# 2022-10-10 Iterative Solvers#

## Last time#

• Request RC accounts

• Sparse direct solvers

• impact of ordering on formulation

• cost scaling

## Today#

• Classical iterative methods

• Concept of preconditioning

• Krylov methods (focus on GMRES)

• PETSc experiments

using Plots
default(linewidth=3)
using LinearAlgebra
using SparseArrays

function my_spy(A)
cmax = norm(vec(A), Inf)
s = max(1, ceil(120 / size(A, 1)))
spy(A, marker=(:square, s), c=:diverging_rainbow_bgymr_45_85_c67_n256, clims=(-cmax, cmax))
end

h = 2 / (n + 1)
rows = Vector{Int64}()
cols = Vector{Int64}()
vals = Vector{Float64}()
idx((i, j),) = (i-1)*n + j
in_domain((i, j),) = 1 <= i <= n && 1 <= j <= n
stencil_advect = [-wind[1], -wind[2], 0, wind[1], wind[2]] / h
stencil_diffuse = [-1, -1, 4, -1, -1] * kappa / h^2
for i in 1:n
for j in 1:n
neighbors = [(i-1, j), (i, j-1), (i, j), (i+1, j), (i, j+1)]
end
end
sparse(rows, cols, vals)
end

advdiff_matrix (generic function with 1 method)


# Request an RC account#

• This gives ssh login access. We’ll use Alpine, which is a modern CPU and GPU cluster.

• This will be good for benchmarking and larger runs. We’ll also use GPUs later in class.

## PETSc#

• We’ll start using PETSc this week.

• You can build PETSc on your laptop. You’ll need C development tools.

• Linux: use package manager (apt install, dnf install, …)

• OSX: install xcode, many people like homebrew as a package manager

• Windows: WSL then follow Linux instructions (or MSYS2 or Cygwin)

• Any: install Docker, then use our image

# How expensive how fast?#

Suppose we have a second order accurate method in 3D.

n = 2. .^ (2:13)
N = n.^3
error = (50 ./ n) .^ 2
seconds = 1e-10 * N.^2
hours = seconds / 3600
cloud_dollars = 3 * hours
kW_hours = 0.2 * hours
barrel_of_oil = kW_hours / 1700
kg_CO2 = kW_hours * 0.709
;

cost = cloud_dollars
plot(cost, error, xlabel="cost", ylabel="error", xscale=:log10, yscale=:log10)


# Outlook on sparse direct solvers#

• Sparse direct works well for 2D and almost-2D problems to medium large sizes

• High order FD methods make sparse direct cry

• High order finite element are okay, but not high-continuity splines

• Almost-2D includes a lot of industrial solid mechanics applications

• The body of a car, the frame of an airplane

• Sparse direct is rarely usable in “fully 3D” problems

• “thick” structures

• soil mechanics, hydrology, building foundations, bones, tires

• fluid mechanics

• aerodynamics, heating/cooling systems, atmosphere/ocean

• Setup cost (factorization) is much more expensive than solve

• Amortize cost in time-dependent problems

• Rosenbrock methods: factorization reused across stages

• “lag” Jacobian in Newton (results in “modified Newton”)

• “lag” preconditioner with matrix-free iterative methods (Sundials, PETSc)

• Factorization pays off if you have many right hand sides

# Why iterative solvers over direct solvers?#

• Less reliable, more leaky abstraction

• More sensitive to problem formulation

• Slower for small problems

• Several different strategies, each with tuning knobs

• Accuracy tolerances needed

## $$O(N)$$ solvers available for many important problems#

• High-order discretization can be okay

Suppose $$A$$ is a symmetric positive definite matrix and consider the scalar functional

$f(u) = \frac 1 2 u^T A u - b^T u .$

$\nabla_u f = A u - b .$
x = LinRange(-4, 4, 40)
A = [2 -1; -1 8]
b = [1, 1]
f(u) = .5 * u' * A * u - b' * u
contour(x, x, (u1, u2) -> f([u1, u2]), aspect_ratio=:equal)


# Aside: Derivative of a dot product#

Let $$f(\boldsymbol x) = \boldsymbol y^T \boldsymbol x = \sum_i y_i x_i$$ and compute the derivative

$\frac{\partial f}{\partial \boldsymbol x} = \begin{bmatrix} y_0 & y_1 & \dotsb \end{bmatrix} = \boldsymbol y^T .$

Note that $$\boldsymbol y^T \boldsymbol x = \boldsymbol x^T \boldsymbol y$$ and we have the product rule,

$\frac{\partial \lVert \boldsymbol x \rVert^2}{\partial \boldsymbol x} = \frac{\partial \boldsymbol x^T \boldsymbol x}{\partial \boldsymbol x} = 2 \boldsymbol x^T .$

Also,

$\frac{\partial \lVert \boldsymbol x - \boldsymbol y \rVert^2}{\partial \boldsymbol x} = \frac{\partial (\boldsymbol x - \boldsymbol y)^T (\boldsymbol x - \boldsymbol y)}{\partial \boldsymbol x} = 2 (\boldsymbol x - \boldsymbol y)^T .$

# Aside: Variational notation#

It’s convenient to express derivatives in terms of how they act on an infinitessimal perturbation. So we might write

$\delta f = \frac{\partial f}{\partial x} \delta x .$

(It’s common to use $$\delta x$$ or $$dx$$ for these infinitesimals.) This makes inner products look like a normal product rule

$\delta(\mathbf x^T \mathbf y) = (\delta \mathbf x)^T \mathbf y + \mathbf x^T (\delta \mathbf y).$

A powerful example of variational notation is differentiating a matrix inverse

$0 = \delta I = \delta(A^{-1} A) = (\delta A^{-1}) A + A^{-1} (\delta A)$

and thus

$\delta A^{-1} = - A^{-1} (\delta A) A^{-1}$

$u_{k+1} = u_k - \omega \nabla_u f$
function grad_descent(loss, grad, u0; omega=1e-3, tol=1e-5)
"""Minimize loss(c) via gradient descent with initial guess u0
using learning rate gamma.  Declares convergence when gradient
is less than tol or after 500 steps.
"""
u = copy(u0)
uhist = [copy(u)]
lhist = [loss(u)]
for it in 1:500
u -= omega * g
push!(uhist, copy(u))
push!(lhist, loss(u))
if norm(g) < tol
break
end
end
(u, hcat(uhist...), lhist)
end

grad_descent (generic function with 1 method)


A = [1 1; 1 30]
loss(u) = .5 * u' * A * u
omega=.06)
plot(lhist, yscale=:log10)

plot(uhist[1, :], uhist[2, :], marker=:circle)
x = LinRange(-1, 1, 30)
contour!(x, x, (x,y) -> loss([x, y]))


# Richardson iteration#

The simplest iterative method is Richardson’s method, which solves $$A u = b$$ by the iteration

$u_{k+1} = u_k + \omega (b - A u_k)$
where $$\omega > 0$$ is a damping parameter and $$u_0$$ is an initial guess (possibly the zero vector).

• Algebraically equivalent to gradient descent when $$A$$ is SPD

• Non-symmetric matrices are harder to visualize

# Richardson convergence#

$u_{k+1} = u_k + \omega (b - A u_k)$

If $$b = A u_*$$, this iteration is equivalent to

(38)#\begin{align} u_{k+1} - u_* &= (u_k - u_*) - \omega A (u_k - u_*) \\ &= (I - \omega A) (u_k - u_*) . \end{align}

It is convenient for convergence analysis to identify the “error” $$e_k = u_k - u_*$$, in which this becomes

$e_{k+1} = (I - \omega A) e_k$
or
$e_k = (I - \omega A)^k e_0$
in terms of the initial error. Evidently powers of the iteration matrix $$I - \omega A$$ tell the whole story.

Suppose that the eigendecomposition

$X \Lambda X^{-1} = I - \omega A$
exists. Then
$(I - \omega A)^k = (X \Lambda X^{-1})^k = X \Lambda^k X^{-1}$
and the convergence (or divergence) rate depends only on the largest magnitude eigenvalue. This analysis isn’t great for two reasons:

1. Not all matrices are diagonalizable.

2. The matrix $$X$$ may be very ill-conditioned.

# Aside: Schur decomposition#

We can repair these weaknesses by using the Schur decomposition

$Q R Q^h = I - \omega A$
where $$R$$ is right-triangular and $$Q$$ is unitary (i.e., orthogonal if real-valued; $$Q^h$$ is the Hermitian conjugate of $$Q$$). The Schur decomposition always exists and $$Q$$ has a condition number of 1.

• Where are the eigenvalues in $$R$$?

Evidently we must find $$\omega$$ to minimize the maximum eigenvalue of $$I - \omega A$$. We can do this if $$A$$ is well conditioned, but not in general.

# Ill-conditioning#

## Question: What is the condition number of the Laplacian on 100 evenly spaced points?#

• How does it scale under grid refinement?

A = advdiff_matrix(20)
cond(Matrix(A))

178.06427461085963

omega = .002
ev = eigvals(Matrix(I - omega * A))
scatter(real.(ev), imag.(ev), xlim=(-2, 2), ylim=(-1, 1))


# Monic polynomials small on the spectrum#

Equivalently to finding $$\omega$$ such that $$\lVert I - \omega A \rVert$$ is minimized, we may seek a monic polynomial $$p(z) = 1 - \omega z$$ that minimizes

$\max_{\lambda \in \sigma(A)} \lvert p(\lambda) \rvert .$

This concept can be extended to higher degree polynomials, which is essentially what Krylov methods do (discovering the polynomial adaptively, weighted by the right hand side).

ev = eigvals(Matrix(A))
scatter(real.(ev), zero.(ev))
plot!(x -> (1 - omega * x)^5)


# Preconditioning#

Preconditioning is the act of creating an “affordable” operation “$$P^{-1}$$” such that $$P^{-1} A$$ (or $$A P^{-1}$$) is is well-conditoned or otherwise has a “nice” spectrum. We then solve the system

$P^{-1} A x = P^{-1} b \quad \text{or}\quad A P^{-1} \underbrace{(P x)}_y = b$

in which case the convergence rate depends on the spectrum of the iteration matrix

$I - \omega P^{-1} A .$

• The preconditioner must be applied on each iteration.

• It is not merely about finding a good initial guess.

There are two complementary techniques necessary for efficient iterative methods:

• “accelerators” or Krylov methods, which use orthogonality to adaptively converge faster than Richardson

• preconditioners that improve the spectrum of the preconditioned operator

Although there is ongoing research in Krylov methods and they are immensely useful, I would say preconditioning is 90% of the game for practical applications, particularly as a research area.

# Krylov subspaces#

All matrix iterations work with approximations in a Krylov subspace, which has the form

$K_n = \big[ b \big| Ab \big| A^2 b \big| \dotsm \big| A^{n-1} b \big] .$

This matrix is horribly ill-conditioned and cannot stably be computed as written. Instead, we seek an orthogonal basis $$Q_n$$ that spans the same space as $$K_n$$. We could write this as a factorization

$K_n = Q_n R_n$

where the first column $$q_0 = b / \lVert b \rVert$$. The $$R_n$$ is unnecessary and hopelessly ill-conditioned, so a slightly different procedure is used.

# Arnoldi iteration#

The Arnoldi iteration applies orthogonal similarity transformations to reduce $$A$$ to Hessenberg form, starting from a vector $$q_0 = b$$,

$A = Q H Q^* .$

Let’s multiply on the right by $$Q$$ and examine the first $$n$$ columns,

$A Q_n = Q_{n+1} H_n$

where $$H_n$$ is an $$(n+1) \times n$$ Hessenberg matrix.

# Aside: Conditioning of Arnoldi process#

The Arnoldi process is well-conditioned because $$Q$$ is orthogonal and

\begin{align}\begin{aligned} \lVert H_n \rVert \le \lVert Q_{n+1}^* \rVert \lVert A \rVert \lVert Q_n \rVert \le \lVert A \rVert .\\For a lower bound, we have\\ \sigma_{\min}(A)^2 \le x^* A^* A x \end{aligned}\end{align}

for all $$x$$ of norm 1. It must also be true for any $$x = Q_n y$$ where $$\lVert y\rVert = 1$$, thus

$\sigma_{\min}(A)^2 \le y^* Q_n^* A^* A Q_n y = y^* H_n^* Q_{n+1}^* Q_{n+1} H_n y = y^* H_n^* H_n y .$

# GMRES#

$A Q_n = Q_{n+1} H_n$

GMRES (Generalized Minimum Residual) minimizes

$\lVert A x - b \rVert$
over the subspace $$Q_n$$. I.e., $$x = Q_n y$$ for some $$y$$. By the Arnoldi recurrence, this is equivalent to
$\lVert Q_{n+1} H_n y - b \lVert$
which can be solved by minimizing
$\lVert H_n y - Q_{n+1}^* b \rVert .$
Since $$q_0 = b/\lVert b \lVert$$, the least squares problem is to minimize
$\Big\lVert H_n y - \lVert b \rVert e_0 \Big\rVert .$
The solution of this least squares problem is achieved by incrementally updating a $$QR$$ factorization of $$H_n$$.

## Notes#

• The solution $$x_n$$ constructed by GMRES at iteration $$n$$ is not explicitly available. If a solution is needed, it must be constructed by solving the $$(n+1)\times n$$ least squares problem and forming the solution as a linear combination of the $$n$$ vectors $$Q_n$$. The leading cost is $$2mn$$ where $$m \gg n$$.

• The residual vector $$r_n = A x_n - b$$ is not explicitly available in GMRES. To compute it, first build the solution $$x_n$$ as above.

• GMRES minimizes the 2-norm of the residual $$\lVert r_n \rVert$$ which is equivalent to the $$A^* A$$ norm of the error $$\lVert x_n - x_* \rVert_{A^* A}$$.

# More notes on GMRES#

• GMRES needs to store the full $$Q_n$$, which is unaffordable for large $$n$$ (many iterations). The standard solution is to choose a “restart” $$k$$ and to discard $$Q_n$$ and start over with an initial guess $$x_k$$ after each $$k$$ iterations. This algorithm is called GMRES(k). PETSc’s default solver is GMRES(30) and the restart can be controlled using the run-time option -ksp_gmres_restart.

• Most implementations of GMRES use classical Gram-Schmidt because it is much faster in parallel (one reduction per iteration instead of $$n$$ reductions per iteration). The PETSc option -ksp_gmres_modifiedgramschmidt can be used when you suspect that classical Gram-Schmidt may be causing instability.

• There is a very similar (and older) algorithm called GCR that maintains $$x_n$$ and $$r_n$$. This is useful, for example, if a convergence tolerance needs to inspect individual entries. GCR requires $$2n$$ vectors instead of $$n$$ vectors, and can tolerate a nonlinear preconditioner. FGMRES is a newer algorithm with similar properties to GCR.

# Experiments in PETSc#

• PETSc = Portable Extensible Toolkit for Scientific computing

• ./configure, make

• Depends on BLAS and LAPACK (default system or package manager)

• or ./configure --download-f2cblaslapack --download-blis

• Depends on MPI for parallelism (package manager)

• or ./configure --download-mpich (or --download-openmpi)

• docker run -it --rm jedbrown/petsc

• Lots of examples (mostly C and Fortran, some Python)

• Experimental bindings in Rust, Julia

• We’ll use src/snes/tutorials/ex15.c

• “p-Bratu”: combines p-Laplacian with Bratu nonlinearity

$-\nabla\cdot\big(\kappa(\nabla u) \nabla u\big) - \lambda e^u = f$

• Newton solver with Krylov on each iteration

# Simple preconditioners#

## Jacobi -pc_type jacobi#

$P_{\text{Jacobi}}^{-1} = D^{-1}$

where $$D$$ is the diagonal of $$A$$.

## Gauss-Seidel -pc_type sor#

$P_{GS}^{-1} = (L+D)^{-1}$

where $$L$$ is the (strictly) lower triangular part of $$A$$. The upper triangular part may be used instead, or a symmetric form

$P_{SGS}^{-1} = (L+U)^{-1} A \Big( I - (L+D)^{-1} \Big) .$

### Over-relaxation#

-pc_sor_omega 1. is default (Gauss-Seidel)

## Run p-Laplacian example#

$cd src/snes/tutorials$ make ex15
$./ex15 -da_refine 2 -dm_view$ ./ex15 -ksp_monitor -pc_type jacobi
\$ ./ex15 -snes_view


## Experiments#

• How does iteration count vary under grid refinement?

• How sensitive is it to parameters

• p-Laplacian -p $$> 1$$ and -epsilon $$> 0$$

• Bratu -lambda $$< 6.8$$

• How sensitive to -ksp_gmres_restart?

• -ksp_monitor_true_residual

• -ksp_view_eigenvalues

# Symmetric problems#

## Lanczos iteration: like GMRES for symmetric problems#

If $$A$$ is symmetric, then $$H = Q^T A Q$$ is also symmetric. Since $$H$$ is Hessenberg, this means $$H$$ is tridiagonal. Instead of storing $$Q_n$$, it is sufficient to store only the last two columns since the iteration satisfies a 3-term recurrence. The analog of GMRES for the symmetric case is called MINRES and is also useful for solving symmetric indefinite problems.

## Conjugate Gradients: changing the norm#

Instead of minimizing the $$A^T A$$ norm of the error, the Conjugate Gradient method minimizes the $$A$$ norm of the error. For $$A$$ to induce a norm, it must be symmetric positive definite. Jeremy Shewchuck’s guide to CG is an excellent resource.