![Page 1: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/1.jpg)
Georg Reuber,
Boris Kaus, Anton Popov
Adjoint based inversion in geodynamic modelling:A guide
![Page 2: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/2.jpg)
Motivation
229/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Geodynamic modelling
![Page 3: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/3.jpg)
Motivation
329/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Geodynamic modelling
Idealized test
Pusok (2015)
• Parameterize real scenario→ Easy to test/quantify effects
![Page 4: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/4.jpg)
Motivation
429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Geodynamic modelling
Idealized test Inversion
Pusok (2015)
• Parameterize real scenario→ Easy to test/quantify effects
• Fit data→ Ultimately represents nature
Baumann (2015)
uobs
ρ,η
![Page 5: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/5.jpg)
Motivation
529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Measurement device
Physical description
Inversion
![Page 6: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/6.jpg)
Motivation
629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Measurement device
Physical description Computer simulation: u
(Field) measurement: uobs
Inversion
![Page 7: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/7.jpg)
Motivation
729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Measurement device
Physical description Computer simulation: u
(Field) measurement: uobs
Cost function:F = 0.5(u-uobs)
T(u-uobs)
Inversion
![Page 8: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/8.jpg)
Motivation
829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Measurement device
Physical description Computer simulation: u
(Field) measurement: uobs
Cost function:F = 0.5(u-uobs)
T(u-uobs)
Inversion
Update computer
Simulation until F = 0
![Page 9: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/9.jpg)
Motivation
929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
E.g: uobs
Inversion
ρ,η
![Page 10: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/10.jpg)
Motivation
1029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
ρ η
F (u, uobs)
Inversion
uobs
ρ,η
![Page 11: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/11.jpg)
Motivation
1129/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Simulation → Find minimum of F
Inversion
uobs
ρ,η
ρ η
F (u, uobs)
![Page 12: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/12.jpg)
Motivation
1229/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Grid search
Inversion
uobs
ρ,η
→ Find minimum of F
ρ η
F (u, uobs)
![Page 13: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/13.jpg)
Motivation
1329/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Statistical
• (Randomized) statistical search
ρ
η → Find minimum of F
F (u, uobs)
Inversion
![Page 14: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/14.jpg)
Motivation
1429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Statistical
• Grid Search Sampling
ρ
η
F (u, uobs)
Inversion
![Page 15: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/15.jpg)
Motivation
1529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Statistical
• Grid Search Sampling→Many simulations→ Broad knowledge of cost function
ρ
η
F (u, uobs)
Inversion
![Page 16: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/16.jpg)
Motivation
1629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• Monte-Carlo Sampling
ρ
η
![Page 17: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/17.jpg)
Motivation
1729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• Monte-Carlo Sampling
ρ
η
![Page 18: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/18.jpg)
Motivation
1829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• Monte-Carlo Sampling→Many simulations→ Broad knowledge of cost function
ρ
η
![Page 19: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/19.jpg)
Motivation
1929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• MCMC Sampling
ρ
η
![Page 20: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/20.jpg)
Motivation
2029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• MCMC Sampling
ρ
η
![Page 21: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/21.jpg)
Motivation
2129/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Inversion
Statistical
• MCMC Sampling→ Few simulations→ Requires fine-tuning
ρ
η
![Page 22: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/22.jpg)
Motivation
2229/8/2019Ada Lovelace Workshop 2019
Georg Reuber
ρ
η
Inversion
Statistical
• Bayes Theorem→ Prior knowledge goes in sampling
![Page 23: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/23.jpg)
Motivation
2329/8/2019Ada Lovelace Workshop 2019
Georg Reuber
ρ
η
Inversion
Statistical
… many more methods
• Bayes Theorem→ Prior knowledge goes in sampling→ Compute posterior→ Uncertainty in parameters
![Page 24: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/24.jpg)
Motivation
2429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Deterministic
∇F
Inversion
ρ
η
Gradient based:
![Page 25: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/25.jpg)
Motivation
2529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Few ‚simulations‘• Sensitive to local minima
Inversion
Deterministic
Gradient based:
ρ
η
![Page 26: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/26.jpg)
Motivation
2629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Gradient based [gradient descent]:
Inversion
Deterministic
ρ
η
![Page 27: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/27.jpg)
Motivation
2729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Gradient based [(Quasi)-Newton]:
… more methods
Hessian matrix→ Relates to the
covariance matrixin Bayesian context
Inversion
Deterministic
ρ
η
![Page 28: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/28.jpg)
Motivation
2829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Few ‚simulations‘• Sensitive to local minima
• How to evaluate gradient?
Inversion
Deterministic
Gradient based:
ρ
η
![Page 29: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/29.jpg)
Motivation
2929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Few ‚simulations‘• Sensitive to local minima
• How to evaluate gradient?
→ Here simple: analytical
→ Or FD:
Inversion
Deterministic
Gradient based:
ρ
η
![Page 30: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/30.jpg)
Motivation
3029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
… n dimensions
• 5 rock phases in thermo-elasto-visco-plastic model = 40 parameters (=dimensions)• Seismology: wavespeed at every node is a free parameter = # nodes parameters (billions)
→ Every viscosity free parameter = # nodes parameters
η η+ρ η+ρ+Kη+ρ+K+…
nD-Inversion
![Page 31: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/31.jpg)
Motivation
3129/8/2019Ada Lovelace Workshop 2019
Georg Reuber
… n dimensions
• Statistical: 40 parameters + 10 samples per parameter = 10^40 simulations• Deterministic: Two solves per parameter with FD = 80 simulations (per descent iteration)
η η+ρ η+ρ+Kη+ρ+K+…
nD-Inversion
![Page 32: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/32.jpg)
Motivation
3229/8/2019Ada Lovelace Workshop 2019
Georg Reuber
… n dimensions
• Statistical: 40 parameters + 10 samples per parameter = 10^40 simulations• Deterministic: Two solves per parameter with FD = 80 simulations (per descent iteration)
→More efficient gradient computation available?
η η+ρ η+ρ+Kη+ρ+K+…
nD-Inversion
![Page 33: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/33.jpg)
Motivation
3329/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Adjoint method:• Independent of # parameters (very efficient)• Requires forward (nonlinear) + adjoint solve (linear) and gradient computation→ Requires formulation of adjoint equation
… n dimensions
η η+ρ η+ρ+Kη+ρ+K+…
nD-Inversion
![Page 34: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/34.jpg)
3429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Appears a lot:
• E.g. Economics: Maximize profit in terms of labor hours and raw material with limited budget
• Logistics• Thermodynamics• …
Inversion
Deterministic
![Page 35: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/35.jpg)
3529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
![Page 36: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/36.jpg)
3629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
![Page 37: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/37.jpg)
3729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
→ Find lowest point of Hike
![Page 38: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/38.jpg)
3829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
I) National Park (with Bears)
![Page 39: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/39.jpg)
3929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
I) National Park (with Bears)II) Foggy day
![Page 40: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/40.jpg)
4029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
I) National ParkII) Foggy day
![Page 41: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/41.jpg)
4129/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Inversion
Deterministic
I) National ParkII) Foggy day
→ Find lowest point of Hike
![Page 42: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/42.jpg)
4229/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• In our example:
1. Function represents mountain belt
2. Find lowest point of hike
Hiking trailValleys
Inversion
Deterministic
x
y
![Page 43: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/43.jpg)
4329/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 44: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/44.jpg)
4429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 45: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/45.jpg)
4529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 46: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/46.jpg)
4629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 47: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/47.jpg)
4729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 48: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/48.jpg)
4829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:• Function F(x,y) - topography• Constraint g(x,y) – hike
→ Find common gradient along constraint
F(x,y)
g(x,y)
Inversion
Deterministic
x
y
![Page 49: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/49.jpg)
4929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Method
Constrained optimization:
Inversion
Deterministic
F(x,y)
g(x,y)
x
y
→ Solve for x,y & λ→ x & y = coordinates of minimum→ [P(x,y) = lowest point→ λ = d(P)/d(Radius)]
![Page 50: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/50.jpg)
5029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Adjoint method(PDE – constrained optimization)
I) Field inversion
II) Vector (p) inversion
![Page 51: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/51.jpg)
Field
51
(1)
Objective function (no regularization):
(2)
(3)
(4)
(5)
(6)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 52: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/52.jpg)
Field
52
(1)
Objective function (no regularization):
(2)
Stokes (inversion constraint, linear, no advection):
(3)
(4)
(5)
(6)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 53: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/53.jpg)
Field
53
(1)
Objective function (no regularization):
(2)
Stokes (inversion constraint, linear, no advection):
(3)
(4)
Lagrangian (constrained optimization):
(5)
(6)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
→ Find critical points of this function
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Field
54
(1)
Objective function (no regularization):
(2)
Stokes (inversion constraint, linear, no advection):
(3)
(4)
Lagrangian (constraint optimization):
(5)
(6)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
→ Find critical points of this function
I. FP multiplied by some Lagrange multipliers (shape functions) is the weak form (FEM) of the FP
II. FP linear – (often) self- adjointIII. If FP nonlinear – additional terms appear (e.g. viscosity
derivative) – like deriving Jacobian
IV. Gradient of parameter at every node – ‚field‘ inversionV. Independent of discretization
VI. Can be derived for any variable that occurs in the equation, e.g. principal stress directions:
Reuber et al. (in prep)
![Page 55: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/55.jpg)
Vector
55
Approach implemented in LaMEM (equal result):
(1)
1) Objective function (no regularization):
(2)
(3)
(4)
5529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 56: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/56.jpg)
Vector
56
Approach implemented in LaMEM (equal result):
(1)
1) Objective function (no regularization):
(2)
2) Objective function derivative (final gradient sought):
(3)
(4)
First term = 0 (!= 0 if regularization)
5629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 57: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/57.jpg)
Vector
57
Approach implemented in LaMEM (equal result):
(1)
1) Objective function (no regularization):
(2)
2) Objective function derivative (final gradient sought):
(3)
2) Residual derivative (R = 0):
(4)
First term = 0 (!= 0 if regularization)
5729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Jacobian
![Page 58: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/58.jpg)
58
3) Substitute:
(5)
(6)
(7)
(8)
Vector
5829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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59
3) Substitute:
(5)
(6)
4) Final:
(7)
(8)
Vector
5929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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60
3) Substitute:
(5)
(6)
4) Final:
(7)
(8)
Vector
6029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Note: Two options to compute gradients with the adjoint method:I) Converge forward problem – compute missing partial derivatives – reuse FP Jacobian - DtOII) Write two codes – one for FP and one for AP – combine solution to compute Gradient - OtD
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6129/8/2019Ada Lovelace Workshop 2019
Georg Reuber
What are these gradients goodfor?
I) Obviously for gradient basedinversion
II) Sensitivity kernels in geodynamics –resolution proxy
III) Automatic derivation of scaling laws
![Page 62: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/62.jpg)
Sensitivity
62
Tromp et al. (2005)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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Sensitivity
63
Tromp et al. (2005)
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Physical resolution test
Why not try in geodynamics as well?
![Page 64: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/64.jpg)
→ What are the velocities at the surface most sensitive to?→ Compute gradient as usual but use:→ Sensitivity of solution to ρ + η at every node→ E.g. resolution test in seismology
Geodynamic sensitivity kernels
6429/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Vx Vz
![Page 65: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/65.jpg)
Sensitivity
6529/8/2019Ada Lovelace Workshop 2019
Georg Reuber
→ What are the velocities at the surface most sensitive to?→ Compute gradient as usual but use:→ Sensitivity of solution to ρ + η at every node→ E.g. resolution test in seismology
![Page 66: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/66.jpg)
Sensitivity
6629/8/2019Ada Lovelace Workshop 2019
Georg Reuber
→ What are the velocities at the surface most sensitive to?→ Compute gradient as usual but use:→ Sensitivity of solution to ρ + η at every node→ E.g. resolution test in seismology
![Page 67: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/67.jpg)
Sensitivity
6729/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Timestep 1
Absolute sensitivity: Density)
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Sensitivity
6829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Timestep 1
Timestep 50
Absolute sensitivity: Density)
Absolute sensitivity: Density)
Cheap! Can be done every timestep
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6929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
Automatic derivation of scalinglaws
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Scaling law
70
• Nondimensional 2 layer Rayleigh-Taylor instability• Random perturbation at interface
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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Scaling law
71
• Nondimensional 2 layer Rayleigh-Taylor instability• Random perturbation at interface
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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No cost function (as before)
Scaling law
72
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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General (multiplicative) scaling law
No cost function!
Scaling law
73
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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General (multiplicative) scaling law
Exponent
Prefactor
No cost function!
Scaling law
74
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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Scaling law
75
→ Exact reproduction of analytical solution
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 76: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/76.jpg)
Scaling law
76
Several ‚domes‘ evolve
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
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Scaling law
77
Several ‚domes‘ evolve
Reuber et al., 2017
29/8/2019Ada Lovelace Workshop 2019
Georg Reuber
![Page 78: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/78.jpg)
Conclusions
7829/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Adjoint method very efficient:• independent of number of parameters (‚field inversion‘)• Can be derived for any parameter for any PDE
![Page 79: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/79.jpg)
Conclusions
7929/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Adjoint method very efficient:• independent of number of parameters (‚field inversion‘)• Can be derived for any parameter for any PDE
• Statistic methods:• Full knowledge (topology) of cost function• (Almost) No additional implementation required• Numerical cost scales with number of parameters
![Page 80: Adjoint based inversion in geodynamic modelling: A guide · Ada Lovelace Workshop 2019 Georg Reuber →Find critical points of this function I. FP multiplied by some Lagrange multipliers](https://reader034.vdocument.in/reader034/viewer/2022042917/5f58f7593367d540c76b8cb0/html5/thumbnails/80.jpg)
Conclusions
8029/8/2019Ada Lovelace Workshop 2019
Georg Reuber
• Adjoint method very efficient:• independent of number of parameters (‚field inversion‘)• Can be derived for any parameter for any PDE
• Statistic methods:• Full knowledge (topology) of cost function• (Almost) No additional implementation required• Numerical cost scales with number of parameters
• Deterministic methods:• Gradient information
→ Reveals sensitivity of e.g. velocity on density – resolution test→ Scaling law computable→ Time integration→ Hessian – link to statistic methods