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)
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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
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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
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Functionals, Dual Spaces, and Adjoints
Tools for Investigating Differential Equations
(32)
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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
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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?
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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
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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
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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
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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‖.
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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
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Linear Functionals and Dual Spaces
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
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Linear Functionals and Dual Spaces
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
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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
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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
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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∗
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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?
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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 ∗
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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
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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
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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>.
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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
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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.
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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
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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
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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))
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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
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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.
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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
).
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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)).
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 30/156
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
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Adjoint Boundary Conditions
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)
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Adjoint Boundary Conditions
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
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Adjoint Boundary Conditions
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.
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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”
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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
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The Usefulness of Duality and Adjoints
(6)
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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
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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∗
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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
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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
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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
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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
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Nonlinear Operators
(13)
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 54/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 55/156
A Point about Linearization
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
))
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 56/156
A Point about Linearization
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 57/156
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 58/156
A Posteriori Error Analysis
(28)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 59/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 60/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 61/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 62/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 63/156
Finite Element Method for Elliptic Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 64/156
Finite Element Method for Elliptic Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 65/156
Finite Element Method for Elliptic Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 66/156
Finite Element Method for Elliptic Problem
U = 0
Th
ΚhΚ
Ω
Triangulation of the domain Ω
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 67/156
Finite Element Method for Elliptic Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 68/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 69/156
A Posteriori Analysis for an Elliptic Problem
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)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 70/156
A Posteriori Analysis for an Elliptic Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 71/156
A Posteriori Analysis for an Elliptic Problem
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)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 72/156
A Posteriori Analysis for an Elliptic Problem
−∆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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 73/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 74/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 75/156
A Posteriori Analysis for an Initial Value Problem
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 )
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 76/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 77/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 78/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 79/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 80/156
A Posteriori Analysis for an Initial Value Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 81/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 82/156
A Posteriori Analysis for a Reaction-Diffusion System
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 83/156
A Posteriori Analysis for a Reaction-Diffusion System
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 84/156
A Posteriori Analysis for a Reaction-Diffusion System
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 85/156
A Posteriori Analysis for a Reaction-Diffusion System
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,
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 86/156
A Posteriori Analysis for a Reaction-Diffusion System
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 87/156
Forward Error Propagation
(4)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 88/156
Forward Error Propagation
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(·)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 89/156
Forward Error Propagation
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 90/156
Forward Error Propagation
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 91/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 92/156
Adjoints and Stability
(8)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 93/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 94/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 95/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 96/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 97/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 98/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 99/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 100/156
The Bistable Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 101/156
The Bistable Problem
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 102/156
Adjoints and Adaptive Error Control
(7)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 103/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 104/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 105/156
Adjoints and Adaptive Error Control
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 106/156
Adjoints and Adaptive Error Control
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 107/156
Adjoints and Adaptive Error Control
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 108/156
Adjoints and Adaptive Error Control
−∇ ·
((.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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 109/156
Adjoints and Adaptive Error Control
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 110/156
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 111/156
Treatment of Multiphysics Problems
(15)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 112/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 113/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 114/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 115/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 116/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 117/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 118/156
A Simple Thermal Actuator
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
A Simple Thermal Actuator
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 120/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 121/156
Analysis for Multiscale, Multiphysics Solutions
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 122/156
Analysis for Multiscale, Multiphysics Solutions
−∆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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 123/156
Analysis for Multiscale, Multiphysics Solutions
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
Analysis for Multiscale, Multiphysics Solutions
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 125/156
Analysis for Multiscale, Multiphysics Solutions
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 126/156
A Simple Thermal Actuator
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 127/156
Extending to Other Discretizations
(16)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 128/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 129/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 130/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 131/156
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 )
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 132/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 133/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 134/156
Operator Splitting for Reaction-Diffusion Equations
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)
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 135/156
Operator Splitting for Reaction-Diffusion Equations
U n-1
URn-1 UR
n
UDn-1 UD
n U n
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 136/156
Operator Splitting for Reaction-Diffusion Equations
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:
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 137/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 138/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 139/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 140/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 141/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 142/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 143/156
Operator Splitting for Reaction-Diffusion Equations
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 144/156
Conclusion
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 145/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 146/156
Limited References
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 147/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 148/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 149/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 150/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 151/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 152/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 153/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 154/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 155/156
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
Donald Estep: Rocky Mountain Summer Workshop on Uncertainty Quantification 156/156