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WIP: Manipulate the assembly #29

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Vectornaut wants to merge 8 commits from tangent-space into main
pull from: tangent-space
merge into: glen:main
Conversation 0 Commits 8 Files Changed 7 +129 -20
5 changed files with 129 additions and 20 deletions
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Showing only changes of commit 2c55a63a6f - Show all commits

2
app-proto/examples/irisawa-hexlet.rs
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@ -2,7 +2,7 @@ use dyna3::engine::{Q, irisawa::realize_irisawa_hexlet};
fn main() {
const SCALED_TOL: f64 = 1.0e-12;
let (config, success, history) = realize_irisawa_hexlet(SCALED_TOL);
let (config, _, success, history) = realize_irisawa_hexlet(SCALED_TOL);
print!("\nCompleted Gram matrix:{}", config.tr_mul(&*Q) * &config);
if success {
println!("Target accuracy achieved!");

2
app-proto/examples/point-on-sphere.rs
View File

@ -18,7 +18,7 @@ fn main() {
]);
let frozen = [(3, 0)];
println!();
let (config, success, history) = realize_gram(
let (config, _, success, history) = realize_gram(
&gram, guess, &frozen,
1.0e-12, 0.5, 0.9, 1.1, 200, 110
);

2
app-proto/examples/three-spheres.rs
View File

@ -21,7 +21,7 @@ fn main() {
])
};
println!();
let (config, success, history) = realize_gram(
let (config, _, success, history) = realize_gram(
&gram, guess, &[],
1.0e-12, 0.5, 0.9, 1.1, 200, 110
);

12
app-proto/src/assembly.rs
View File

@ -5,7 +5,7 @@ use std::{collections::BTreeSet, sync::atomic::{AtomicU64, Ordering}};
use sycamore::prelude::*;
use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
use crate::engine::{realize_gram, PartialMatrix};
use crate::engine::{realize_gram, ConfigSubspace, PartialMatrix};
// the types of the keys we use to access an assembly's elements and constraints
pub type ElementKey = usize;
@ -109,7 +109,6 @@ impl Element {
}
}
}
#[derive(Clone)]
pub struct Constraint {
@ -127,6 +126,9 @@ pub struct Assembly {
pub elements: Signal<Slab<Element>>,
pub constraints: Signal<Slab<Constraint>>,
// solution variety tangent space
pub tangent: Signal<ConfigSubspace>,
// indexing
pub elements_by_id: Signal<FxHashMap<String, ElementKey>>
}
@ -136,6 +138,7 @@ impl Assembly {
Assembly {
elements: create_signal(Slab::new()),
constraints: create_signal(Slab::new()),
tangent: create_signal(ConfigSubspace::zero()),
elements_by_id: create_signal(FxHashMap::default())
}
}
@ -247,7 +250,7 @@ impl Assembly {
}
// look for a configuration with the given Gram matrix
let (config, success, history) = realize_gram(
let (config, tangent, success, history) = realize_gram(
&gram, guess, &[],
1.0e-12, 0.5, 0.9, 1.1, 200, 110
);
@ -271,6 +274,9 @@ impl Assembly {
|rep| rep.set_column(0, &config.column(elt.column_index))
);
}
// save the tangent space
self.tangent.set_silent(tangent);
}
}
}

131
app-proto/src/engine.rs
View File

@ -1,5 +1,5 @@
use lazy_static::lazy_static;
use nalgebra::{Const, DMatrix, DVector, Dyn};
use nalgebra::{Const, DMatrix, DVector, DVectorView, Dyn, SymmetricEigen};
use web_sys::{console, wasm_bindgen::JsValue}; /* DEBUG */
// --- elements ---
@ -85,6 +85,47 @@ impl PartialMatrix {
}
}
// --- configuration subspaces ---
pub struct ConfigSubspace(Vec<DMatrix<f64>>);
impl ConfigSubspace {
pub fn zero() -> ConfigSubspace {
ConfigSubspace(Vec::new())
}
// approximate the kernel of a symmetric endomorphism of the configuration
// space for `assembly_dim` elements. we consider an eigenvector to be part
// of the kernel if its eigenvalue is smaller than the constant `THRESHOLD`
fn symmetric_kernel(a: DMatrix<f64>, assembly_dim: usize) -> ConfigSubspace {
const ELEMENT_DIM: usize = 5;
const THRESHOLD: f64 = 1.0e-9;
let eig = SymmetricEigen::new(a);
let eig_vecs = eig.eigenvectors.column_iter();
let eig_pairs = eig.eigenvalues.iter().zip(eig_vecs);
let basis = eig_pairs.filter_map(
|(λ, v)| (λ.abs() < THRESHOLD).then_some(
Into::<DMatrix<f64>>::into(
v.reshape_generic(Dyn(ELEMENT_DIM), Dyn(assembly_dim))
)
)
);
ConfigSubspace(basis.collect())
}
// find the projection onto this subspace of the motion where the element
// with the given column index has velocity `v`
/* TO DO */
// for the zero subspace, this method's behavior doesn't match its name: it
// panics rather than returning zero
fn proj(&self, v: &DVectorView<f64>, column_index: usize) -> DMatrix<f64> {
let ConfigSubspace(basis) = self;
basis.into_iter().map(
|b| b.column(column_index).dot(&v) * b
).sum()
}
}
// --- descent history ---
pub struct DescentHistory {
@ -181,7 +222,7 @@ pub fn realize_gram(
reg_scale: f64,
max_descent_steps: i32,
max_backoff_steps: i32
) -> (DMatrix<f64>, bool, DescentHistory) {
) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
// start the descent history
let mut history = DescentHistory::new();
@ -201,12 +242,8 @@ pub fn realize_gram(
// use Newton's method with backtracking and gradient descent backup
let mut state = SearchState::from_config(gram, guess);
let mut hess = DMatrix::zeros(element_dim, assembly_dim);
for _ in 0..max_descent_steps {
// stop if the loss is tolerably low
history.config.push(state.config.clone());
history.scaled_loss.push(state.loss / scale_adjustment);
if state.loss < tol { break; }
// find the negative gradient of the loss function
let neg_grad = 4.0 * &*Q * &state.config * &state.err_proj;
let mut neg_grad_stacked = neg_grad.clone().reshape_generic(Dyn(total_dim), Const::<1>);
@ -229,7 +266,7 @@ pub fn realize_gram(
hess_cols.push(deriv_grad.reshape_generic(Dyn(total_dim), Const::<1>));
}
}
let mut hess = DMatrix::from_columns(hess_cols.as_slice());
hess = DMatrix::from_columns(hess_cols.as_slice());
// regularize the Hessian
let min_eigval = hess.symmetric_eigenvalues().min();
@ -249,6 +286,11 @@ pub fn realize_gram(
hess[(k, k)] = 1.0;
}
// stop if the loss is tolerably low
history.config.push(state.config.clone());
history.scaled_loss.push(state.loss / scale_adjustment);
if state.loss < tol { break; }
// compute the Newton step
/*
we need to either handle or eliminate the case where the minimum
@ -256,7 +298,7 @@ pub fn realize_gram(
singular. right now, this causes the Cholesky decomposition to return
`None`, leading to a panic when we unrap
*/
let base_step_stacked = hess.cholesky().unwrap().solve(&neg_grad_stacked);
let base_step_stacked = hess.clone().cholesky().unwrap().solve(&neg_grad_stacked);
let base_step = base_step_stacked.reshape_generic(Dyn(element_dim), Dyn(assembly_dim));
history.base_step.push(base_step.clone());
@ -269,10 +311,16 @@ pub fn realize_gram(
state = better_state;
history.backoff_steps.push(backoff_steps);
},
None => return (state.config, false, history)
None => return (state.config, ConfigSubspace::zero(), false, history)
};
}
(state.config, state.loss < tol, history)
let success = state.loss < tol;
let tangent = if success {
ConfigSubspace::symmetric_kernel(hess, assembly_dim)
} else {
ConfigSubspace::zero()
};
(state.config, tangent, success, history)
}
// --- tests ---
@ -291,7 +339,7 @@ pub mod irisawa {
use super::*;
pub fn realize_irisawa_hexlet(scaled_tol: f64) -> (DMatrix<f64>, bool, DescentHistory) {
pub fn realize_irisawa_hexlet(scaled_tol: f64) -> (DMatrix<f64>, ConfigSubspace, bool, DescentHistory) {
let gram = {
let mut gram_to_be = PartialMatrix::new();
for s in 0..9 {
@ -399,7 +447,7 @@ mod tests {
fn irisawa_hexlet_test() {
// solve Irisawa's problem
const SCALED_TOL: f64 = 1.0e-12;
let (config, _, _) = realize_irisawa_hexlet(SCALED_TOL);
let (config, _, _, _) = realize_irisawa_hexlet(SCALED_TOL);
// check against Irisawa's solution
let entry_tol = SCALED_TOL.sqrt();
@ -409,6 +457,61 @@ mod tests {
}
}
#[test]
fn tangent_test() {
const SCALED_TOL: f64 = 1.0e-12;
const ELEMENT_DIM: usize = 5;
const ASSEMBLY_DIM: usize = 3;
let gram = {
let mut gram_to_be = PartialMatrix::new();
for j in 0..3 {
for k in j..3 {
gram_to_be.push_sym(j, k, if j == k { 1.0 } else { -1.0 });
}
}
gram_to_be
};
let guess = DMatrix::from_columns(&[
sphere(0.0, 0.0, 0.0, -2.0),
sphere(0.0, 0.0, 1.0, 1.0),
sphere(0.0, 0.0, -1.0, 1.0)
]);
let frozen: [_; 5] = std::array::from_fn(|k| (k, 0));
let (config, tangent, success, history) = realize_gram(
&gram, guess.clone(), &frozen,
SCALED_TOL, 0.5, 0.9, 1.1, 200, 110
);
assert_eq!(config, guess);
assert_eq!(success, true);
assert_eq!(history.scaled_loss.len(), 1);
// confirm that the tangent space has dimension five or less
let ConfigSubspace(ref tangent_basis) = tangent;
assert_eq!(tangent_basis.len(), 5);
// confirm that the tangent space contains all the motions we expect it
// to. since we've already bounded the dimension of the tangent space,
// this confirms that the tangent space is what we expect it to be
let tangent_motions = vec![
basis_matrix((0, 1), ELEMENT_DIM, ASSEMBLY_DIM),
basis_matrix((1, 1), ELEMENT_DIM, ASSEMBLY_DIM),
basis_matrix((0, 2), ELEMENT_DIM, ASSEMBLY_DIM),
basis_matrix((1, 2), ELEMENT_DIM, ASSEMBLY_DIM),
DMatrix::<f64>::from_column_slice(ELEMENT_DIM, 3, &[
0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, -1.0, -0.25, -1.0,
0.0, 0.0, -1.0, 0.25, 1.0
])
];
let tol_sq = ((ELEMENT_DIM * ASSEMBLY_DIM) as f64) * SCALED_TOL * SCALED_TOL;
for motion in tangent_motions {
let motion_proj: DMatrix<_> = motion.column_iter().enumerate().map(
|(k, v)| tangent.proj(&v, k)
).sum();
assert!((motion - motion_proj).norm_squared() < tol_sq);
}
}
// at the frozen indices, the optimization steps should have exact zeros,
// and the realized configuration should match the initial guess
#[test]
@ -428,7 +531,7 @@ mod tests {
]);
let frozen = [(3, 0), (3, 1)];
println!();
let (config, success, history) = realize_gram(
let (config, _, success, history) = realize_gram(
&gram, guess.clone(), &frozen,
1.0e-12, 0.5, 0.9, 1.1, 200, 110
);