Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q)...

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Computing the Shadow Hamiltonian Andreas Smolenko | M. Sc. MathCCES | Rheinisch-Westf¨ alische Technische Hochschule IPAM | University of California Los Angeles 28 th of September 2017

Transcript of Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q)...

Page 1: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Computing the Shadow Hamiltonian

Andreas Smolenko | M. Sc.MathCCES | Rheinisch-Westfalische Technische Hochschule

IPAM | University of California Los Angeles

28th of September 2017

Page 2: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Motivation

• Approximation by numerical integration scheme

x(t0)Φt [x(t0)]

Φt [x(t0)]

• Transform error into slightly different equation of motion

• Knowledge of how to increase accuracy

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Page 3: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Hamiltonian Equations of Motion• Quantum mechanical model

reduction→ classical force-fields

• Long phase-space vector x = (q, p) of positions q andmomenta p in a molecule

• Hamiltonian H(x) = T (p) + V (q) describing the dynamics

d

dtx(t) = J∇xH(x(t))

• Poisson matrix

J =

(0 1−1 0

)• Symplecticity

d

dtH(x(t)) = 0

2010, Lelievre, Rousset, Stoltz

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Page 4: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Hamiltonian Flow Operator

• Lie-derivative of observable through chain rule

LH f (x(t)) =d

dtf (x(t)) = ([J∇xH(x(t))] · ∇x)f (x(t))

• Definition by exponential function

Φt [f (x(t0))] = etLH f (x(t0))

• Symplecticity follows from exponential form and f = H

2015, Leimkuhler

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Page 5: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Numerical Integration Approximation Schemes

• Small temporal discretisation steps τ

• Symplectic Euler

ΦSEτ = eτLV eτLT

• Position Verlet

ΦPVτ = eτLT/2eτLV eτLT/2

• Velocity Verlet

ΦVVτ = eτLV/2eτLT eτLV/2

2015, Leimkuhler

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Page 6: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Baker-Campbell-Hausdorff (BCH) Expansion

• Summarise exponential functions into one

• Separation of Hamiltonian into two contributions

H = H1 + H2

• Rearranging of two derivatives into one exponent

eτLH1 eτLH2 = eτLH

• BCH expansion → shadow Hamiltonian

H − H =τ

2{H1,H2}+O

(τ 2)

• Poisson-Braket

{H1,H2} = ∇qH1 · ∇pH2 −∇qH2 · ∇pH1

2015, Leimkuhler

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Page 7: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Approximated Trajectory vs.Approximated Hamiltonian

x(t) H(x(t))

x(t) H(x(t))

ddt x(t) = J∇xH(x(t))

ddt x(t) = J∇xH(x(t))

maxx|H(x)− H(x)|= O(τ r)

maxt∈[0,T ]

||x(t)− x(t)||2= O(τ r)

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Visualisation of the Shadow Hamiltonian

q

p

H(Φt [x(t0)]) = const

H(Φt [x(t0)]) = const

x(t0)

2015, Leimkuhler

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Analytical Example for the Shadow Hamiltonian

• Harmonic oscillator in one dimension

H(x) =q2

2+

p2

2

• Symplectic Euler

• Linear correction computation

τ

2{V (q),T (p)} =

τ

2qp

• Linear term leads to an undecomposable shadow Hamiltonian

H(x) = H(x) +τ

2qp +O

(τ 2)

2015, Leimkuhler

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Page 10: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Ansatz for the Shadow Hamiltonian

• Shadow Hamiltonian as a superposition of basis functions

H(x) = H(x) +∑i

ci fi(x)

• Hermite polynomials in one dimension

g0(ξ) =1

g1(ξ) =ξ

g2(ξ) =ξ2 − 1

· · ·

• Hermite polynomials in two dimensions

fi(q, p) =fiq,ip(q, p) = giq(q) gip(p)

2013, Noe, Nuske

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Variational Approach

• Idea: constant shadow Hamiltonian for temporal evolution

• Expectation of vanishing functional

F (H) =1

T

∫ T

0

(H(x(t))− H(Φτ [x(t)])

)2

dt

• Vanishing derivative with respect to variational coefficients

d

dciF (H) = 0 ∀i

• Reformulation into linear matrix vector optimization problem

c = argminc′||Ac ′ − b||2

2013, Noe, Nuske

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Variational Approach Variation

• Idea: vanishing temporal Hamiltonian derivative

• Expectation of vanishing functional

F (1)(H) =1

T

∫ T

0

(d

dτH(Φτ [x(t)])

)2

dt

• Efficient computation of derivative by tangent modeautomatic differentiation

• Vanishing derivative with respect to variational coefficients

d

dciF (1)(H) = 0 ∀i

• Potential of multiobjective minimisation

2012, Naumann

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Page 13: Computing the Shadow Hamiltonian · momenta p in a molecule Hamiltonian H(x) = T(p) + V(q) describing the dynamics d dt x(t) = Jr xH(x(t)) Poisson matrix J = 0 1 1 0 Symplecticity

Numerical Results

• Harmonic oscillator in one dimension

• Parallelisation on laptop running ≈ 1 minute

• Symplectic Euler

• Coefficient results including statistics of 5 launches⟨cnum

1,1

ctheo1,1

⟩diff

= 1.003± 0.003⟨cnum

1,1

ctheo1,1

⟩deriv

= 1.00 ± 0.01⟨cnum

1,1

ctheo1,1

⟩diff+deriv

= 0.998± 0.004

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Applications

• Study of error sources of complex numerical integrationschemes, like integration algorithms with splitting of slow andfast modes

• Studying a general system, where the dynamics is known butthe Hamiltonian is not known

• Assisting algorithms, which rely on the knowledge of theHamiltonian → symplectic reduced-order modelling schemes

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Reduced-Order Modelling

• Assuming the case of dynamics x ∈ Rn, which are centeredaround linear k < n dominant modes

x(t0)x(t1) x(t2) x(t3)

• Storing of phase-space snapshot matrix

M = [x(t0), x(t1), · · · ]

• Singular value decomposition leads to dominant modescorresponding to columns of U with greatest singular values σj

M = U diag(σj) WT

2017, Afkham, Hesthaven

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Symplectic Reduced-Order Modelling

• Method described exponentially increases total energy

• Energy conserving reduced-order modelling by inducingsymplectic property of the matrix U

UTJU = J

• Using the Hamiltonian of reduced system to compute in agreedy style the dominant modes of the system

2017, Afkham, Hesthaven

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Conclusion

• Variational ansatz for computing the shadow Hamiltonian

• Numerical results discussed for small-dimensional system

• Applications for the computation of shadow Hamiltonians

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Acknowledgements

• Professor Dr. Benjamin Stamm, RWTH Aachen University,MATHCCES

• Dr. Virginie Ehrlacher, Ecole des Ponts ParisTech, CERMICS

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Thank you for your attention.

Are there propositions for further applications?

Was something comparable done so far?

Is there expertise available in efficient variational computations?

Are different basis functions preferable?

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