Computing the Shadow Hamiltonian
Andreas Smolenko | M. Sc.MathCCES | Rheinisch-Westfalische Technische Hochschule
IPAM | University of California Los Angeles
28th of September 2017
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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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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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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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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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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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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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
Computing the Shadow Hamiltonian | 28th of September 2017 | 10/19
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
Computing the Shadow Hamiltonian | 28th of September 2017 | 11/19
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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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
Computing the Shadow Hamiltonian | 28th of September 2017 | 17/19
Acknowledgements
• Professor Dr. Benjamin Stamm, RWTH Aachen University,MATHCCES
• Dr. Virginie Ehrlacher, Ecole des Ponts ParisTech, CERMICS
Computing the Shadow Hamiltonian | 28th of September 2017 | 18/19
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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