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Optimization and gradient methods
Contents
Plans for the week of February 17-21, 2025
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, Fokker-Planck and Langevin equations
Importance sampling, programming elements
Importance sampling, program elements
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Importance sampling
Top-down optimization view
Motivation
Simple example and demonstration
Simple example and demonstration
Exercise 1: Find the local energy for the harmonic oscillator
Variance in the simple model
Computing the derivatives
Expressions for finding the derivatives of the local energy
Derivatives of the local energy
Exercise 2: General expression for the derivative of the energy
Python program for 2-electrons in 2 dimensions
Using Broyden's algorithm in scipy
Brief reminder on Newton-Raphson's method
The equations
Simple geometric interpretation
Extending to more than one variable
Taylor expansion
Jacobian
Inverse of Jacobian
Jacobian
Defining the Jacobian matrix
ˆ
J
we have
ˆ
J
=
(
∂
f
1
/
∂
x
1
∂
f
1
/
∂
x
2
∂
f
2
/
∂
x
1
∂
f
2
/
∂
x
2
)
,
we can rephrase Newton's method as
(
x
n
+
1
1
x
n
+
1
2
)
=
(
x
n
1
x
n
2
)
+
(
h
n
1
h
n
2
)
,
where we have defined
(
h
n
1
h
n
2
)
=
−
ˆ
J
−
1
(
f
1
(
x
n
1
,
x
n
2
)
f
2
(
x
n
1
,
x
n
2
)
)
.
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