408 lines
14 KiB
Markdown
408 lines
14 KiB
Markdown
---
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title: Evaluation of the Performance of Randomized FFD Control Grids
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subtitle: Master Thesis
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author: Stefan Dresselhaus
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affiliation: Graphics & Geometry Group
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...
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# Introduction
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- Many modern industrial design processes require advanced optimization methods
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due to increased complexity
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- Examples are
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- physical domains
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- aerodynamics (i.e. drag)
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- fluid dynamics (i.e. throughput of liquid)
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- NP-hard problems
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- layouting of circuit boards
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- stacking of 3D--objects
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-----
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# Motivation
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- Evolutionary algorithms cope especially well with these problem domains
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- But formulation can be tricky
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-----
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# Motivation
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- Problems tend to be very complex
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- i.e. a surface with $n$ vertices has $3\cdot n$ Degrees of Freedom (DoF).
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- Need for a small-dimensional representation that manipulates the
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high-dimensional problem-space.
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- We concentrate on smooth deformations ($C^3$-continuous)
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- But what representation is good?
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-----
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# What representation is good?
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- In biological evolution this measure is called *evolvability*.
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- no consensus on definition
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- meaning varies from context to context
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- measurable?
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- Measure depends on representation as well.
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-----
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# RBF and FFD
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- Andreas Richter uses Radial Basis Functions (RBF) to smoothly deform meshes
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-----
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# RBF and FFD
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- My master thesis transferred his idea to Freeform-Deformation (FFD)
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- same setup
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- same measurements
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- same results?
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-----
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# Outline
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- **What is FFD?**
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- What is evolutionary optimization?
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- How to measure evolvability?
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- Scenarios
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- Results
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-----
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# What is FFD?
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- Create a function $s : [0,1[^d \mapsto \mathbb{R}^d$ that is parametrized
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by some special control--points $p_i$ with coefficient functions $a_i(u)$:
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$$
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s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}
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$$
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- All points inside the convex hull of $\vec{p_i}$ accessed by the right $u \in [0,1[^d$.
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-----
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# Definition B-Splines
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- The coefficient functions $a_i(u)$ in $s(\vec{u}) = \sum_i a_i(\vec{u}) \vec{p_i}$ are different for each control-point
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- Given a degree $d$ and position $\tau_i$ for the $i$th control-point $p_i$ we
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define
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\begin{equation}
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N_{i,0,\tau}(u) = \begin{cases} 1, & u \in [\tau_i, \tau_{i+1}[ \\ 0, & \mbox{otherwise} \end{cases}
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\end{equation}
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and
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\begin{equation} \label{eqn:ffd1d2}
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N_{i,d,\tau}(u) = \frac{u-\tau_i}{\tau_{i+d}} N_{i,d-1,\tau}(u) + \frac{\tau_{i+d+1} - u}{\tau_{i+d+1}-\tau_{i+1}} N_{i+1,d-1,\tau}(u)
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\end{equation}
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- The derivatives of these coefficients are also easy to compute:
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$$\frac{\partial}{\partial u} N_{i,d,r}(u) = \frac{d}{\tau_{i+d} - \tau_i} N_{i,d-1,\tau}(u) - \frac{d}{\tau_{i+d+1} - \tau_{i+1}} N_{i+1,d-1,\tau}(u)$$
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# Properties of B-Splines
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- Coefficients vanish after $d$ differentiations
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- Coefficients are continuous with respect to $u$
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- A change in prototypes only deforms the mapping locally
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(between $p_i$ to $p_{i+d+1}$)
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![Example of Basis-Functions for degree $2$. [Brunet, 2010]<br /> Note, that Brunet starts his index at $-d$ opposed to our definition, where we start at $0$.](../arbeit/img/unity.png)
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# Definition FFD
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- FFD is a space-deformation resulting based on the underlying B-Splines
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- Coefficients of space-mapping $s(u) = \sum_j a_j(u) p_j$ for an initial vertex
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$v_i$ are constant
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- Set $u_{i,j}~:=~N_{j,d,\tau}$ for each $v_i$ and $p_j$ to get the projection:
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$$
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v_i = \sum_j u_{i,j} \cdot p_j = \vec{u}_i^{T} \vec{p}
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$$
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or written with matrices:
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$$
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\vec{v} = \vec{U} \vec{p}
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$$
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- $\vec{U}$ is called **deformation matrix**
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# Implementation of FFD
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- As we deal with 3D-Models we have to extend the introduced 1D-version
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- We get one parameter for each dimension: $u,v,w$ instead of $u$
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- Task: Find correct $u,v,w$ for each vertex in our model
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- We used a gradient-descent (via the gauss-newton algorithm)
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# Implementation of FFD
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- Given $n,m,o$ control-points in $x,y,z$--direction each Point inside the
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convex hull is defined by
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$$V(u,v,w) = \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot C_{ijk}.$$
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- Given a target vertex $\vec{p}^*$ and an initial guess $\vec{p}=V(u,v,w)$
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we define the error--function for the gradient--descent as:
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$$Err(u,v,w,\vec{p}^{*}) = \vec{p}^{*} - V(u,v,w)$$
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# Implementation of FFD
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- Derivation is straightforward
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$$
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\scriptsize
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\begin{array}{rl}
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\displaystyle \frac{\partial Err_x}{\partial u} & p^{*}_x - \displaystyle \sum_i \sum_j \sum_k N_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x \\
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= & \displaystyle - \sum_i \sum_j \sum_k N'_{i,d,\tau_i}(u) N_{j,d,\tau_j}(v) N_{k,d,\tau_k}(w) \cdot {c_{ijk}}_x
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\end{array}
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$$
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yielding a Jacobian:
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$$
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\scriptsize
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J(Err(u,v,w)) =
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\left(
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\begin{array}{ccc}
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\frac{\partial Err_x}{\partial u} & \frac{\partial Err_x}{\partial v} & \frac{\partial Err_x}{\partial w} \\
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\frac{\partial Err_y}{\partial u} & \frac{\partial Err_y}{\partial v} & \frac{\partial Err_y}{\partial w} \\
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\frac{\partial Err_z}{\partial u} & \frac{\partial Err_z}{\partial v} & \frac{\partial Err_z}{\partial w}
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\end{array}
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\right)
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$$
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# Implementation of FFD
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- Armed with this we iterate the formula
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$$J(Err(u,v,w)) \cdot \Delta \left( \begin{array}{c} u \\ v \\ w \end{array} \right) = -Err(u,v,w)$$
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using Cramer's rule for inverting the small Jacobian.
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- Usually terminates after $3$ to $5$ iteration with an $\epsilon := \vec{p^*} -
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V(u,v,w) < 10^{-4}$
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- self-intersecting grids can invalidate the results
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- no problem, as these get not generated and contradict some properties we
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want (like locality)
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-----
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# Outline
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- What is FFD?
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- **What is evolutionary optimization?**
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- How to measure evolvability?
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- Scenarios
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- Results
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-----
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# What is evolutionary optimization?
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## 1 1
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~~~
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$t := 0$;
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initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$;
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evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$;
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while($c(F(t)) \neq$ true) {
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recombine: $P’(t) := r(P(t))$;
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mutate: $P''(t) := m(P’(t))$;
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evaluate $F(t) : \{\Phi(x) | x \in P''(t)\}$
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select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$;
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$t := t + 1$;
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}
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~~~
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~~~
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$t$: Iteration-step
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$I$: Set of possible Individuals
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$P$: Population of Individuals
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$F$: Fitness of Individuals
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$Q$: Either set of parents or $\emptyset$
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$r(..) : I^\mu \mapsto I^\lambda$
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$m(..) : I^\lambda \mapsto I^\lambda$
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$s(..) : I^{\lambda + \mu} \mapsto I^\mu$
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~~~
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- Algorithm to model simple inheritance
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- Consists of three main steps
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- recombination
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- mutation
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- selection
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- An "individual" in our case is the displacement of control-points
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# Evolutional loop
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- **Recombination** generates $\lambda$ new individuals based on the characteristics of the $\mu$ parents.
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- This makes sure that the next guess is close to the old guess.
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- **Mutation** introduces new effects that cannot be produced by mere recombination of the
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parents.
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- Typically these are minor defects to individual members of the population
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i.e. through added noise
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- **Selection** selects $\mu$ individuals from the children (and optionally the
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parents) using a *fitness--function* $\Phi$.
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- Fitness could mean low error, good improvement, etc.
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- Fitness not solely determines who survives, there are many possibilities
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# Outline
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- What is FFD?
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- What is evolutionary optimization?
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- **How to measure evolvability?**
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- Scenarios
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- Results
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# How to measure evolvability?
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- Different (conflicting) optimization targets
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- convergence speed?
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- convergence quality?
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- As $\vec{v} = \vec{U}\vec{p}$ is linear, we can also look at $\Delta \vec{v} = \vec{U}\,
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\Delta \vec{p}$
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- We only change $\Delta \vec{p}$, so evolvability should only use $\vec{U}$
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for predictions
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# Evolvability criteria
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- **Variability**
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- roughly: "How many actual Degrees of Freedom exist?"
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- Defined by
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$$\mathrm{variability}(\vec{U}) := \frac{\mathrm{rank}(\vec{U})}{n} \in [0..1]$$
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- in FFD this is $1/\#\textrm{CP}$ for the number of control-points used for
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parametrization
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# Evolvability criteria
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- **Regularity**
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- roughly: "How numerically stable is the optimization?"
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- Defined by
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$$\mathrm{regularity}(\vec{U}) := \frac{1}{\kappa(\vec{U})} = \frac{\sigma_{min}}{\sigma_{max}} \in [0..1]$$
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with $\sigma_{min/max}$ being the least/greatest right singular value.
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- high, when $\|\vec{Up}\| \propto \|\vec{p}\|$
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# Evolvability criteria
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- **Improvement Potential**
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- roughly: "How good can the best fit become?"
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- Defined by
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$$\mathrm{potential}(\vec{U}) := 1 - \|(\vec{1} - \vec{UU}^+)\vec{G}\|^2_F$$
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with a unit-normed guessed gradient $\vec{G}$
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# Outline
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- What is FFD?
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- What is evolutionary optimization?
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- How to measure evolvability?
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- **Scenarios**
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- Results
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# Scenarios
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- 2 Testing Scenarios
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- 1-dimensional fit
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- $xy$-plane to $xyz$-model, where only the $z$-coordinate changes
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- can be solved analytically with known global optimum
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- 3-dimensional fit
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- fit a parametrized sphere into a face
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- cannot be solved analytically
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- number of vertices differ between models
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# 1D-Scenario
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{width=70%}
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# 3D-Scenarios
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# Outline
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- What is FFD?
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- What is evolutionary optimization?
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- How to measure evolvability?
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- Scenarios
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- **Results**
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# Variability 1D
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- Should measure Degrees of Freedom and thus quality
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- $5 \times 5$, $7 \times 7$ and $10 \times 10$ have *very strong* correlation ($-r_S = 0.94, p = 0$) between the *variability* and the evolutionary error.
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# Variability 3D
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- Should measure Degrees of Freedom and thus quality
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- $4 \times 4 \times 4$, $5 \times 5 \times 5$ and $6 \times 6 \times 6$ have *very strong* correlation ($-r_S = 0.91, p = 0$) between the *variability* and the evolutionary error.
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# Varying Variability
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## 1 1
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# Regularity 1D
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- Should measure convergence speed
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{width=70%}
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- Not in our scenarios - maybe due to the fact that a better solution simply
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takes longer to converge, thus dominating.
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# Regularity 3D
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- Should measure convergence speed
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{width=70%}
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- Only *very weak* correlation
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- Point that contributes the worst dominates regularity by lowering the least
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right singular value towards 0.
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# Improvement Potential in 1D
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- Should measure expected quality given a gradient
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{width=70%}
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- *very strong* correlation of $- r_S = 1.0, p = 0$.
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- Even with a distorted gradient
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# Improvement Potential in 3D
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- Should measure expected quality given a gradient
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{width=70%}
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- *weak* to *moderate* correlation within each group.
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# Summary
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- *Variability* and *Improvement Potential* are good measurements in our cases
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- *Regularity* does not work well because of small singular right values
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- But optimizing for regularity *could* still lead to a better grid-setup
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(not shown, but likely)
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- Effect can be dominated by other factors (i.e. better solutions just take
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longer)
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# Outlook / Further research
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- Only focused on FFD, but will DM-FFD perform better?
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- for RBF the indirect manipulation also performed worse than the direct one
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- Do grids with high regularity indeed perform better?
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# Thank you
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Any questions?
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