed

  • ut or enhanced due to

realization scatter

  • Construct smooth power

spectra using a process convolution model (Higdon 2002)

  • Basic idea: calculate moving

average using a kernel whose width is allowed to change to account for non-stationarity

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SLIDE 10

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Next step: Smooth Power Spectrum

Baryon wiggles

∆2(k) = k3P(k) 2π2 ; P( ~ k) = hδ2( ~ k)i

Gadget PM, 2048³ PM, 1024³

Coyote III, Process convolution

  • Each simulation represents one

possible realization of the Universe in a finite volume

  • Need smooth prediction for

building the emulator for each model

  • Major challenge: Make sure that

baryon features are not washed

  • ut or enhanced due to

realization scatter

  • Construct smooth power

spectra using a process convolution model (Higdon 2002)

  • Basic idea: calculate moving

average using a kernel whose width is allowed to change to account for non-stationarity

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SLIDE 11

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

!3 !2 !1 1 .5 !6 !4 !2

z

mean

log k ln P

!3 !2 !1 1 .5 !0.2 !0.1 0.1

z

PC basis 1

log k ! ln P

!3 !2 !1 1 .5 !0.2 !0.1 0.1

z

PC basis 2

log k ! ln P

!3 !2 !1 1 .5 !0.2 !0.1 0.1

z

PC basis 3

log k ! ln P

!3 !2 !1 1 .5 !0.2 !0.1 0.1

z

PC basis 4

log k ! ln P

!3 !2 !1 1 .5 !0.2 !0.1 0.1

z

PC basis 5

log k ! ln P

  • After simulation design

specification: Build interpolation scheme that yields predictions for any cosmology within the priors

  • Model simulation outputs using

a - dimensional basis representation

  • Find suitable set of orthogonal

basis vectors , here: Principal Component Analysis

  • 5 PC bases needed, fifth PC

basis pretty flat

  • Next step: modeling the

weights

  • Here: Gaussian Process

modeling (non-parametric regression approach, local interpolator; specified by mean function and covariance function)

pη

θ [0,1]pθ

∈

ln ⇢∆2(k,z) 2πk3/2

  • =

pη

∑

i=1

φi(k,z)wi(θ)+ε

Number of basis functions, here: 5 Basis functions, here: PC basis Weights, here: GP model Cosmological parameters Number of parameters, 5

φi(k,z)

The Interpolation Scheme: Gaussian Processing

Basis functions

time

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SLIDE 12

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Cosmic Emu(lator)

  • Prediction tool for matter power

spectrum has been constructed

  • Accuracy within specified priors

between z=0 and z=1 out to k=1 h/ Mpc at the 1% level achieved

  • Emulator has been publicly released,

C code (Lawrence et al., 2010)

  • Extension: Includes a additional

parameter, covers smaller scales and earlier times (Heitmann et al., 2014)

  • Nested simulations to cover large

k-range

  • Approach degrades accuracy to ~3%

Emulator performance: Comparison of prediction and simulation output for a model not used to build emulator at 6 redshifts. 1% 1%

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SLIDE 13

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Cosmic Emu(lator)

  • Prediction tool for matter power

spectrum has been constructed

  • Accuracy within specified priors

between z=0 and z=1 out to k=1 h/ Mpc at the 1% level achieved

  • Emulator has been publicly released,

C code (Lawrence et al., 2010)

  • Extension: Includes a additional

parameter, covers smaller scales and earlier times (Heitmann et al., 2014)

  • Nested simulations to cover large

k-range

  • Approach degrades accuracy to ~3%

Emulator performance: Comparison of prediction and simulation output for a model not used to build emulator at 6 redshifts. 1% 1%

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SLIDE 14

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The FrankenEmu Challenge

0.01 0.1 1 10 100 1000 10000 1e+05 P(k) [Mpc

3]

0.001 0.01 0.1 1 10 k [Mpc

  • 1]

0.01 0.1 1 10 100 1000 10000 1e+05 P(k) [Mpc

3]

RPT 1300 Mpc 365 Mpc 180 Mpc 90 Mpc 0.001 0.01 0.1 1 10 k [Mpc

  • 1]

M019 z=0 z=4 z=0 z=4 M037

Nyquist limit Shot noise limit

kNy = πNp/L

Pshot(k) = (L/Np)3

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SLIDE 15

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

FrankenEmu Results

10

−3

10

−2

10

−1

10 0.94 0.96 0.98 1 1.02 1.04 1.06

k[1/Mpc] ∆2

emu / ∆2 sim

10

−3

10

−2

10

−1

10 0.94 0.96 0.98 1 1.02 1.04 1.06

k[1/Mpc] ∆2

emu / ∆2 sim

Extension in k- and z-range Hubble treated as free parameter

High-z predictions

Ratio emulator/simulation 1% 2% 4%

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SLIDE 16

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

FrankenEmu Concentration Emulator

Kwan et al. 2013 Emulator Simulation fit LCDM, M000

  • Nested simulation provide high

mass resolution and allow us to measure halo concentration for small halo masses

  • Due to large variance in

concentration measures, accuracy requirements are not as daunting

  • Emulator for z-range 0-1,

concetration variation between c~2 to c~8

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SLIDE 17

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

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SLIDE 18

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

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SLIDE 19

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

  • Extend parameter space to include varying

w(z) and massive neutrinos

  • Build “nested designs”: enable to build

emulator from first set of 26 models, improve with additional 29 models, final precision with 101 models overall

  • Various emulators for P(k), mass function, c-M

relation, RSD predictions, derived quantities...

LCDM simulation Parameters

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k ∆2

emu / ∆2 sim

+/-1% 26 models, emulator for linear P(k), 8 parameters w0+wa=-0.01 w0+wa=-0.02

Heitmann et al. 2015, proxy models

Key advance, Bergner PhD thesis, SFU, 2011

slide-20
SLIDE 20

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

  • Extend parameter space to include varying

w(z) and massive neutrinos

  • Build “nested designs”: enable to build

emulator from first set of 26 models, improve with additional 29 models, final precision with 101 models overall

  • Various emulators for P(k), mass function, c-M

relation, RSD predictions, derived quantities...

LCDM simulation Parameters

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k ∆2

emu / ∆2 sim

+/-1% 26 models, emulator for linear P(k), 8 parameters w0+wa=-0.01 w0+wa=-0.02

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k 2

emu / 2 sim

+/-1% 55 models, emulator for linear P(k), 8 parameters

Heitmann et al. 2015, proxy models

Key advance, Bergner PhD thesis, SFU, 2011

slide-21
SLIDE 21

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

  • Extend parameter space to include varying

w(z) and massive neutrinos

  • Build “nested designs”: enable to build

emulator from first set of 26 models, improve with additional 29 models, final precision with 101 models overall

  • Various emulators for P(k), mass function, c-M

relation, RSD predictions, derived quantities...

LCDM simulation Parameters

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k ∆2

emu / ∆2 sim

+/-1% 26 models, emulator for linear P(k), 8 parameters w0+wa=-0.01 w0+wa=-0.02

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k 2

emu / 2 sim

+/-1% 55 models, emulator for linear P(k), 8 parameters

Heitmann et al. 2015, proxy models

Key advance, Bergner PhD thesis, SFU, 2011

slide-22
SLIDE 22

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Next Step: The Mira-Titan Universe

  • Extend parameter space to include varying

w(z) and massive neutrinos

  • Build “nested designs”: enable to build

emulator from first set of 26 models, improve with additional 29 models, final precision with 101 models overall

  • Various emulators for P(k), mass function, c-M

relation, RSD predictions, derived quantities...

LCDM simulation Parameters

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k ∆2

emu / ∆2 sim

+/-1% 26 models, emulator for linear P(k), 8 parameters w0+wa=-0.01 w0+wa=-0.02

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k 2

emu / 2 sim

+/-1% 55 models, emulator for linear P(k), 8 parameters

10

−3

10

−2

10

−1

10 10

1

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

k 2

emu / 2 sim

Parameters +/-1% 101 models, emulator for linear P(k), 8 parameters

Heitmann et al. 2015, proxy models

Key advance, Bergner PhD thesis, SFU, 2011

slide-23
SLIDE 23

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

0-15+7053/0-15+7058/0-15+7059/0-15+7056/0-15+7057/0-15+7054/0-15+7055/0-15+8304/0-15+8305/0-15+8302/0-15+8303/0-15+8300/0-15+8301/0-15+8306/0-15+8307/0-15+8588/0-15+8589/0-15+8594/0-15+8595/0-15+8592/0-15+8593/0-15+8590/0-15+8591/0-15+9840/0-15+9841/0-15+9842/0-15+9843/0-15+9836/0-15 +9837/0-15+9838/0-15+9839/0-15+9888/0-15+9889/0-15+9982/0-15+9983/0-15+9980/0-15+9981/0-15+9890/0-15+9891/0-15+8540/0-15+8541/0-15+8450/0-15+8451/0-15+8448/0-15+8449/0-15+8542/0-15+8543/0-15+8352/0-15+8353/0-15+8446/0-15+8447/0-15+6912/0-15+6913/0-15+7006/0-15+7007/0-15+7004/0-15+70 05/0-15+8354/0-15+8355/0-15+8444/0-15+8445/0-15+6914/0-15+6915/0-15+6916/0-15+6917/0-15+7002/0-15+7003/0-15+7000/0-15+7001/0-15+6918/0-15+6919/0-15+8440/0-15+8441/0-15+8358/0-15+8359/0-15+8356/0-15+8357/0-15+8442/0-15+8443/0-15+8452/0-15+8453/0-15+8536/0-15+8537/0-15+8454/0-15+8455/0

  • 15+9976/0-15+9977/0-15+9978/0-15+9979/0-15+9892/0-15+9893/0-15+9894/0-15+9895/0-15+9896/0-15+9897/0-15+9974/0-15+9975/0-15+8533/0-15+8458

/ 0-15+8459/0-15+8456/0-15+8457/0-15+8534/0-15+8535/0-15+8360/0-15+8361/0-15+8438/0-15+8439/0-15+6920/0-15+6921/0-15+6998/0-15+6999/0-15+6996/0-15+6997/0-15+8362/0-15+8436/0-15+8437/0-15+6922/0-15+6923/0-15+6992/0-15+6993/0-15+6926/0-15+6927/0-15+8432/0-15+8433/0-15+8366/0-15+8367/0-15 +8364/0-15+8365/0-15+8434/0-15+8435/0-15+8460/0-15+8461/0-15+8530/0-15+8531/0-15+8528/0-15+8529/0-15+8462/0-15+8463/0-15+9968/0-15+9969/0-15+9970/0-15+9971/0-15+9900/0-15+9901/0-15+9902/0-15+9903/0-15+9904/0-15+9905/0-15+9966/0-15+9967/0-15+9964/0-15+9965/0-15+9906/0-15+9907/0-15+85 24/0-15+8525/0-15+8466/0-15+8467/0-15+8464/0-15+8465/0-15+8526/0-15+8527/0-15+8368/0-15+8369/0-15+8430/0-15+8431/0-15+6928/0-15+6929/0-15+6990/0-15+6991/0-15+6988/0-15+6989/0-15+8370/0-15+8371/0-15+8428/0-15+8429/0-15+6930/0-15+6931/0-15+6932/0-15+6933/0-15+6986/0-15+6987/0-15+6984/0

  • 15+6985/0-15+6934/0-15+6935/0-15+8424/0-15+8425/0-15+8374/0-15+8375/0-15+8372/0-15+8373/0-15+8426/0-15+8427/0-15+8468/0-15+8469/0-15+852

2/0-15+8523/0-15+8520/0-15+8521/0-15+8470/0-15+8471/0-15+9960/0-15+9961/0-15+9962/0-15+9908/0-15+9909/0-15+9911/0-15+9912/0-15+9913/0-15+9958/0-15+9959/0-15+9956/0-15+9957/0-15+9914/0-15+9915/0-15+8516/0-15+8517/0-15+8474/0-15+8475/0-15+8472/0-15+8473/0-15+8518/0-15+8519/0-15+8376/0- 15+8377/0-15+8422/0-15+8423/0-15+6936/0-15+6937/0-15+6982/0-15+6983/0-15+6980/0-15+6981/0-15+8378/0-15+8379/0-15+8420/0-15+8421/0-15+6938/0-15+6939/0-15+6940/0-15+6941/0-15+6978/0-15+6979/0-15+6976/0-15+6977/0-15+6942/0-15+6943/0-15+8416/0-15+8417/0-15+8382/0-15+8383/0-15+8380/0-15+ 8381/0-15+8418/0-15+8419/0-15+8476/0-15+8477/0-15+8514/0-15+8515/0-15+8512/0-15+8513/0-15+8478/0-15+8479/0-15+9952/0-15+9953/0-15+9954/0-15+9955/0-15+9916/0-15+9917/0-15+9918/0-15+9919/0-15+9920/0-15+9921/0-15+9950/0-15+9951/0-15+8508/0-15+8509/0-15+8482/0-15+8483/0-15+8480/0-15+848 1/0-15+8510/0-15+8511/0-15+8384/0-15+8385/0-15+8414/0-15+8415/0-15+6944/0-15+6945/0-15+6974/0-15+6975/0-15+6972/0-15+6973/0-15+8386/0-15+83 87/0-15+8412/0-15+8413/0-15+6946/0-15+6947/0-15+6968/0-15+6969/0-15+6950/0-15+6951/0-15+8408/0-15+8409/0-15+8390/0-15+8391/0-15+8388/0-15+8389/0-15+8410/0-15+8411/0-15+8484/0-15+8485/0-15+8506/0-15+8507/0-15+8504/0-15+8505/0-15+8486/0-15+8487/0-15+9944/0-15+9945/0-15+9946/0-15+9947/0

  • 15+9924/0-15+9925/0-15+9926/0-15+9927/0-15+9928/0-15+9929/0-15+9942/0-15+9943/0-15+9940/0-15+9941/0-15+9930/0-15+9931/0-15+8500/0-15+8501/0-15+8490/0-15+8491/0-15+8488/0-15+8489/0-15+8502/0-15+8503/0-15+8392/0-15+8393/0-15+8406/0-15+8407/0-15+6952/0-15+6953/0-15+6966/0-15+6967/0-15+

6964/0-15+6965/0-15+8394/0-15+8395/0-15+8404/0-15+8405/0-15+6954/0-15+6955/0-15+6956/0-15+6957/0-15+6962/0-15+6963/0-15+6960/0-15+6961/0-15+6958/0-15+6959/0-15+8400/0-15+8401/0-15+8398/0-15+8399/0-15+8396/0-15+8397/0-15+8402/0-15+8403/0-15+8492/0-15+8493/0-15+8498/0-15+8499/0-15+849 6/0-15+8497/0-15+8494/0-15+8495/0-15+9936/0-15+9937/0-15+9938/0-15+9939/0-15+9932/0-15+9933/0-15+9934/0-15+9935/0-15+6144/0-15+6145/0-15+6 238/0-15+6239/0-15+6236/0-15+6237/0-15+6146/0-15+6147/0-15+6140/0-15+6141/0-15+6050/0-15+6051/0-15+6048/0-15+6049/0-15+6142/0-15+6143/0-15+4608/0-15+4609/0-15+4702/0-15+4703/0-15+4512/0-15+4513/0-15+4606/0-15+4607/0-15+4604/0-15+4605/0-15+4610/0-15+4611/0-15+4700/0-15+4701/0-15+4514/ 0-15+4515/0-15+4516/0-15+4517/0-15+4602/0-15+4603/0-15+4600/0-15+4601/0-15+4518/0-15+4519/0-15+4696/0-15+4697/0-15+4614/0-15+4615/0-15+4612/0-15+4613/0-15+4698/0-15+4699/0-15+6052/0-15+6053/0-15+6138/0-15+6139/0-15+6136/0-15+6137/0-15+6054/0-15+6055/0-15+6232/0-15+6233/0-15+6234/0-15 +6235/0-15+6148/0-15+6149/0-15+6150/0-15+6151/0-15+6152/0-15+6153/0-15+6230/0-15+6231/0-15+6228/0-15+6229/0-15+6154/0-15+6155/0-15+6132/0-15+6133/0-15+6058/0-15+6059/0-15+6056/0-15+6057/0-15+6134/0-15+6135/0-15+4616/0-15+4617/0-15+4694/0-15+4695/0-15+4520/0-15+4521/0-15+4598/0-15+45 99/0-15+4596/0-15+4597/0-15+4618/0-15+4619/0-15+4692/0-15+4693/0-15+4522/0-15+4523/0-15+4524/0-15+4525/0-15+4594/0-15+4595/0-15+4592/0-15+ 5+6084/0-15+6085/0-15+6106/0-15+6107/0-15+6104/0-15+6105/0-15+6086/0-15+6087/0-15+6200/0-15+6201/0-15+6202/0-15+6203/0-15+6180/0-15+6181/0-15+6182/0-15+6183/0-15+6184/0-15+6185/0-15+6198/0-15+6199/0-15+6196/0-15+6197/0-15+6186/0-15+6187/0-15+6100/0-15+6101/0-15+6090/0-15+6091/0-15+6 088/0-15+6089/0-15+6102/0-15+6103/0-15+4648/0-15+4649/0-15+4662/0-15+4663/0-15+4552/0-15+4553/0-15+4566/0-15+4567/0-15+4564/0-15+4565/0-15+4554/0-15+4555/0-15+4556/0-15+4557/0-15+4562/0-15+4563/0-15+4560/0-15+4561/0-15+4558/0-15+4559/0-15+4656/0-15+4657/0-15+4654/0-15+4655/0-15+4652/ 0-15+4653/0-15+4658/0-15+4659/0-15+6092/0-15+6093/0-15+6098/0-15+6096/0-15+6097/0-15+6094/0-15+6095/0-15+6192/0-15+6193/0-15+6194/0-15+6195/0-15+6188/0-15+6189/0-15+6190/0-15+6191/0-15+6240/0-15+6241/0-15+6334/0-15+6335/0-15+6332/0-15+6333/0-15+6242/0-15+6243/0-15+6044/0-15+6045/0-15 +5954/0-15+5955/0-15+5952/0-15+5953/0-15+6047/0-15+4704/0-15+4705/0-15+4798/0-15+4799/0-15+4416/0-15+4417/0-15+4510/0-15+4511/0-15+4508/0- 15+4509/0-15+4706/0-15+4707/0-15+4796/0-15+4797/0-15+4418/0-15+4419/0-15+4420/0-15+4421/0-15+4506/0-15+4507/0-15+4504/0-15+4505/0-15+4422/0-15+4423/0-15+4792/0-15+4793/0-15+4710/0-15+4711/0-15+4708/0-15+4709/0-15+4794/0-15+4795/0-15+5956/0-15+5957/0-15+6042/0-15+6043/0-15+6040/0-15+ 6041/0-15+5958/0-15+5959/0-15+6328/0-15+6329/0-15+6330/0-15+6331/0-15+6244/0-15+6245/0-15+6246/0-15+6247/0-15+6248/0-15+6249/0-15+6326/0-15+6327/0-15+6324/0-15+6325/0-15+6250/0-15+6251/0-15+6036/0-15+6037/0-15+5962/0-15+5963/0-15+5960/0-15+5961/0-15+6038/0-15+6039/0-15+4712/0-15+471 3/0-15+4790/0-15+4791/0-15+4500/0-15+4501/0-15+4714/0-15+4715/0-15+4788/0-15+4789/0-15+4426/0-15+4427/0-15+4428/0-15+4429/0-15+4498/0-15+4499/0-15+4496/0-15+4497/0-15+4430/0-15+4431/0-15+4784/0-15+4785/0-15+4718/0-15+4719/0-15+4716/0-15+4717/0-15+4786/0-15+4787/0-15+5964/0-15+5965/0- 15+6034/0-15+6035/0-15+6320/0-15+6321/0-15+6322/0-15+6323/0-15+6252/0-15+6253/0-15+6254/0-15+6255/0-15+6256/0-15+6257/0-15+6318/0-15+6319/0

  • 15+6316/0-15+6317/0-15+6258/0-15+6259/0-15+6028/0-15+6029/0-15+5970/0-15+5971/0-15+5968/0-15+5969/0-15+6030/0-15+6031/0-15+4720/0-15+4721/0-15+4782/0-15+4783/0-15+4432/0-15+4433/0-15+4494/0-15+4495/0-15+4492/0-15+4493/0-15+4722/0-15+4723/0-15+4780/0-15+4781/0-15+4434/0-15+4435/0-15+

4436/0-15+4437/0-15+4490/0-15+4491/0-15+4488/0-15+4489/0-15+4438/0-15+4439/0-15+4776/0-15+4777/0-15+4726/0-15+4727/0-15+4724/0-15+4725/0-15+4778/0-15+4779/0-15+5972/0-15+5973/0-15+6026/0-15+6027/0-15+6024/0-15+6025/0-15+5974/0-15+5975/0-15+6312/0-15+6313/0-15+6314/0-15+6315/0-15+626 0/0-15+6261/0-15+6262/0-15+6263/0-15+6264/0-15+6265/0-15+6310/0-15+6311/0-15+6308/0-15+6309/0-15+6266/0-15+6267/0-15+6020/0-15+6021/0-15+5978/0-15+5979/0-15+5976/0-15+5977/0-15+6022/0-15+6023/0-15+4728/0-15+4729/0-15+4774/0-15+4775/0-15+4440/0-15+4441/0-15+4486/0-15+4487/0-15+4484/0- 15+4485/0-15+4730/0-15+4731/0-15+4772/0-15+4773/0-15+4442/0-15+4443/0-15+4444/0-15+4445/0-15+4482/0-15+4483/0-15+4480/0-15+4481/0-15+4446/ 0-15+4447/0-15+4768/0-15+4769/0-15+4734/0-15+4735/0-15+4732/0-15+4733/0-15+4770/0-15+4771/0-15+5980/0-15+5981/0-15+6018/0-15+6019/0-15+6016/0-15+6017/0-15+5982/0-15+5983/0-15+6304/0-15+6305/0-15+6306/0-15+6307/0-15+6268/0-15+6269/0-15+6270/0-15+6271/0-15+6272/0-15+6273/0-15+6302/0-15 +6303/0-15+6300/0-15+6301/0-15+6274/0-15+6275/0-15+6012/0-15+6013/0-15+5986/0-15+5987/0-15+5984/0-15+5985/0-15+6014/0-15+6015/0-15+4736/0-15+4737/0-15+4766/0-15+4767/0-15+4476/0-15+4477/0-15+4738/0-15+4739/0-15+4764/0-15+4765/0-15+4450/0-15+4451/0-15+4452/0-15+4453/0-15+4474/0-15+44 75/0-15+4472/0-15+4473/0-15+4454/0-15+4455/0-15+4760/0-15+4761/0-15+4742/0-15+4743/0-15+4740/0-15+4741/0-15+4762/0-15+4763/0-15+5988/0-15+5989/0-15+6010/0-15+6011/0-15+6296/0-15+6297/0-15+6298/0-15+6299/0-15+6276/0-15+6277/0-15+6278/0-15+6279/0-15+6280/0-15+6281/0-15+6294/0-15+6295/0

  • 15+6292/0-15+6293/0-15+6282/0-15+6283/0-15+6004/0-15+6005/0-15+5994/0-15+5995/0-15+5992/0-15+5993/0-15+6006/0-15+6007/0-15+4744/0-15+4745

/ 0-15+4758/0-15+4759/0-15+4456/0-15+4457/0-15+4470/0-15+4471/0-15+4468/0-15+4469/0-15+4746/0-15+4747/0-15+4756/0-15+4757/0-15+4458/0-15+4459/0-15+4460/0-15+4461/0-15+4466/0-15+4467/0-15+4464/0-15+4465/0-15+4462/0-15+4463/0-15+4752/0-15+4753/0-15+4750/0-15+4751/0-15+4748/0-15+4749/0-15 +4754/0-15+4755/0-15+5996/0-15+5997/0-15+6002/0-15+6003/0-15+6000/0-15+6001/0-15+5998/0-15+5999/0-15+6288/0-15+6289/0-15+6290/0-15+6291/0-15+6284/0-15+6285/0-15+6286/0-15+6287/0-15+6336/0-15+6337/0-15+6430/0-15+6431/0-15+6428/0-15+6429/0-15+6338/0-15+6339/0-15+5948/0-15+5949/0-15+58 58/0-15+5859/0-15+5856/0-15+5857/0-15+5950/0-15+5951/0-15+4800/0-15+4801/0-15+4894/0-15+4895/0-15+4320/0-15+4321/0-15+4414/0-15+4415/0-15+4412/0-15+4413/0-15+4802/0-15+4803/0-15+4892/0-15+4893/0-15+4322/0-15+4323/0-15+4324/0-15+4325/0-15+4410/0-15+4411/0-15+4408/0-15+4409/0-15+4326/0

  • 15+4327/0-15+4888/0-15+4889/0-15+4806/0-15+4807/0-15+4804/0-15+4805/0-15+4890/0-15+4891/0-15+5860/0-15+5861/0-15+5946/0-15+5947/0-15+594

4/0-15+5945/0-15+5862/0-15+5863/0-15+6426/0-15+6427/0-15+6340/0-15+6341/0-15+6344/0-15+6345/0-15+6422/0-15+6423/0-15+6420/0-15+6421/0-15+6346/0-15+6347/0-15+5940/0-15+5941/0-15+5866/0-15+5867/0-15+5864/0-15+5865/0-15+5942/0-15+5943/0-15+4808/0-15+4809/0-15+4886/0-15+4887/0-15+4328/0- 15+4329/0-15+4406/0-15+4407/0-15+4404/0-15+4405/0-15+4810/0-15+4811/0-15+4884/0-15+4885/0-15+4330/0-15+4331/0-15+4332/0-15+4333/0-15+4402/0-15+4403/0-15+4400/0-15+4401/0-15+4334/0-15+4335/0-15+4880/0-15+4881/0-15+4814/0-15+4815/0-15+4812/0-15+4813/0-15+4882/0-15+4883/0-15+5868/0-15+ 5869/0-15+5938/0-15+5939/0-15+5936/0-15+5937/0-15+5870/0-15+5871/0-15+6416/0-15+6417/0-15+6418/0-15+6419/0-15+6348/0-15+6349/0-15+6350/0-15+6351/0-15+6352/0-15+6353/0-15+6414/0-15+6415/0-15+6412/0-15+6413/0-15+6354/0-15+6355/0-15+5932/0-15+5933/0-15+5874/0-15+5875/0-15+5872/0-15+587 3/0-15+5934/0-15+5935/0-15+4816/0-15+4817/0-15+4878/0-15+4879/0-15+4336/0-15+4337/0-15+4398/0-15+4399/0-15+4396/0-15+4397/0-15+4818/0-15+48 19/0-15+4876/0-15+4877/0-15+4338/0-15+4339/0-15+4340/0-15+4341/0-15+4394/0-15+4395/0-15+4392/0-15+4393/0-15+4342/0-15+4343/0-15+4872/0-15+4873/0-15+4822/0-15+4823/0-15+4820/0-15+4821/0-15+4874/0-15+4875/0-15+5876/0-15+5877/0-15+5930/0-15+5931/0-15+5928/0-15+5929/0-15+5878/0-15+5879/0

  • 15+6408/0-15+6409/0-15+6410/0-15+6411/0-15+6356/0-15+6357/0-15+6358/0-15+6359/0-15+6360/0-15+6361/0-15+6406/0-15+6407/0-15+6404/0-15+6405/0-15+6362/0-15+6363/0-15+5924/0-15+5925/0-15+5882/0-15+5883/0-15+5880/0-15+5881/0-15+5926/0-15+5927/0-15+4824/0-15+4825/0-15+4870/0-15+4871/0-15+

4344/0-15+4345/0-15+4390/0-15+4391/0-15+4388/0-15+4389/0-15+4826/0-15+4827/0-15+4868/0-15+4869/0-15+4346/0-15+4347/0-15+4348/0-15+4349/0-15+4386/0-15+4387/0-15+4864/0-15+4865/0-15+4830/0-15+4831/0-15+4828/0-15+4829/0-15+4866/0-15+4867/0-15+5884/0-15+5885/0-15+5922/0-15+5923/0-15+592 0/0-15+5921/0-15+5886/0-15+5887/0-15+6402/0-15+6403/0-15+6364/0-15+6365/0-15+6368/0-15+6369/0-15+6398/0-15+6399/0-15+6396/0-15+6397/0-15+6 370/0-15+6371/0-15+5916/0-15+5917/0-15+5890/0-15+5891/0-15+5888/0-15+5889/0-15+5918/0-15+5919/0-15+4832/0-15+4833/0-15+4862/0-15+4863/0-15+4352/0-15+4353/0-15+4382/0-15+4383/0-15+4380/0-15+4381/0-15+4834/0-15+4835/0-15+4860/0-15+4861/0-15+4354/0-15+4355/0-15+4356/0-15+4357/0-15+4378/ 0-15+4379/0-15+4376/0-15+4377/0-15+4358/0-15+4359/0-15+4856/0-15+4857/0-15+4838/0-15+4839/0-15+4836/0-15+4837/0-15+4858/0-15+4859/0-15+5892/0-15+5893/0-15+5914/0-15+5915/0-15+5912/0-15+5913/0-15+5894/0-15+5895/0-15+6392/0-15+6393/0-15+6394/0-15+6395/0-15+6372/0-15+6373/0-15+6374/0-15 +6375/0-15+6376/0-15+6377/0-15+6390/0-15+6391/0-15+6388/0-15+6389/0-15+6378/0-15+6379/0-15+5908/0-15+5909/0-15+5898/0-15+5899/0-15+5896/0-15+5897/0-15+5910/0-15+5911/0-15+4840/0-15+4841/0-15+4854/0-15+4855/0-15+4360/0-15+4361/0-15+4374/0-15+4375/0-15+4372/0-15+4373/0-15+4842/0-15+48 43/0-15+4852/0-15+4853/0-15+4362/0-15+4363/0-15+4364/0-15+4365/0-15+4370/0-15+4371/0-15+4368/0-15+4369/0-15+4366/0-15+4367/0-15+4848/0-15+ 4849/0-15+4846/0-15+4847/0-15+4844/0-15+4845/0-15+4850/0-15+4851/0-15+5900/0-15+5901/0-15+5906/0-15+5907/0-15+5904/0-15+5905/0-15+5902/0-15+5903/0-15+6384/0-15+6385/0-15+6387/0-15+6380/0-15+6381/0-15+6382/0-15+6383/0-15+6432/0-15+6433/0-15+6526/0-15+6527/0-15+6524/0-15+6525/0-15+643 4/0-15+6435/0-15+5852/0-15+5853/0-15+5762/0-15+5763/0-15+5760/0-15+5761/0-15+5854/0-15+5855/0-15+4896/0-15+4897/0-15+4990/0-15+4991/0-15+4224/0-15+4225/0-15+4318/0-15+4319/0-15+4316/0-15+4317/0-15+4898/0-15+4899/0-15+4988/0-15+4989/0-15+4226/0-15+4227/0-15+4228/0-15+4229/0-15+4314/0- 15+4315/0-15+4312/0-15+4313/0-15+4230/0-15+4231/0-15+4984/0-15+4985/0-15+4902/0-15+4903/0-15+4900/0-15+4901/0-15+4986/0-15+4987/0-15+5764/0-15+5765/0-15+5850/0-15+5851/0-15+5848/0-15+5849/0-15+5766/0-15+5767/0-15+6520/0-15+6521/0-15+6522/0-15+6523/0-15+6436/0-15+6437/0-15+6438/0-15+ 6439/0-15+6440/0-15+6441/0-15+6518/0-15+6519/0-15+6516/0-15+6517/0-15+6442/0-15+6443/0-15+5844/0-15+5845/0-15+5770/0-15+5771/0-15+5768/0-15 +5769/0-15+5846/0-15+5847/0-15+4904/0-15+4905/0-15+4982/0-15+4983/0-15+4232/0-15+4233/0-15+4310/0-15+4311/0-15+4308/0-15+4309/0-15+4906/0-15+4907/0-15+4980/0-15+4981/0-15+4234/0-15+4235/0-15+4236/0-15+4237/0-15+4306/0-15+4307/0-15+4304/0-15+4305/0-15+4238/0-15+4239/0-15+4976/0-15+49 77/0-15+4910/0-15+4911/0-15+4908/0-15+4909/0-15+4978/0-15+4979/0-15+5772/0-15+5773/0-15+5842/0-15+5843/0-15+5840/0-15+5841/0-15+5774/0-15+5775/0-15+6512/0-15+6513/0-15+6514/0-15+6515/0-15+6444/0-15+6445/0-15+6446/0-15+6447/0-15+6448/0-15+6449/0-15+6510/0-15+6511/0-15+6508/0-15+6509/0

  • 15+6450/0-15+6451/0-15+5836/0-15+5837/0-15+5778/0-15+5779/0-15+5776/0-15+5777/0-15+5838/0-15+5839/0-15+4912/0-15+4913/0-15+4974/0-15+4975/0-15

Execution terminated Exit_status=0 resources_used.cput=00:00:30 resources_used.energy_used=0 resources_used.mem=80560kb resources_used.vmem=291496kb resources_used.walltime=08:18:43 Error_Path: titan-ext4:/lustre/atlas2/hep100/current/TitanU/Grid/M038/L2100/HACC000/run/run003/hacc_gpu_m038.err Output_Path: titan-ext4:/lustre/atlas2/hep100/current/TitanU/Grid/M038/L2100/HACC000/run/run003/hacc_gpu_m038.log

The Next Step: The Mira-Titan Universe

Mass function

26 model emulator 55 model emulator 101 model emulator TimeRG 16 PM runs High-res

32003particles 2100 Mpc

slide-24
SLIDE 24

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Emulating the Galaxy Power Spectrum

Ncen(M) = 1 2erfc ln(Mcut/M) √ 2σ

  • Nsat(M) =

✓M − κMcut M1 ◆α

  • First paper: Keep cosmology fixed and only vary 5 HOD parameters
  • Emulators for: Galaxy-galaxy auto, galaxy-dark matter cross power

spectra and correlation function based on 100 HOD models

  • Accuracy: 1-2% between z=0 and z=1 out to k=1/Mpc
  • Currently in preparation: Currently extended emulators to take into

account cosmology dependence

Kwan et al. 2015

slide-25
SLIDE 25

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

The Mira-Titan Universe: Power Spectrum

10-3 10-2 10-1 100

k

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

∆2

emu / ∆2 sim

M038 M039 M040 M041 M042 M043 M044 M000

Lawrence et al. 2017

  • 26 + 10 models
  • ~4% accuracy out to k~5/Mpc

Models not used to build the emulator

slide-26
SLIDE 26

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

10-3 10-2 10-1 100

k

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

2 emu / 2 Takahashi

M038, wa=0 M039, wa=0 M040, wa=0 M041, wa=0 M042, wa=0 M043, wa=0 M044, wa=0 M000

10-3 10-2 10-1 100

k

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

2 emu / 2 Mead

M038 M039 M040 M041 M042 M043 M044 M000

Comparison with Other Methods

Lawrence et al. 2017 Takahashi et al. 2012 Mead et al. 2015

slide-27
SLIDE 27

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Mira-Titan Universe Release in Preparation

in collaboration with Tom Uram

slide-28
SLIDE 28

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Mira-Titan Universe Release in Preparation

slide-29
SLIDE 29

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Mira-Titan Universe Release in Preparation

slide-30
SLIDE 30

Katrin Heitmann, Los Alamos National Laboratory Benasque Cosmology Workshop, August 2010

Mira-Titan Universe Release in Preparation

slide-31
SLIDE 31

Future Work and Open Questions

  • More Emulators: Mass function, galaxy power spectrum and correlation

function across cosmologies (real and redshift space), dark matter halo bias, …

  • Discrepancy Modeling: What happens if our forward model isn’t

correct?

  • Nested/Adaptive Sampling: Convergent/Learning approach to

emulation

  • Covariance Emulation: Emulate covariances rather than just the mean

(observations are only for one realization!)

  • Accuracy Limits: Theory for convergence (a posteriori so far)
  • Limits of Dimensionality: How high can we go?
  • Cross-Correlations: Optical X CMB, lensing X galaxy distribution, etc.
  • Galaxy Catalogs: Emulation of statistics from galaxy formation models
slide-32
SLIDE 32

Key References

Derek

  • General Method:
  • K. Heitmann, D. Higdon, C. Nakhleh, and S. Habib, ApJ Lett 646, 1 (2006) [short]
  • S. Habib, K. Heitmann, D. Higdon, C. Nakhleh, and B. Williams, Phys. Rev. D 76, 083503 (2007)

[technical]

  • D. Higdon, K. Heitmann, C. Nakhleh, and S. Habib, in the Oxford Handbook of Applied Bayesian

Analysis edited by O’ Hagan and West (Oxford, 2010) [review with a worked out inverse problem]

  • K. Heitmann, D. Bingham, E. Lawrence, S. Bergner, S. Habib, D. Higdon, A. Pope, R. Biswas, H.

Finkel, N. Frontiere, and S. Bhattacharya, ApJ 820, 108 (2016) [nested sampling, strong convergence]

  • Power Spectra:
  • K. Heitmann, M. White, C. Wagner, S. Habib, and D. Higdon, ApJ 715, 104 (2010) [Coyote Universe I];
  • K. Heitmann, D. Higdon, M. White, S. Habib, B.J. Williams, and C. Wagner, ApJ 705, 156 (2009)

[Coyote Universe II]; E. Lawrence, K. Heitmann, M. White, D. Higdon, C. Wagner, S. Habib, and B. Williams, ApJ 713, 1322 (2010) [Coyote Universe III]; K. Heitmann, E. Lawrence, J. Kwan, S. Habib, and D. Higdon, ApJ 780, 111 (2014) [Coyote Universe IV]

  • J. Kwan, K. Heitmann, S. Habib, N. Padmanabhan, E. Lawrence, H. Finkel, N. Frontiere, and A. Pope,
  • Phys. Rev. D 810, 35 (2015) [galaxy power spectrum]
  • E. Lawrence, K. Heitmann, J. Kwan, A. Upadhye, D. Bingham, S. Habib, D. Higdon, A. Pope, H.

Finkel, and Nicholas Frontiere, ApJ 847, 50 (2017) [Mira-Titan Universe II];

  • Other Examples:
  • J. Kwan, S. Bhattacharya, K. Heitmann, and S. Habib, ApJ 768, 123 (2013) [Halo shape emulation]
  • T

. Holsclaw, U. Alam, B. Sanso, H. Lee, K. Heitmann, S. Habib, and D. Higdon, Phys. Rev. Lett. 105, 241302 (2010) [Sn constraints on w(z) using GPs]

  • T

. Holsclaw, U. Alam, B. Sanso, H. Lee, K. Heitmann, S. Habib, and D. Higdon, Phys. Rev. D 84, 083501 (2011) [combining data sets]