# Fast numerical methods for solving linear PDEs Fast numerical methods for solving linear PDEs P.G....

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Fast numerical methods for solving linear PDEs

P.G. Martinsson, The University of Colorado at Boulder

Acknowledgements: Some of the work presented is joint work with Vladimir Rokhlin and Mark Tygert at Yale University.

In this talk, we will discuss numerical methods for solving the equation {−∆u(x) = g(x), x ∈ Ω,

u(x) = f(x), x ∈ Γ,

where Ω is a domain in R2 or R3 with boundary Γ.

More generally, we will consider stationary linear Boundary Value Problems

(BVP)

{ Au(x) = g(x), x ∈ Ω, B u(x) = f(x), x ∈ Γ,

such as:

• The equations of linear elasticity. • Stokes’ equation. • Helmholtz’ equation (at least at low and intermediate frequencies). • The Yukawa equation.

Outline of talk:

• Background:

– “Fast” methods and scaling of computational cost.

– “Iterative” vs. “direct” methods.

– Existing methodology for “fast” PDE solvers.

• New: Fast and direct methods for solving PDEs numerically:

– Techniques for representing functions and operators.

– Hierarchical computation of inverse operators.

– Matrix approximation via randomized sampling.

• Numerical examples.

Computational science — background

One of the principal developments in science and engineering over the last couple of decades has been the emergence of computational simulations.

We have achieved the ability to computationally model a wide range of phenomena. As a result, complex systems such as cars, micro-chips, new materials, city infra-structures, etc, can today be designed more or less entirely in a computer, with little or no need for physical prototyping.

This shift towards computational modelling has in many areas led to dramatic cost savings and improvements in performance.

What enabled all this was the development of faster more powerful computers, and the development of faster algorithms.

Growth of computing power and the importance of algorithms

1980 2000

1

10

100

1000

CPU speed

Year

Consider the computational task of solving a linear system of N algebraic equations with N unknowns.

Classical methods such as Gaussian elimination require O(N3) operations.

Using an O(N3) method, an increase in computing power by a factor of 1000 enables the solution of problems that are (1000)1/3 = 10 times larger.

Using a method that scales as O(N), problems that are 1000 times larger can be solved.

Growth of computing power and the importance of algorithms

1980 2000

1

10

100

1000

CPU speed

Year

1980 2000

103

104

Problem size

YearYear

Consider the computational task of solving a linear system of N algebraic equations with N unknowns.

Classical methods such as Gaussian elimination require O(N3) operations.

Using an O(N3) method, an increase in computing power by a factor of 1000 enables the solution of problems that are (1000)1/3 = 10 times larger.

Using a method that scales as O(N), problems that are 1000 times larger can be solved.

Growth of computing power and the importance of algorithms

1980 2000

1

10

100

1000

CPU speed

Year

1980 2000

103

104

105

106

Problem size

Year

O(N3) method

O(N) method

Year

Consider the computational task of solving a linear system of N algebraic equations with N unknowns.

Classical methods such as Gaussian elimination require O(N3) operations.

Using an O(N3) method, an increase in computing power by a factor of 1000 enables the solution of problems that are (1000)1/3 = 10 times larger.

Using a method that scales as O(N), problems that are 1000 times larger can be solved.

Definition of the term “fast”:

We say that a numerical method is “fast” if its computational speed scales as O(N) as the problem size N grows.

Methods whose complexity is O(N log(N)) or O(N(log N)2) are also called “fast”.

Caveat: It appears that Moore’s law is no longer operative.

Processor speed is currently increasing quite slowly.

The principal increase in computing power is coming from parallelization.

Successful algorithms must scale well both with problem size and with the number of processors that a computer has.

The methods of this talk all parallelize naturally.

“Iterative” versus ”direct” solvers

Two classes of methods for solving an N ×N linear algebraic system Ax = b.

Direct methods:

Examples: Gaussian elimination, LU factorizations, matrix inversion, etc.

Directly access elements or blocks of A.

Always give an answer. Robust.

Deterministic.No convergence analysis.

Have often been considered too slow for high performance computing.

Iterative methods:

Examples: GMRES, conjugate gradi- ents, Gauss-Seidel, etc.

Construct a sequence of vectors x1, x2, x3, . . . that (hopefully!) con- verge to the exact solution.

Many iterative methods access A only via its action on vectors.

High performance when they work well. O(N) solvers.

Often require problem specific pre- conditioners.

Solvers for linear boundary value problems.

↙

Direct discretization of the differ- ential operator via Finite Elements, Finite Differences, . . .

↓

N ×N discrete linear system. Very large, sparse, ill-conditioned.

↓

Fast solvers: iterative (multigrid), O(N), direct (nested dissection), O(N3/2).

↘

Conversion of the BVP to a Bound- ary Integral Operator (BIE).

↓

Discretization of (BIE) using Nyström, collocation, BEM, . . . .

↓

N ×N discrete linear system. Moderate size, dense, (often) well-conditioned.

↓

Iterative solver accelerated by fast matrix-vector multiplier, O(N).

Solvers for linear boundary value problems.

↙

Direct discretization of the differ- ential operator via Finite Elements, Finite Differences, . . .

↓

N ×N discrete linear system. Very large, sparse, ill-conditioned.

↓

Fast solvers: iterative (multigrid), O(N), direct (nested dissection), O(N3/2). O(N) direct solvers.

↘

Conversion of the BVP to a Bound- ary Integral Operator (BIE).

↓

Discretization of (BIE) using Nyström, collocation, BEM, . . . .

↓

N ×N discrete linear system. Moderate size, dense, (often) well-conditioned.

↓

Iterative solver accelerated by fast matrix-vector multiplier, O(N). O(N) direct solvers.

Reformulating a BVP as a Boundary Integral Equation.

The idea is to convert a linear partial differential equation

(BVP)

{ Au(x) = g(x), x ∈ Ω, B u(x) = f(x), x ∈ Γ,

to an “equivalent” integral equation

(BIE) v(x) + ∫

Γ k(x, y) v(y) ds(y) = h(x), x ∈ Γ.

Example:

Let us consider the equation

(BVP)

{−∆u(x) = 0, x ∈ Ω, u(x) = f(x), x ∈ Γ.

We make the following Ansatz:

u(x) = ∫

Γ

( n(y) · ∇y log |x− y|

) v(y) ds(y), x ∈ Ω,

where n(y) is the outward pointing unit normal of Γ at y. Then the boundary charge distribution v satisfies the Boundary Integral Equation

(BIE) v(x) + 2 ∫

Γ

( n(y) · ∇y log |x− y|

) v(y) ds(y) = 2f(x), x ∈ Γ.

• (BIE) and (BVP) are in a strong sense equivalent. • (BIE) is appealing mathematically (2nd kind Fredholm equation).

The BIE formulation has powerful arguments in its favor (reduced dimension, well-conditioned, etc) that we will return to, but it also has a major drawback:

Discretization of integral operators typically results in dense matrices.

In the 1950’s when computers made numerical PDE solvers possible, researchers faced a grim choice:

PDE-based: Ill-conditioned, N is too large, low accuracy.

Integral Equations: Dense system.

In most environments, the integral equation approach turned out to be simply too expensive.

(A notable exception concerns methods for dealing with scattering problems.)

The situation changed dramatically in the 1980’s. It was discovered that while KN (the discretized integral operator) is dense, it is possible to evaluate the matrix-vector product

v 7→ KN v in O(N) operations — to high accuracy and with a small constant.

A very succesful such algorithm is the Fast Multipole Method by Rokhlin and Greengard (circa 1985).

Combining such methods with iterative solvers (GMRES / conjugate gradient / . . . ) leads to very fast solvers for the integral equations, especially when second kind Fredholm formulations are used.

A prescription for rapidly solving BVPs:

(BVP)

{−∆ v(x) = 0, x ∈ Ω, v(x) = f(x), x ∈ Γ.

Convert (BVP) to a second kind Fredholm equation:

(BIE) u(x) + ∫

Γ

( n(y) · ∇y log |x− y|

) u(y) ds(y) = f(x), x ∈ Γ.

Discretize (BIE) into the discrete equation

(DISC) (I + KN ) uN = fN

where KN is a (typically de

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