minitutorial on a posteriori error estimation

Minitutorial on A Posteriori Error Estimation

Donald Estep

University Interdisciplinary Research ScholarEditor in Chief, SIAM/ASA Journal on Uncertainty Quantification

SIAM FellowChalmers Jubilee Professor

Department of StatisticsColorado State University

Research supported by Defense Threat Reduction Agency (HDTRA1-09-1-0036),Department of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699,

DE-FC02-07ER54909, DE-SC0001724, DE-SC0005304, INL00120133,DE0000000SC9279), Engility, Inc. (BY13-050SP), Idaho National Laboratory(00069249, 00115474), Lawrence Livermore National Laboratory (B573139,

B584647, B590495), National Science Foundation (DMS-0107832,DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559,

DMS-1065046, DMS-1016268, DMS-FRG-1065046, DMS-1228206), NationalInstitutes of Health (#R01GM096192), Sandia Corporation (PO299784)

Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 1/156

Collaborators and Colleagues

This minitutorial describes developments in a posteriori erroranalysis over a period of more than twenty years

The approach is built on a foundation of functional analysis thathas a broader application than just numerical analysis

My own contributions result from collaborations with a numberof senior people, students, and postdocs

Significant contributions have been made by many othercolleagues

I give a few references at the end and these contain furtherreferences


A Priori and A Posteriori Analysis

Two general approaches to analyzing numerical methods:

a priori The goal is to derive the general convergence andaccuracy properties of a numerical method for awide class of solutions

a posteriori The goal is derive a computational estimate thatevaluates the accuracy of information computedfrom a particular numerical solution

A priori analysis is the “classic” error analysis that dominatesthe research literature

A priori analysis is generally useless for determining theaccuracy of a particular computation

A posteriori analysis techniques yield accurate error estimatesbut do not describe general approximation properties well


Functionals, Dual Spaces, and Adjoints

Tools for Investigating Differential Equations

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Banach Spaces

We investigate properties of operators (maps) between spaces

We often assume the spaces in question are Banach spaces

A Banach space is a normed vector space such that Cauchysequences converge to a limit in the space

A sequence xn in a space X is a Cauchy sequence if forevery ε > 0 there is an N such that ‖xn − xm‖ < ε for alln,m > N

This is a computable criterion for checking convergence


What Information is to be Computed?

The starting point is the computation of particular information,or a quantity of interest, obtained from a numerical solution of amodel

Considering a particular quantity of interest is importantbecause obtaining solutions that are accurate everywhere isoften impossible

The application should begin by answering

What do we want to compute from the model?


Definition of Linear Functionals

Let X be a vector space with norm ‖ ‖

A bounded linear functional ` is a continuous linear map from Xto the reals R, ` ∈ L(X,R)

A linear functional is a one dimensional “snapshot” of a vector

For v in Rn fixed, the map

`(x) = v · x = (x, v)

is a linear functional on Rn

The linear functional on Rn given by the inner product with thebasis vector ei gives the ith component of a vector


Definition of Linear Functionals

For a continuous function f on [a, b],

`(f) =

∫ b

af(x) dx and `(f) = f(y) for a ≤ y ≤ b

are linear functionals

Statistical moments like the expected value E(X) of a randomvariable X are linear functionals

The Fourier coefficients of a continuous function f on [0, 2π],

cj =

∫ 2π

0f(x) e−ijx dx

are linear functionals of f


Sets of Linear Functionals

Presumably, it is easier to compute accurate snapshots thanthe entire functions

We may use a set of snapshots to attempt to describe afunction

In many situations, we settle for an “incomplete” set ofsnapshots

We are often happy with a small set of moments of a randomvariable

In practical applications of Fourier series, we truncate theinfinite series to a finite number of terms,

∞∑j=−∞

cjeijx →

J∑j=−J

cjeijx


Are There Many linear functionals?

Are there many bounded linear functionals on an arbitrarynormed vector space?

The celebrated Hahn-Banach theorem provides a way togenerate a large number

Hahn-Banach Theorem Let X be a Banach space and X0 asubspace of X. Suppose that F0(x) is a bounded linearfunctional defined on X0. There is a linear functional F definedon X such that F (x) = F0(x) for x in X0 and ‖F‖ = ‖F0‖.


Linear Functionals and Dual Spaces

We are interested in the structure of the set of all linearfunctionals

If X is a normed vector space, the vector space L(X,R) ofcontinuous linear functionals on X is called the dual space ofX, and is denoted by X∗

The dual space is a normed vector space under the dual normdefined for y ∈ X∗ as

‖y‖X∗ = supx∈X‖x‖X=1

|y(x)| = supx∈Xx 6=0

|y(x)|‖x‖

size = largest value of the snapshot on vectors of length 1



Consider X = Rn. Every vector v in Rn is associated with alinear functional Fv(·) = (·, v). This functional is bounded since|(x, v)| ≤ ‖v‖ ‖x‖ = C‖x‖

A classic result in linear algebra is that all linear functionals onRn have this form, i.e., we can make the identification(Rn)∗ ' Rn



Recall Holder’s inequality: if f ∈ Lp(Ω) and g ∈ Lq(Ω) withp−1 + q−1 = 1 for 1 ≤ p, q ≤ ∞, then

‖fg‖L1(Ω) ≤ ‖f‖Lp(Ω)‖g‖Lq(Ω)

Each g in Lq(Ω) is associated with a bounded linear functionalon Lp(Ω) when p−1 + q−1 = 1 and 1 ≤ p, q ≤ ∞ by

F (f) =

∫Ωg(x)f(x) dx

We can “identify” (Lp)∗ with Lq for 1 < p, q <∞, p−1 + q−1 = 1

The cases p = 1, q =∞ and p =∞, q = 1 are trickier


Duality for Hilbert Spaces

Hilbert spaces are Banach spaces with an inner product ( , )

Rn and L2 are Hilbert spaces

If X is a Hilbert space, then ψ ∈ X determines a boundedlinear functional via the inner product

`ψ(x) = (x, ψ), x ∈ X

The Riesz Representation theorem says this is the only kind oflinear functional on a Hilbert space

We can identify the dual space of a Hilbert space with itself

Linear functionals are commonly represented as inner products


Riesz Representors for Hilbert Spaces

Some useful choices of Riesz representors ψ for functions f :

I ψ = χω/|ω| gives the error in the average value of f over asubset ω ⊂ Ω, where χω is the characteristic function of ω

I ψ = δc gives the average value∮c f(s) ds of f on a curve c

in Rn, n = 2, 3, and ψ = δs gives the average value of fover a plane surface s in R3 (δ denotes the correspondingdelta function)

I We can obtain average values of derivatives using dipolessimilarly

I ψ = f/‖f‖ gives the L2 norm of f

Only some of these ψ have spatially local support


The Bracket Notation

We “borrow” the Hilbert space notation for the general case:

If x is in X and y is in X∗, we denote the value

y(x) = 〈x, y〉

This is called the bracket notation

The generalized Cauchy inequality is

|〈x, y〉| ≤ ‖x‖X ‖y‖X∗ , x ∈ X, y ∈ X∗


Motivation for the Adjoint Operator

Let X, Y be normed vector spaces

Assume that L ∈ L(X,Y ) is a continuous linear map

The goal is to compute a snapshot or functional value of theoutput

`(L(x)

), some x ∈ X

Some important questions:I Can we find a way to compute the snapshot value

efficiently?I What is the error in the snapshot value if approximations

are involved?I Given a collection of snapshot values, what can we say

about L?I Given a snapshot value, what can we say about x?


Definition of the Adjoint Operator

Let X, Y be normed vector spaces with dual spaces X∗, Y ∗

Assume that L ∈ L(X,Y ) is a continuous linear map

For each y∗ ∈ Y ∗ there is an x∗ ∈ X∗ defined by

x∗(x) = y∗(L(x)

)snapshot of x in X= snapshot of image L(x) of x in Y

The adjoint map L∗ : Y ∗ → X∗ satisfies the bilinear identity

〈L(x), y∗〉 = 〈x, L∗(y∗)〉, x ∈ X, y∗ ∈ Y ∗


Adjoint of a Matrix

Let X = Rm and Y = Rn with the standard inner product andnorm

L ∈ L(Rm,Rn) is associated with a n×m matrix A:

A =

a11 · · · a1m...

...an1 · · · anm

, y =

y1...yn

, x =

x1...xm

and

yi =m∑j=1

aijxj , 1 ≤ i ≤ n


Adjoint of a Matrix

The bilinear identity reads

(Lx, y) = (x, L∗y), x ∈ Rm, y ∈ Rn.

For a linear functional y∗ = (y∗1, · · · , y∗n)> ∈ Y ∗

L∗y∗(x) = y∗(L(x)) =

(y∗1, · · · , y∗n),

∑m

j=1 a1jxj...∑m

j=1 anjxj

=

m∑j=1

y∗1a1jxj + · · ·m∑j=1

y∗nanjxj

=

m∑j=1

( n∑i=1

y∗i aij)xj


Adjoint of a Matrix

L∗(y∗) is given by the inner product with y = (y1, · · · , ym)>

where

yj =

n∑i=1

y∗i aij .

The matrix A∗ of L∗ is

A∗ =

a∗11 · · · a∗1n...

...a∗m1 · · · a∗mn

=

a11 a21 · · · an1...

...a1m a2m · · · anm

= A>.


Practical Point about Defining an Adjoint

We can define L∗ for a linear operator L : X → Y byconsidering L on a dense subset of X

We define the adjoint to the restriction of L on a dense subsetof X

If this “restricted” adjoint is uniformly continuous on the densesubset, then the Hahn-Banach theorem implies that therestriction can be extended continuously to all of X

This is useful in the weak formulation of differential equations inSobolev spaces


Properties of Adjoint Operators

Theorem Let X, Y , and Z be normed linear spaces. ForL1, L2 ∈ L(X,Y ):

L∗1 ∈ L(Y ∗, X∗)

‖L∗1‖ = ‖L1‖0∗ = 0

(L1 + L2)∗ = L∗1 + L∗2

(αL1)∗ = αL∗1, all α ∈ R

If L2 ∈ L(X,Y ) and L1 ∈ L(Y,Z), then (L1L2)∗ ∈ L(Z∗, X∗)and

(L1L2)∗ = L∗2L∗1.


Adjoints for Differential Operators

The adjoint of the differential operator L

(Lu, v)→ (u, L∗v)

is obtained by a succession of integration by parts

Boundary terms involving functions and derivatives arise fromeach integration by parts

We use a two step process

1. We first compute the formal adjoint by assuming that allfunctions have compact support and ignoring boundaryterms

2. We then compute the adjoint boundary and data conditionsto make the bilinear identity hold


Formal Adjoints

Consider

Lu(x) = − d

dx

(a(x)

d

dxu(x)

)+

d

dx(b(x)u(x))

on [0, 1]. Integration by parts gives

−∫ 1

0

d

dx

(a(x)

d

dxu(x)

)v(x) dx

=

∫ 1

0a(x)

d

dxu(x)

d

dxv(x) dx− a(x)

d

dxu(x)v(x)

∣∣∣∣10

= −∫ 1

0u(x)

d

dx

(a(x)

d

dxv(x)

)dx+ u(x)a(x)

d

dxv(x)

∣∣∣∣10


Formal Adjoints

∫ 1

0

d

dx(b(x)u(x))v(x) dx

= −∫ 1

0u(x)b(x)

d

dxv(x) dx+ b(x)u(x)v(x)

∣∣∣∣10

,

Neglecting boundary terms gives the formal adjoint

Lu(x) = − d

dx

(a(x)

d

dxu(x)

)+

d

dx(b(x)u(x))

=⇒ L∗v = − d

dx

(a(x)

d

dxv(x)

)− b(x)

d

dx(v(x))


Formal Adjoints

In higher space dimensions, we use the divergence theorem

A general linear second order differential operator L in Rn:

L(u) =

n∑i=1

n∑j=1

aij∂2u

∂xi∂xj+

n∑i=1

bi∂u

∂xi+ cu,

where aij, bi, and c are functions of x1, x2, · · · , xn. Then,

L∗(u) =

n∑i=1

n∑j=1

∂2(aijv)

∂xi∂xj−

n∑i=1

∂(biv)

∂xi+ cv


Formal Adjoints

It can be verified directly that

vL(u)− uL∗(v) =

n∑i=1

∂pi∂xi

,

where

pi =

n∑j=1

(aijv

∂u

∂xj− u∂(aijv)

∂xj

)+ biuv.

The divergence theorem yields∫Ω

(vL(u)− uL∗(v)) dx =

∫∂Ωp · nds = 0,

where p = (p1, · · · , pn) and n is the outward normal in ∂Ω.


Formal Adjoints

A typical term,

va11∂2u

∂x21

= va11∂

∂x1

(∂u

∂x1

)=

∂

∂x1

(va11

∂u

∂x1

)− ∂(a11v)

∂x1

∂u

∂x1

=∂

∂x1

(va11

∂u

∂x1

)− ∂

∂x1

(u∂(a11v)

∂x1

)+ u

∂2(a11v)

∂x21

yielding

va11∂2u

∂x21

− u∂2(a11v)

∂x21

=∂

∂x1

(a11v

∂u

∂x1− u∂(a11v)

∂x1

).


Formal Adjoints

Important examples:

grad∗ = −div

div∗ = −grad

curl∗ = curl

Lu =∑|α|≤p

aα(x)Dαu

thenL∗v =

∑|α|≤p

(−1)|α|Dα(aα(x)v(x)).


Adjoint Boundary Conditions

In the second stage, we deal with the boundary terms that ariseduring integration by parts

The standard adjoint boundary conditions are the minimalconditions required so the bilinear identity holds true

The form of the boundary conditions imposed on the differentialoperator is important, but not the values

We assume homogeneous boundary values for the differentialoperator when determining the adjoint conditions

We may need to use other boundary conditions, e.g. if thequantity of interest is determined on a boundary



Consider s′′(x) = f(x), 0 < x < 1,

s(0) = s′(0) = 0

Integrating by parts,∫ 1

0(s′′v − sv′′) dx = (vs′ − sv′)

∣∣10

The boundary conditions imply the contributions at x = 0vanish, while at x = 1 we have

v(1)s′(1)− v′(1)s(1)

The adjoint boundary conditions are v(1) = v′(1) = 0 since wecannot specify s′(1) or s(1)



Recall ∫Ω

(u∆v − v∆u) dx =

∫∂Ω

(u∂v

∂n− v ∂u

∂n

)ds,

The Dirichlet and Neumann boundary value problems for theLaplacian are their own adjoints



Let Ω ⊂ R2 be bounded with a smooth boundary and let s =arclength along the boundary

Consider −∆u = f, x ∈ Ω,∂u∂n + ∂u

∂s = 0, x ∈ ∂Ω

Since∫Ω

(u∆v − v∆u) dx =

∫∂Ω

(u

(∂v

∂n− ∂v

∂s

)− v

(∂u

∂n+∂u

∂s

))ds,

the adjoint problem is−∆v = g, x ∈ Ω,∂v∂n −

∂v∂s = 0, x ∈ ∂Ω.


Adjoint for an Evolution Operator

For an initial value problem, we have ddt and an initial condition

Now ∫ T

0

du

dtv dt = u(t)v(t)

∣∣T0−∫ T

0udv

dtdt

The boundary term at 0 vanishes because u(0) = 0

The adjoint is a final-value problem with “initial” conditionv(T ) = 0

The adjoint problem has −dvdt and time “runs backwards”


Adjoint for an Evolution Operator

Lu = du

dt −∆u = f, x ∈ Ω, 0 < t ≤ T,u = 0, x ∈ ∂Ω, 0 < t ≤ T,u = u0, x ∈ Ω, t = 0

=⇒

L∗v = −dv

dt −∆v = ψ, x ∈ Ω, T > t ≥ 0,

v = 0, x ∈ ∂Ω, T > t ≥ 0,

v = 0, x ∈ Ω, t = T

A useful alternative is

=⇒

L∗v = −dv

dt −∆v = 0, x ∈ Ω, T > t ≥ 0,

v = 0, x ∈ ∂Ω, T > t ≥ 0,

v = ψ, x ∈ Ω, t = T


The Usefulness of Duality and Adjoints

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The Dual Space is Well Behaved

The dual space can be better behaved than the original normedvector space

If X is a normed vector space over R, then X∗ is a Banachspace whether or not X is a Banach space

This can be exploited in analysis in X, e.g. “weak” convergenceresults


Condition of an Operator

There is an intimate connection between the adjoint operatorand the stability properties of the original operator

The singular values of a matrix L are the square roots of theeigenvalues of the square, symmetric transformations L∗L orLL∗

This connects the condition number of a matrix L to L∗


Solving Problems for Operators

Given normed vector spaces X and Y , an operator L(X,Y ),and b ∈ Y , find x ∈ X such that

Lx = b

A necessary condition that b is in the range of L is y∗(b) = 0 forall y∗ in the null space of L∗

This is a sufficient condition if the range of L is closed in Y

If A is an n×m matrix, a necessary and sufficient condition forthe solvability of Ax = b is b is orthogonal to all linearlyindependent solutions of A>y = 0


Solving Problems for Operators

When X is a Hilbert space and L ∈ L(X,Y ), then the range ofL∗ is a subset of the orthogonal complement of the null spaceof L

If the range of L∗ is “large”, then the orthogonal complement ofthe null space of L must be “large” and the null space of L mustbe “small”

The existence of sufficiently many solutions of thehomogeneous adjoint equation L∗φ = 0 implies there is at mostone solution of Lu = b for a given b

This is the basis for the Holmgren Uniqueness Theorem inPDEs


Green’s Functions

Suppose we wish to compute a functional `(x) of the solutionx ∈ Rn of a n× n system

Lx = b

For a linear functional `(·) = (·, ψ), we define the adjointproblem

L∗φ = ψ

Variational analysis yields the representation formula

`(x) = (x, ψ) = (x,L∗φ) = (Lx, φ) = (b, φ)

We can compute many solutions by computing one adjointsolution and taking inner products


Green’s Functions

For −∆u = f, x ∈ Ω,

u = 0, x ∈ ∂Ω

the Green’s function solves

−∆φ = δx (delta function at x)

This yieldsu(x) = (u, δx) = (f, φ)

The generalized Green’s function solves the adjoint problemwith general functional data, rather than just δx

The imposition of the adjoint boundary conditions is crucial


Nonlinear Operators

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Nonlinear Operators

There is no unique “natural” adjoint for a general nonlinearoperator

We assume that the Banach spaces X and Y are Sobolevspaces and use ( , ) for the L2 inner product, and so forth

We define the adjoint for a specific kind of nonlinear operator

We assume f is a nonlinear map from X into Y , where thedomain of f is a convex set


A Perturbation Operator

Consider a “true” u and “approximation” U in the domain of fand define the “error” e = U − u

u and e are “unknown”

DefineF (e) = f(u+ e)− f(u),

The domain of F is

v ∈ X| v + u ∈ domain of f

Assume the domain of F is independent of e and dense in X

Note that 0 is in the domain of F and F (0) = 0


A Perturbation Operator

Two reasons to work with functions of this form:

I This is the kind of nonlinearity that arises when estimatingthe error of a numerical solution or studying the effects ofperturbations

I Nonlinear problems typically do not enjoy the globalsolvability that characterizes linear problems, only a localsolvability


Definition 1

A common definition is based on the bilinear identity

An operator A∗(e) is an adjoint operator corresponding to F if

(F (e), w) = (e,A∗(e)w) for all e ∈ domain of F, w ∈ domain of A∗

This is an adjoint operator associated with F , not the adjointoperator to F


Definition 1

Suppose that F can be represented as F (e) = A(e)e, whereA(e) is a linear operator with the domain of F contained in thedomain of A

For a fixed e in the domain of F , define the adjoint of Asatisfying

(A(e)w, v) = (w,A∗(e)v)

for all w ∈ domain of A, v ∈ domain of A∗

Substituting w = e shows this defines an adjoint of F as well

If there are several such linear operators A, then there will beseveral different possible adjoints.


Definition 1

Let (t, x) ∈ Ω = (0, 1)× (0, 1), with X = X∗ = Y = Y ∗ = L2

denoting the space of periodic functions in t and x, with periodequal to 1

Consider a periodic problem

F (e) =∂e

∂t+ e

∂e

∂x+ ae = f

where a > 0 is a constant and the domain of F is the set ofcontinuously differentiable functions.


Definition 1

We can write F (e) = Ai(e)e where

A1(e)v =∂v

∂t+ e

∂v

∂x+ av =⇒ A∗1(e)w = −∂w

∂t− ∂(ew)

∂x+ aw

A2(e)v =∂v

∂t+

(a+

∂e

∂x

)v =⇒ A∗2(e)w = −∂w

∂t+

(a+

∂e

∂x

)w

A3(e)v =∂v

∂t+

1

2

∂(ev)

∂x+ av =⇒ A∗3(e)w = −∂w

∂t− e

2

∂w

∂x+ aw


Definition 2

If the nonlinearity is Frechet differentiable, we base the seconddefinition of an adjoint on the integral mean value theorem

The integral mean value theorem states

f(U) = f(u) +

∫ 1

0f ′(u+ se) ds e

where e = U − u and f ′ is the Frechet derivative of f


Definition 2

We rewrite this as

F (e) = f(U)− f(u) = A(e)e

with

A(e) =

∫ 1

0f ′(u+ se) ds

Note that we can apply the integral mean value theorem to F :

A(e) =

∫ 1

0F ′(se) ds

To be precise, we should discuss the smoothness of F


Definition 2

For a fixed e, the adjoint operator A∗(e), defined in the usualway for the linear operator A(e), is said to be an adjoint for F

Continuing the previous example,

F ′(e)v =∂v

∂t+ e

∂v

∂x+

(a+

∂e

∂x

)v.

After some technical analysis of the domains of the operatorsinvolved,

A∗(e)w = −∂w∂t− e

2

∂w

∂x+ aw.

This coincides with the third adjoint computed above


A Point about Linearization

The linearization approach is used in many practicalapplications

Typically the linearization in the integral mean value theorem isapproximated by linearization at a “known” point

This raises the issue of the effect of such linearization “error” onany subsequent use of the adjoint

For many analysis purposes, the issue is the accuracy of theinverse of the approximation



Consider F : R2 → R2:

F (u) =

(u2

1 + 3u2

u1 eu2

)

For perturbation ε = (ε1, ε2)>,

E(ε) = F (u+ ε)− F (u) =

(2u1 + ε1 3

eu2+ε2 u1 eu2(eε2−1ε2

)) (ε1

ε2

)

This yields

E(ε)∗ =

(2u1 + ε1 eu2+ε2

3 u1 eu2(eε2−1ε2

))



For small ε,E(ε)∗ ≈ (F ′(u))∗

where F ′(u) is the Jacobian,

F ′(u) =

(2u1 3eu2 u1 e

u2

)

In practical computations, we use F ′(u)∗

For |u1| bounded away from√

3/2,(F ′(u)∗

)−1 ≈(E(v)∗

)−1 forall ε with sufficiently small norm

If |u1| ≈√

3/2,(F ′(u)∗

)−1 may not be close to(E(ε)∗

)−1


A Posteriori Error Analysis

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A Posteriori Error Analysis

Problem: Estimate the error in a quantity of interest computedusing a numerical solution of a differential equation

We assume that the quantity of information can be representedas a linear functional of the solution

We use the adjoint problem associated with the linear functional


What about Convergence Analysis?

Recall the standard a priori convergence result for an initialvalue problem

y = f(y), 0 < t,

y(0) = y0

Let Y ≈ y be an approximation associated with time step ∆t

A typical a priori (Gronwall argument) bound is

‖Y − y‖L∞(0,t) ≤ C eLt ∆tp∥∥∥∥dp+1y

dtp+1

∥∥∥∥L∞(0,t)

L is often large in practice, e.g. L ∼ 100− 10000

It is typical for an a priori convergence bound to be orders ofmagnitude larger than the error


A Linear Algebra Problem

We compute a quantity of interest (u, ψ) from a solution of

Au = b

If U is an approximate solution, we estimate the error

(e, ψ) = (u− U,ψ)

We can compute the residual

R = AU − b

Using the adjoint problem A>φ = ψ, variational analysis gives

|(e, ψ)| = |(e,A>φ)| = |(Ae, φ)| = |(R,φ)|

We solve for φ numerically to compute the estimate


Finite Element Method for Elliptic Problem

The a posteriori analysis naturally applies to finite elementdiscretizations

Variational form of the equation is formed by multiplying by atest function, integrating over space and time, and usingintegration by parts to reduce derivative orders

Appropriate function spaces must be chosen

The finite element approximation uses finite dimensionalfunction spaces, e.g. piecewise polynomials

Finite volume and finite difference methods are written as finiteelements + quadrature



Approximate u : Rn → R solvingLu = f, x ∈ Ω,

u = 0, x ∈ ∂Ω,

where

L(D,x)u = −∇ · a(x)∇u+ b(x) · ∇u+ c(x)u(x),

I Ω ⊂ Rn, n = 1, 2, 3, is a convex polygonal domainI a = (aij), where ai,j are continuous and there is a a0 > 0

such that v>av ≥ a0 for all v ∈ Rn \ 0 and x ∈ Ω

I b = (bi) where bi is continuousI c and f are continuous



The variational formulation reads

Find u ∈ H10 (Ω) such that

A(u, v) = (a∇u,∇v) + (b · ∇u, v) + (cu, v) = (f, v)

for all v ∈ H10 (Ω)

H10 (Ω) is the space of L2(Ω) functions whose first derivatives

are in L2(Ω)

This says that the solution solves the “average” form of theproblem for a large number of weights v



We construct a triangulation of Ω into simplices, or elements,such that boundary nodes of the triangulation lie on ∂Ω

Th denotes a simplex triangulation of Ω that is locallyquasi-uniform, i.e. no arbitrarily long, skinny triangles

We use the length of the longest edge hK of K ∈ Th to quantifysize



U = 0

Th

ΚhΚ

Ω

Triangulation of the domain Ω



Vh denotes the space of functions that areI continuous on Ω

I piecewise linear with respect to ThI zero on the boundary

Vh ⊂ H10 (Ω), and for smooth functions, the error of interpolation

into Vh is O(h2) in ‖ ‖

The finite element method is:

Compute U ∈ Vh such that A(U, v) = (f, v) for all v ∈ Vh

This is Galerkin orthogonality since it is equivalent toA(U − u, v) = 0 for all v ∈ Vh


A Posteriori Analysis for an Elliptic Problem

We assume that quantity of interest is the functional (u, ψ)

The generalized Green’s function φ solves the weak adjointproblem : Find φ ∈ H1

0 (Ω) such that

A∗(v, φ) = (∇v, a∇φ)− (v,div (bφ)) + (v, cφ) = (v, ψ)

for all v ∈ H10 (Ω),

corresponding to the adjoint problem L∗(D,x)φ = ψ



We now estimate the error e = U − u:

(e, ψ) = (∇e, a∇φ)− (e,div (bφ)) + (e, cφ)

= (a∇e,∇φ) + (b · ∇e, φ) + (ce, φ)

= (a∇u,∇φ) + (b · ∇u, φ) + (cu, φ)

− (a∇U,∇φ)− (b · ∇U, φ)− (cU, φ)

= (f, φ)− (a∇U,∇φ)− (b · ∇U, φ)− (cU, φ)

The weak residual of U is

R(U, v) = (f, v)− (a∇U,∇v)− (b · ∇U, v)− (cU, v), v ∈ H10 (Ω)

R(U, v) = 0 for v ∈ Vh but not for general v ∈ H10 (Ω)



πhφ denotes an approximation of φ in Vh

Theorem The error representation is,

(e, ψ) = (f, φ− πhφ)− (a∇U,∇(φ− πhφ))

− (b · ∇U, φ− πhφ)− (cU, φ− πhφ),

Subtracting the projection φ− πhφ is not needed theoreticallybut useful in practice

The subtraction is required for standard adaptive error control



We approximate φ using a higher order finite element methodor on a significantly finer mesh

For a second order elliptic problem, good results are obtainedusing the space V 2

h

The approximate generalized Green’s function Φ ∈ V 2h solves

A∗(v,Φ) = (∇v, a∇Φ)−(v,div (bΦ))+(v, cΦ) = (v, ψ) for all v ∈ V 2h

Theorem The approximate error representation is

(e, ψ) ≈ (f,Φ− πhΦ)− (a∇U,∇(Φ− πhΦ))

− (b · ∇U,Φ− πhΦ)− (cU,Φ− πhΦ)



−∆u = 200π2 sin(10πx) sin(10πy), (x, y) ∈ Ω = [0, 1]× [0, 1],

u = 0, (x, y) ∈ ∂Ω

The solution is u = sin(10πx) sin(10πy)

-1

1

0u

x y 0.0 0.1 1.0 10.0 100.0percent error

0

1

2

3

erro

r/est

imat

e


A Posteriori Analysis for an Initial Value Problem

We develop an estimate fory′ = f(y), 0 < t ≤ T,y(0) = y0

where f is smooth

Discretization of domain:

0 = t0 < t1 < · · · < tN = T,

In = (tn−1, tn], kn = tn − tn−1, k|In = kn

Space of approximation:

Pq(In) = polynomials of degree q and less on InW qn = w : w ∈ Pq(In), Wn = w : w|In ∈W q

nπk = interpolant/projection operator into Pq(In) for each n



The continuous Galerkin method of order q (CG(q)): Forn = 1, · · · , N ,∫

In(Y ′, v) dt =

∫In

(f(Y ), v) dt all v ∈W q−1n

Y +n−1 = Y −n−1

with Y0 = y0, U+n = limt↓tn U(t), U−n = limt↑tn U(t)

There is a discontinuous Galerkin method

Quantity of interest:∫ T

0 (y, ψ) dt

ψ = 1/T gives the average value



Linearization to define adjoint:

f ′ = f ′(y, Y ) =

∫ 1

0

∂f

∂y(sy + (1− s)Y ) ds

Adjoint problem: −φ′ = (f ′)∗φ, T > t ≥ 0,

φ(T ) = 0

time runs “backwards” but note the “-” on the time derivative

In practice, f ′ → f ′(Y )



Substituting∫ T

0(e, ψ) dt =

N∑i=1

∫In

(e,−φ′ − (f ′)∗φ) dt

∫In

(e,−φ′) dt = −(e−n , φn) + (e+n−1, φn) +

∫In

(e′, φ) dt∫In

(e, (f ′)∗φ) dt =

∫In

(f(y)− f(Y ), φ) dt

Theorem The error representation is∫ T

0(e, ψ) dt =

N∑n=1

∫In

(Rn, πkφ− φ) dt

Rn = Y ′ − f(Y ) on In



The chaotic Lorenz problemu′1 = −10u1 + 10u2,

u′2 = 28u1 − u2 − u1u3, 0 < t,

u′3 = −83u3 + u1u2,

Chaotic behavior affects numerical solutions as well



0

50

-10

30 -20

30

100% error at t=18

U3

0

50

-10

30 -20

30

2% error on [0,30]

U1U2

U3

U1U2



10

.001

.002

.003

.004

.005

.006

Time0 10 15 20 25 30

1E-7

1E-5

1E-3

0.11

10100

Dis

tanc

e B

etw

een

Solu

tion

s

T ime

ErrorsDecrease

ErrorsIncrease

5 0 5

Separat ix

The distance between the two numerical solutions

The errors follow an increasing trend, but decrease as well asincrease



Time

Com

pone

nt E

rror

U1U2U3

0 5 10 15 20

0

10

-10 0

0.8

1

1.2

1.4

Simulation Final TimeC

ompo

nent

wis

e Er

ror/E

stim

ate

0 189

U1 U2 U3

The accuracy of the estimate in the pointwise error


A Posteriori Analysis for a Reaction-Diffusion System

A system of D reaction-diffusion equations consisting of dparabolic equations and D − d ordinary equations for u ∈ RD:

u′i −∇ · (εi(u)∇ui) = fi(u), (x, t) ∈ Ω× R+, 1 ≤ i ≤ D,ui(x, t) = 0, (x, t) ∈ ∂Ω× R+, 1 ≤ i ≤ d,u(x, 0) = u0(x), x ∈ Ω,

Assumptions:

εi(u) ≥ ε0 > 0 for 1 ≤ i ≤ d and εi(u) ≡ 0 for the restΩ is a convex polygonal domain with boundary ∂Ω

ε, f have smooth second derivatives

upi = ui for 1 ≤ i ≤ d and upi = 0 for d < i ≤ D



Discretization:

0 = t0 < t1 < t2 < · · · < tN = T, In = (tn−1, tn], kn = tn − tn−1

Tn is a triangulation of Ω for t ∈ In

Space

Tim

e

Ω



Discrete space: polynomials in time and piecewise polynomialsin space on each space-time “slab” Sn = Ω× In

Vn ⊂ (H10 (Ω))d × (H1(Ω))D−d denotes the space of piecewise

linear continuous vector-valued functions v(x) ∈ RD defined onTn, where the first d components of v are zero on ∂Ω

Define

W qn =

w(x, t) : w(x, t) =

q∑j=0

tjvj(x), vj ∈ Vn, (x, t) ∈ Sn.

W q denotes the space of functions defined on the space-timedomain Ω× R+ such that v|Sn ∈W

qn for n ≥ 1



The cG(q)-cG(1) approximation U ∈W q satisfies U−0 = P0u0

and for n ≥ 1,∫ tn

tn−1

((U ′i , vi) + (εi(U)∇Ui,∇vi)

)dt =

∫ tn

tn−1

(fi(U), vi) dt

for all v ∈W q−1n , 1 ≤ i ≤ D,

U+n−1 = PnU

−n−1

Pn is the L2 projection into Vn

The approximation is continuous across time nodes wherethere is no mesh change



Linearization for adjoint:

εi = εi(u, U) =

∫ 1

0εi(us+ U(1− s)

)ds,

βij = βij(u, U) =

∫ 1

0

∂εj∂ui

(us+ U(1− s))∇(uis+ Ui(1− s)

)ds,

fij = fij(u, U) =

∫ 1

0

∂fj∂ui

(us+ U(1− s))ds.

Adjoint problem:−φ′i −∇ ·

(εi∇φi

)+∑D

j=1 βji · ∇φj −∑D

j=1 fijφj = ψi,

(x, t) ∈ Ω× (tn, 0], 1 ≤ i ≤ D,φi (x, t) = 0, (x, t) ∈ ∂Ω× (tN , 0], 1 ≤ i ≤ d,φ(x, tN ) = 0, x ∈ Ω,



Theorem∫ T

0(e, ψ) dt = (u0 − P0u0, φ(0))

+

N∑n=1

∫In

(U ′, πkπhφ− πhφ) + (ε(U)∇U,∇(πkπhφ− πhφ))

− (f(U), πkπhφ− πhφ))dt

+

N∑n=1

∫In

(U ′, πhφ− φ) + (ε(U)∇U,∇(πhφ− φ))

− (f(U), πhφ− φ))dt


Forward Error Propagation

(4)



We briefly describe the classic alternative for error estimation

For the initial value problem

y′ = f(y)

write the equation for a numerical solution Y ≈ y

Y ′ = F (Y )

whereF (·) ≈ f(·)



Subtraction yields an equation for the error e = y − Y

e′ = f(y)− F (Y ) = f(y)− f(Y ) + f(Y )− F (Y )

Linearizinge′ ≈ f ′(Y )e+R(Y )

with the defect R(Y ) = f(Y )− F (Y )

We can solve the systemY ′ = F (Y ),

e′ = f ′(Y )e+R(Y )

to compute the numerical solution and an approximation to theerror simultaneously



Technical issues that must be addressed:I The same issue regarding the choice of linearization point

that affects the adjoint-based approachI The defect must be interpreted in a correct mathematical

senseI The defect can be approximated in multiple ways without a

clearly best choice

Computational evaluation of the two approaches:I The forward propagation is cheaper when the goal is to

estimate the error in the solution itselfI The adjoint-based method is cheaper when the goal is to

estimate the error in a handful of quantity of interestfunctionals


My Preference

I prefer the adjoint-based approach becauseI I have rarely encountered situations in which error in the

solution is importantI The dual problem provides quantitative information about

stabilityI It seems easier to quantify relative contributions of different

sources of errorI It seems easier to adapt the method to deal with such

things as multiphysics, finite iteration, quadrature, etc.

A lot of nonsense is written about relative merits of the twoapproaches


Adjoints and Stability

(8)


The Adjoint and Stability

The solution of the adjoint problem scales local perturbations toglobal effects on a solution

The adjoint problem carries stability information about thequantity of interest computed from the solution

We can use the adjoint problem to investigate stability


Condition Numbers and Stability Factors

The classic error bound for an approximate solution U ofAu = b is

‖e‖ ≤ Cκ(A)‖R‖, R = AU − b

The condition number κ(A) = ‖A‖ ‖A−1‖ measures stability

κ(A) =1

distance from A to singular matrices

The a posteriori estimate |(e, ψ)| = |(R,φ)| yields

|(e, ψ)| ≤ ‖φ‖ ‖R‖

The stability factor ‖φ‖ is a weak condition number for thequantity of interest

We can obtain κ from ‖φ‖ by taking the sup over all ‖ψ‖ = 1


Condition Numbers and Stability Factors

We consider the problem of computing (u, e1) from the solutionof

Au = b

where A is a random 800× 800 matrix

The condition number is 1.7× 105

The stability factor is 16


The Condition of the Lorenz Problem

u10

0

Orbiting

OrbitingLeaving orbit

u 2

×10-7

7.6

7

Res

idua

l

10 20t

1

106

10t

20

Stab

ility

Fac

tor


The Quantity of Interest is Important

The rate that errors grow depends strongly on the informationbeing computed

We consider the average distance from a solution to the originover a long time interval

u3

distance

0 10 20 30 40 50 60 70 800

10

20

30

40

50

Time

Dis

tanc

e fr

om O

rigin

CoarseFine


The Quantity of Interest is Important

We compare to an ensemble average of 100 accurate solutionscomputed using time step .0001 for 15 time units

Coarse Solutions Fine Solutions Ensemble AveEnd Time Ave Var Ave Var Ave Var

20 27.620 52.011 27.622 51.94780 26.470 79.461 26.467 79.231

320 26.3 83.7 26.3 83.0 26.3 83.7


The Bistable Problem

A parabolic problem:u′ − ε∆u = u− u3, x ∈ Ω, t > 0

Neumann boundary conditions x ∈ ∂Ω, t > 0

u(x, 0) = u0(x), x ∈ Ω

Long time dynamics depends on space dimension1D Metastable behavior of long periods of nearly

motionless behavior punctuated by rapidtransients

2D Approximate motion by mean curvature

The solutions in two dimensions are much more stable withrespect to perturbations



Example of metastable solution in one space dimension

1

-10

1 0

167

timex

U

t

.024

0

0 170

Space

Evolution of Solution Norms of Adjoint Components

Time



Example of solution in two space dimensions

t = 18

11

-1

1t = 54

11

-1

1

t = 180t = 144

11

-1

1

11

-1

1

time0 150

100 1D

2D

.1

Norms of Adjoint SolutionsEvolution of Solution


Adjoints and Adaptive Error Control

(7)


Idea Behind Adaptive Error Control

The possibility of accurate error estimation suggests thepossibility of optimizing discretizations

Unfortunately, cancellation of errors significantly complicatesthe optimization problem

In fact, there is no good theory for adaptive control of error

There is good theory for adaptive control of error bounds

The standard approach is based on optimal control theory

The stability information in adjoint-based a posteriori errorestimates is useful for this


Bounds and Estimates

We consider the problem of computing (u, e1) from the solutionof

Au = b

where A is a random 800× 800 matrix

The condition number of A is 1.7× 105

estimate of the error in the quantity of interest ≈ 1.4× 10−14

a posteriori error bound for the quantity of interest ≈ 2.2× 10−10

The traditional error bound for the error ≈ 5.7× 10−5



An abstract a posteriori error estimate has the form

|(e, ψ)| =∣∣(Residual, Adjoint Weight

)∣∣Given a tolerance TOL, a given discretization is refined if∣∣(Residual, Adjoint Weight

)∣∣ ≥ TOL

Refinement decisions are based on a bound consisting of asum of element contributions

|(e, ψ)| ≤∑

elements K

∣∣(Residual, Adjoint Weight)K

∣∣where ( , )K is the inner product on K

The element contributions in the bound do not cancel



There is no cancellation of errors across elements in thebound, so optimization theory yields

Principle of Equidistribution: The optimal discretization is one inwhich the element contributions are equal

The adaptive strategy is to refine some of the elements with thelargest element contributions

The adjoint weighted residual approach is different thantraditional approaches because the element residuals arescaled by an adjoint weight, which measures how much error inthat element affects the solution on other elements

An optimal solution is approximated iteratively starting with acoarse mesh



1: Choose error tolerance TOL and compute contributiontolerances

2: Choose initial coarse discretization parameters3: Compute initial coarse solution4: Compute error estimate for initial solution5: while Error Estimate > TOL do6: Adjust discretization parameters depending on the

contributions relative to the contribution tolerances7: Compute solution8: Compute error estimate for solution9: end while



−∇ ·

((.05 + tanh(10(x− 5)2 + 10(y − 1)2))∇u

)+

(−100

0

)· ∇u = 1, (x, y) ∈ Ω = [0, 10]× [0, 2],

u = 0, (x, y) ∈ ∂Ω

Convection



1

01

2 02

46

8100

2

4

6x 10-4

yx

01

2 02

46

8100

2

4

6x 10-4

yx

Final meshes for an average error of 4% 24,000 elementsversus 3500 elements


Treatment of Multiphysics Problems

(15)


Multiphysics, Multiscale Systems

Multiphysics, multiscale systems couple different physicalprocesses interacting across a wide range of scales

Such systems abound in science and engineering applicationdomains

Computational modeling plays a critical role in the predictivestudy of such systems

Some applications where multiphysics systems arise:I Fusion and fission reactorsI Reacting fluids and fluid-solid interactionsI Advanced materials, nano-manufacturingI Biological systems, drug design and deliveryI Environmental, climate, ecological modelsI Weather models


Example: A MEMs Thermal Actuator

Vsig

Contact ContactContact

Conductor Conductor

250 mm

Vswitch

∇ · (σ(d)∇V ) = 0 electrostatic current∇ · (κ(T )∇T ) = σ(∇V · ∇V ) energy

∇ ·(λ tr(E)I + 2µE − β(T − Tref )I

)= 0

E =(∇d+ ∇d>

)/2

displacement


Solution of Multiscale, Multiphysics Systems

Multiscale, multiphysics systems pose severe challenges forcomputational solution:

I Scale effectsI Complex stabilityI Coupling between physicsI Different kinds of physical descriptionsI Mixtures of discretization methods and scalesI Complex HPC discretization techniques

Significant extensions of the a posteriori theory are required


Solution of a Coupled Elliptic System

A simplified Thermal Actuator model:−∇ · a1∇u1 + b1 · ∇u1 + c1u1 = f1(x), x ∈ Ω,

−∇ · a2∇u2 + b2 · ∇u2 + c2u2 = f2(x, u1, Du1), x ∈ Ω,

u1 = u2 = 0, x ∈ ∂Ω,

Ω is a bounded domain with boundary ∂Ω

The coefficients are smooth functions and a1, a2 are boundedaway from zero

We wish to compute information depending on u2


Solution of a Coupled Elliptic System

AlgorithmI Construct discretizations Th,1, Th,2 and finite element

spaces Vh,1, Vh,2I Compute a finite element solution U1 ∈ Vh,1(Ω) of the first

equationI Project U1 ∈ Vh,1(Ω) into the space Vh,2(Ω)

I Compute a finite element solution U2 ∈ Vh,2(Ω) of thesecond equation

The projection between the discretization spaces is a crucialstep

In a fully coupled system, U2 is projected into Vh,1(Ω) for thenext iteration


A Simple Thermal Actuator

Consider−∆u1 = sin(4πx) sin(πy), x ∈ Ω

−∆u2 = b · ∇u1 = 0, x ∈ Ω,

u1 = u2 = 0, x ∈ ∂Ω,

b =2

π

(sin(4πx)

25 sin(πy)

)

where Ω = [0, 1]× [0, 1]

We consider the quantity of interest

u2(.25, .25)

We solve for u1 first and then solve for u2 using independentmeshes



Using uniform meshes, an a posteriori error estimate yields

estimate of the error in the quantity of interest ≈ .0042

true error ≈ .0048

discrepancy in estimate ≈ .0006 (≈ 13%)

0 0.2 0.4 0.6 0.8 1

0

0.5

1−1.5−1

−0.50

0.51

u1

0 0.2 0.4 0.6 0.8 1

0

0.5

1−0.8−0.6−0.4−0.2

00.20.4

u2

This arises from the operator decompositionDonald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 119/156


Adapting the mesh using only an error estimate for the secondcomponent causes the discrepancy to become alarminglyworse

estimate of the error in the quantity of interest ≈ .0001

true error ≈ .2244

0 0.2 0.4 0.6 0.8 10

0.5

1−0.6

−0.4

−0.2

0

0.2

0.4u1

0 0.2 0.4 0.6 0.8 1

0

0.5

1−2

−1.5

−1

−0.5

0

0.5

Pointwise Error of u2


Analysis for Multiscale, Multiphysics Solutions

Physics 1 Physics 2

ComputedInformation

Error

We define auxiliary quantities of interest corresponding toinformation passed between components

We solve auxiliary adjoint problems to estimate the error in thatinformation

In an iterative scheme, we also estimate the “history” of errorspassed from one iteration level to the next

We can also estimate the effect of processing, e.g. up anddown scaling, the information



Recall the “triangular” problem−∆u1 = sin(4πx) sin(πy), x ∈ Ω

−∆u2 = b · ∇u1 = 0, x ∈ Ω,

u1 = u2 = 0, x ∈ ∂Ω,

b =2

π

(25 sin(4πx)

sin(πx)

)

where Ω = [0, 1]× [0, 1]

We consider the quantity of interest

u2(.25, .25)

We solve for u1 first and then solve for u2 using independentmeshes



−∆u1 = f1(x), x ∈ Ω,

−∆u2 = f2(x, u1, Du1), x ∈ Ω

u1 = 0, u2 = 0, x ∈ ∂Ω

We compute a quantity of interest(ψ(1), u

)that only depends

on u2, thus

ψ(1) =

(0

ψ(1)2

)

We require the linearization Lf2(w) of f2 with respect to u1

around the function w



The weak form of the primary adjoint problem is(∇φ(1)

1 ,∇v1

)+(Lf2(U1)φ

(1)2 , v1

)= 0,(

∇φ(1)2 ,∇v2

)=(ψ

(1)2 , v2

),

all test functions v1, v2

This yields the first error representation(ψ(1), e

)=(f2(u1, Du1), (I − πh)φ

(1)2

)−(∇U2,∇(I − πh)φ

(1)2

)The residual depends on the unknown true solution

We write

f2(u1, Du1) = f2(U2, DU2) +(f2(u1, Du1)− f2(U2, DU2)

)Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 124/156


We have a new linear functional of the error(f2(u1, Du1)− f2(U1, DU1), φ

(1)2

)≈(Lf(U1)e1, φ

(1)2

)=(Lf(U1)∗φ

(1)2 , e1

)=(ψ(2), e

)We pose a secondary adjoint problem(

∇φ(2)1 ,∇v1

)+(Lf2(U1)φ

(2)2 , v1

)=(Lf2(U1)∗φ

(1)2 , v1

)(∇φ(2)

2 ,∇v2

)= 0

all test functions v1, v2



Theorem(ψ(1), e

)=(f2(U1, DU1), (I − πh)φ

(1)2

)−(∇U2,∇(I − πh)φ

(1)2

)+(f1, (I − πh)φ

(2)1

)−(∇U1,∇(I − πh)φ

(2)1

)φ(1), φ(2) are the primary and secondary adjoint solutions

The new analysis estimates the error in the information passedbetween components

There is a tertiary adjoint problem and another term in theestimate if the different discretizations are used for the twocomponents



If we adapt the meshes using all the terms in the estimate, wecan drive the error below .0001

0 0.2 0.4 0.6 0.8 1

0

0.5

1−1

−0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 100.5

1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

We actually refine the mesh for u1 more than the mesh for u2


Extending to Other Discretizations

(16)


Finite Elements and Other Discretizations

The a posteriori error analysis applies to wide classes ofdiscretization

The key is to write a chosen discretization method as somekind of finite element method with modifications such asquadrature to evaluate integrals

Examples of extensions includeI Finite difference schemesI Finite volume methodsI Explicit and IMEX time integration methodsI Operator split discretizations

Extending the analysis always involves identifying newresiduals and may involve defining additional adjoint problems


Finite Difference Schemes for an Initial Value Problem

Let Qr(g, n) denote a quadrature formula of order r for thefunction g on In

CG(q) method with quadrature: For n = 1, · · · , N ,∫In

(Y ′, v) dt = Qr((f(Y ), n), v), n

)all v ∈W q−1

n

Y +n−1 = Y −n−1

with Y0 = y0, U+n = limt↓tn U(t), U−n = limt↑tn U(t)

This yields a system of discrete equations determining Y

The discrete equations yield a finite difference scheme for Y


Finite Difference Schemes for an Initial Value Problem

Consider q = 1 and the trapezoidal rule quadrature

On In,

Y = Ynt− tn−1

kn+ Yn−1

tn − tkn

Q1(g, n) =1

2(g(tn) + g(tn−1))kn

The equation for the unknown coefficient Yn is

Yn −1

2f(Yn)kn = Yn−1 +

1

2f(Yn−1)kn


Treatment of Quadrature in A Posteriori Analysis

Key observation: Galerkin orthogonality usesQr((f(Y ), n), v), n

)instead of exact integration

We add and subtract Qr((f(Y ), n), v), n

)in the argument

Theorem The error representation is

∫ T

0(e, ψ) dt =

N∑n=1

∫In

(Rn, πkφ− φ) dt

+

N∑n=1

(∫In

(f(Y ), φ) dt−Qr((f(Y ), n), φ), n

))Rn = Y ′ − f(Y )


Treatment of Quadrature in A Posteriori Analysis

Observations about the contribution to the error fromquadrature (∫

In

(f(Y ), φ) dt−Qr((f(Y ), n), φ), n

))

I The stability factor is φ not πkφ− φ, so this contributionaccumulates at a different rate in general

I Exactly evaluating this expression requires exactintegration of the nonlinear term, and this has to beapproximated in general

Generically, such estimates include terms that cannot beevaluated exactly


Operator Splitting for Reaction-Diffusion Equations

Estimation of the effects of methods such as quadrature isimportant

We consider operator splitting for a reaction-diffusion problemdudt = ∆u+ F (u), 0 < t,

u(0) = u0

The diffusion component ∆u induces stability and change overlong time scales

The reaction component F induces instability and change overshort time scales



t0 t1 t2 t3 t4 t5k1 k2 k3 k4 k5 ...

On (tn−1, tn], we numerically solveduR

dt = F (uR), tn−1 < t ≤ tn,uR(tn−1) = uD(tn−1)

On (tn−1, tn], we numerically solveduD

dt −∆(uD), tn−1 < t ≤ tn,uD(tn−1) = uR(tn)

The operator split approximation is u(tn) ≈ uD(tn)



U n-1

URn-1 UR

n

UDn-1 UD

n U n



We discretize using a finite element in time

To account for the fast reaction, we approximate ur using manytime steps inside each diffusion step

t0 t1 t2 t3 t4 t5

s1,0 s1,M

s2,0 s2,M

s3,0 s3,M

s4,0 s4,Mk1

K1 K2 K3 K4 K5

k2

k3

......

......

Diffusion Integration:

Reaction Integration:



Decompose the error using an exact operator split solution us:

u− U = (u− us) + (us − U)

I u− us: error of analytic operator splittingThis is determined by properties of adjoint operators

I us − U : error of numerical component solvesThis is determined by the numerical errors made in eachcomponent

Estimates of the numerical error in each component do notindicate the effects of splitting in general



The Brusselator problem∂ui∂t − 0.025 ∂2ui

∂x2= fi(u1, u2) i = 1, 2

f1(u1, u2) = 0.6− 2u1 + u21u2

f2(u1, u2) = 2u1 − u21u2

I Use a linear finite element method in space with 500elements

I Use a standard first order splitting schemeI Use Trapezoidal Rule with time step of .2 for the diffusion

and Backward Euler with time step of .004 for the reaction



10-3 10-2 10-1 100 10110-5

10-4

10-3

10-2

10-1

100

Dt

L 2 nor

m o

f err

or

t = 6.4t = 16t = 32t = 64t = 80

slope 1

Spatial Location

Op

erat

orSp

litSo

lutio

nInstability in the Brusselator Operator Splitting

For large times, there is a critical step size above which there isno convergence. The instability is a direct consequence of the

operator splitting



Operator splitting complicates the definition of adjoint operators

I In a linear problem, the adjoint operators associated withthe original problem and an operator split discretization arefundamentally different

I Additionally in a nonlinear problem, the issue of choice oflinearization point becomes complicated

Estimates of the effects on adjoints cannot be entirelycomputable



We derive a hybrid a priori - a posteriori estimate

(e(tN ), ψ) = Q1 +Q2 +Q3

I Q1 estimates the contribution of the numerical solution ofeach component in an standard a posteriori way

I Q2 ≈∑N

n=1(Un−1, En−1), E ≈ a computable contributionfor the error in the adjoint arising from operator splitting

I Q3 is a noncomputable a priori expression that is provablyhigher order



Accuracy of the error estimate for the Brusselator example over[0, 2]

Component

Erro

r

k = 0 .01, M = 1 0

Time

Erro

r

∆ t = 0 .01, M = 1 0



Accuracy of the error estimate for the Brusselator example atT = 8 and T = 40

Component

Erro

r

k = 0 .01, M = 1 0

Component

Erro

r

k = 0 .01, M = 1 0


Conclusion


Conclusion

A posteriori error analysis based on duality, variational analysis,and adjoint equations provides a functional analytic approachto accurate computational error estimation

Deriving the estimates and quantifying sources of discretizationerrors provides significant insight into the behavior of thenumerical scheme

Consideration of the adjoint problem provides insight into thestability, e.g. accumulation, propagation, and transmission oferror

Having accurate estimates not only is a key aspect ofuncertainty quantification, but it also is useful for computation inseveral different ways


Limited References


Limited References

Duality and adjoints:I Linear Differential Operators, C. Lanczos, SIAM, 1996I Applied Functional Analysis, E. Zeidler, Springer, 1995I Adjoint Equations and Perturbation Algorithms in Nonlinear Problems,

G. Marchuk, V. Agoshkov, and V. Shutyaev, CRC Press, 1996I Real Analysis, G. Folland, J. Wiley and Sons, 1999I Applied Functional Analysis, J. Aubin, J. Wiley and Sons, 2000I Analysis for Applied Mathematics, W. Cheney, Springer, 2001

General a posteriori error analysis and adaptive error control:I Global error control for the continuous Galerkin finite element method

for ordinary differential equations, D. Estep and D. French, RAIROMMAN, 1994

I A posteriori error bounds and global error control for approximations ofordinary differential equations, D. Estep, SIAM. J. Numer. Analysis,1995

I Introduction to adaptive methods for differential equations,K. Eriksson,D. Estep, P. Hansbo, and C. Johnson, Acta Numerica, 1995


Limited References

General a posteriori error analysis and adaptive error control:I Computational Differential Equations, K. Eriksson, D. Estep, P. Hansbo,

and C. Johnson, Cambridge University Press, 1996I Accurate parallel integration of large sparse systems of differential

equations, D. Estep and R. Williams, MMMAS, 1996I Estimating the error of numerical solutions of systems of nonlinear

reaction-diffusion equations, D. Estep, M. Larson and R. Williams,Memoirs of the AMS, 2000

I An optimal control approach to a posteriori error estimation in finiteelement methods, R. Becker and R. Rannacher, Acta Numerica, 2001

I Adjoint methods for PDEs: a posteriori error analysis andpost-processing by duality, M. Giles and E. Suli, Acta Numerica, 2002

I Accounting for stability: a posteriori estimates based on residuals andvariational analysis, D. Estep, M. Holst, D. Mikulencak, Communicationsin Numerical Methods in Engineering, 2002

I Adaptive finite element methods for differential equations, W. Bangerthand R. Rannacher, Birkhauser, 2003


Limited References

General a posteriori error analysis and adaptive error control:I Generalized Green’s functions and the effective domain of influence, D.

Estep, M. Holst, and M. Larson, SIAM J. Sci. Comput., 2005I A posteriori error estimation of approximate boundary fluxes, T. Wildey,

S. Tavener, and D. Estep, Comm. Num. Meth. Engin., 24 (2008)I A posteriori error analysis of a cell-centered finite volume method for

semilinear elliptic problems, D. Estep, M. Pernice, D. Pham, S. Tavener,H. Wang, J. Computational and Applied Mathematics, 2009

I Bridging the Scales in Science and Engineering, Editor J. Fish, OxfordUniversity Press, Chapter 11: Error Estimation for Multiscale OperatorDecomposition for Multiphysics Problems, D. Estep, 2010

I Blockwise adaptivity for time dependent problems based on coarsescale adjoint solutions, V. Carey, D. Estep, A. Johansson, M. Larson,and S. Tavener, SIAM J. Sci. Comput., 2010

I A posteriori error analysis of two stage computation methods withapplication to efficient resource allocation and the Parareal Algorithm, J.H. Chaudhry, D. Estep, S. Tavener, V. Carey, and J . Sandelin, SIAM J.Numerical Analysis, 2014, submitted


Limited References

Our work on multiphysics problems:I A posteriori - a priori analysis of multiscale operator splitting, D. Estep,

V. Ginting, D. Ropp, J. Shadid, and S. Tavener, SINUM, 46 (2008)I A posteriori analysis and improved accuracy for an operator

decomposition solution of a conjugate heat transfer problem, D. Estep,S. Tavener, T. Wildey, SINUM, 46 (2008)

I A posteriori error analysis of multiscale operator decompositionmethods for multiphysics models, D. Estep, V. Carey, V. Ginting, S.Tavener, T. Wildey, J. Physics: Conf. Ser. 125 (2008)

I A posteriori analysis and adaptive error control for multiscale operatordecomposition methods for coupled elliptic systems I: One way coupledsystems, V. Carey, D. Estep, and S. Tavener, SINUM 47 (2009)

I A posteriori error analysis for a transient conjugate heat transferproblem, D. Estep, S. Tavener, T. Wildey, Fin. Elem. Analysis Design 45(2009)

I Bridging the Scales in Science and Engineering, Editor J. Fish, OxfordUniversity Press, Chapter 11: Error Estimation for Multiscale OperatorDecomposition for Multiphysics Problems, D. Estep, 2010


Limited References

Our work on multiphysics problems:I A posteriori error estimation and adaptive mesh refinement for a

multi-discretization operator decomposition approach to fluid-solid heattransfer, D. Estep, S. Tavener, T. Wildey, JCP, 2010

I A posteriori analysis of multirate numerical methods for ordinarydifferential equations, SINUM, D. Estep, V. Ginting, S. Tavener, Comput.Meth. Appl. Mech. Eng., 2012

I A posteriori analysis and adaptive error control for operatordecomposition solution of coupled semilinear elliptic systems, V. Carey,D. Estep, S. Tavener, Int. J. Numer. Meth. Engin., 2013

I A posteriori analysis of an iterative multi-discretization method forreaction-diffusion systems, D. Estep. V. Ginting, J. Hameed, and S.Tavener, Comp. Meth. Appl. Mech. Engin., 2013

I A-posteriori error estimates for mixed finite element and finite volumemethods for problems coupled through a boundary with non-matchinggrids, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, IMA J. Numer.Analysis, 2013


Limited References

Our work on multiphysics problems:I A posteriori error estimates for mixed finite element and finite volume

methods for parabolic problems coupled through a boundary withnon-matching discretizations, T. Arbogast, D. Estep, B. Sheehan, and S.Tavener, SIAM J. Uncertainty Quantification, 2015

I Adaptive finite element solution of multiscale PDE-ODE systems, A.Johansson, J. H. Chaudhry, V. Carey, D. Estep, V. Ginting, M. Larson,and S. Tavener, CMAME, 2015

Treatment of various discretization “crimes”:I A posteriori error analysis for a cut cell finite volume method, D. Estep,

S. Tavener, M. Pernice and H. Wang, Computer Methods in AppliedMechanics and Engineering, 2011

I A posteriori error estimation for the Lax-Wendroff finite differencescheme, J. B. Collins, D. Estep, and S. Tavener, J. Comput. Appl. Math.263C (2014)

I A posteriori error analysis for nite element methods with projectionoperators as applied to explicit time integration techniques, J. Collins, D.Estep and S. Tavener, BIT, 2014


Limited References

Treatment of various discretization “crimes”:I A posteriori error analysis of IMEX time integration schemes for

advection-diffusion-reaction equations, J. Chaudry, D. Estep, V. Ginting,J. Shadid, and S. Tavener, CMAME, 2014

I A posteriori analysis for iterative solvers for non-autonomous evolutionproblems, J. H. Chaudry, D. Estep, V. Ginting, and S. Tavener, SIAM J.Uncertainty Quantification, 2015

Optimal control problems:I Computational error estimation and adaptive mesh refinement for a

finite element solution of launch vehicle trajectory problems, D. Estep,D. Hodges and M. Warner, SIAM Journal on Scientific Computing, 2000

I The solution of a launch vehicle trajectory problem by an adaptive finiteelement method, D. Estep, D. H. Hodges, M. Warner, ComputerMethods in Applied Mechanics and Engineering, 2001


Limited References

Use of adjoints to investigate stability:I The computability of the Lorenz system, D. Estep and C. Johnson,

Mathematical Models and Methods in Applied Sciences, 1998I Analysis of the sensitivity pr operties of a model of vector-borne

bubonic plague, M. Buzby, D. Neckels, M. Antolin, and D. Estep, RoyalSociety Journal Interface, 5 (2008), 1099-1107

A posteriori error analysis for uncertainty quantification:I Fast and reliable methods for determining the evolution of uncertain

parameters in differential equations, D. Estep and D. Neckels, J.Comput. Phys., 2006

I Fast methods for determining the evolution of uncertain parameters inreaction-diffusion equations, D. Estep and D. Neckels, Comput. Meth.Appl. Mech. Engin., 2007

I Nonparametric density estimation for randomly perturbed ellipticproblems I: Computational methods, a posteriori analysis, and adaptiveerror control, D. Estep, A. Malqvist, and S. Tavener, SISC, 2009

I Nonparametric density estimation for randomly perturbed ellipticproblems II: Applications and adaptive modeling, D. Estep, A. Malqvist,S. Tavener, Int. J. Numer. Meth. Engin., 2009


Limited References

A posteriori error analysis for uncertainty quantification:I A measure-theoretic computational method for inverse sensitivity

problems I: Method and analysis, J. Breidt, T. Butler and D. Estep, SIAMJ. Numer. Analysis, 2011

I A computational measure theoretic approach to inverse sensitivityproblems II: A posteriori error analysis, T. Butler, D. Estep and J.Sandelin, SIAM J. Numerical Analysis, 2012

I Uncertainty quantification for approximate p-quantiles for physicalmodels with uncertain inputs, D. Elfverson, D. Estep, F. Hellman and A.Malqvist, SIAM J. Uncertainty Quantification, 2014


minitutorial on a posteriori error estimation

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