\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Variational and Topological Methods:
Theory, Applications, Numerical Simulations, and Open Problems (2012).
{\em Electronic Journal of Differential Equations},
Conference 21 (2014), pp. 235--246.
ISSN: 1072-6691. http://ejde.math.txstate.edu,
http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document} \setcounter{page}{235}
\title[\hfilneg EJDE-2014/Conf/21 \hfil Methods for numerical differentiation]
{Methods for numerical differentiation of \\ noisy data}
\author[I. Knowles, R. J. Renka \hfil EJDE-2014/Conf/21\hfilneg]
{Ian Knowles, Robert J. Renka} % in alphabetical order
\address{Ian Knowles \newline
Department of Mathematics, University of Alabama at Birmingham,
Birmingham, AL 35294-1170, USA}
\email{knowles@math.uab.edu}
\address{Robert J. Renka \newline
Department of Computer Science \& Engineering,
University of North Texas, Denton, TX 76203-1366, USA}
\email{robert.renka@unt.edu}
\thanks{Published February 10, 2014.}
\subjclass[2000]{65D10, 65D25, 65R32}
\keywords{Ill-posed problem; numerical differentiation; smoothing spline;
\hfill\break\indent Tikhonov regularization; total variation}
\begin{abstract}
We describe several methods for the numerical approximation of a first
derivative of a smooth real-valued univariate function for which only
discrete noise-contaminated data values are given. The methods allow
for both uniformly distributed and non-uniformly distributed abscissae.
They are compared for accuracy on artificial data sets constructed by
adding Gaussian noise to simple test functions. We also display
results associated with an experimental data set.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
The problem of approximating a derivative of a function defined by
error-con\-taminated data points arises in several scientific computing and
engineering disciplines. Among the applications are image reconstruction,
probability density estimation, satellite orbit determination, and
seismic profiling. The problem is notable for the large variety of
methods that have been developed for its treatment. These include
polynomial regression, spline smoothing, filtering with Fourier and wavelet
transforms, total variation denoising, and convolution with a mollifier,
as well as local methods and other global methods.
We describe the problem in Section 2, discuss methods in Section 3, and
present test results in Section 4.
\section{The problem}
Given a set of $m$ discrete data points (observations) $\{(x_i,y_i)\}$
taken from a smooth function $g:[a,b] \to \mathbb{R}$ with
\begin{equation}
a \leq x_1 \leq x_2 \leq \dots \leq x_m \leq b,
\end{equation}
we wish to construct an approximation $u$ to the first derivative $g'$.
The differentiation problem is the inverse of the problem of computing an
integral. As with most inverse problems, it is ill-posed because the solution
$u \approx g'$ does not depend continuously on $g$. If we use divided
difference approximations to derivative values, then arbitrarily small
relative perturbations in the data can lead to arbitrarily large relative
changes in the solution, and the discretized problem is ill-conditioned.
We therefore require some form of regularization in order to avoid
overfitting.
In addition to constructing the derivative $u$, we compute a smooth
function $f$ which approximates $g$. If $u$ is computed directly from
the data, it may be integrated to produce $f$. Alternatively, we may view
the problem as that of constructing a smooth approximation $f$ which may
then be differentiated to obtain $u = f'$.
We assume that the data values are contaminated by independent identically
distributed zero-mean noise values $\eta_i$:
\[
y_i = g(x_i) + \eta_i, \;\; E[\eta_i] = 0, \quad (i = 1,\dots,m).
\]
Treating values of the fitting function $f(x)$ as random variables, the
mean squared error at each point $x \in [a,b]$ is
\[
\operatorname{MSE}(x) = E[(g(x)-f(x))^2] = (g(x)-E[f(x)])^2
+ E[(f(x)-E[f(x)])^2],
\]
where the first term is the squared bias and the second term is the
variance. A small error requires that both terms be small, but generally
there is a tradeoff between the two terms. We can
minimize the bias by choosing a very flexible model with a large number
of free parameters --- enough freedom to interpolate the data points as
the extreme case. However, this would result in large variance due to
too much sensitivity to the data. The fit would be poor on another sample
data set. The inflexibility of too few parameters, on the other hand,
would result in large error due to large bias. Most smoothing
methods incorporate the bias-variance tradeoff into a single smoothing
parameter which must be chosen to achieve the proper balance.
\section{Methods}
We implemented and tested one or more methods in each of the following
categories:
\begin{itemize}
\item[(i)] Least squares polynomial approximation.
\item[(ii)] Tikhonov regularization.
\item[(iii)] Smoothing spline.
\item[(iv)] Convolution smoothing with a Friedrichs mollifier.
\item[(v)] Knowles and Wallace variational method.
\item[(vi)] Total variation regularization.
\end{itemize}
Each category is described in a separate subsection below.
\subsection{Least squares polynomial approximation}
A simple smoothing method consists of fitting the data points in a
least squares sense with a sequence of polynomials f of degree $d$ for
$d = 0, 1, \dots, k$, where $k$ is the smallest value for which the
residual norm is bounded by a tolerance. We then define $u = f'$.
The reciprocal of the polynomial degree serves as a regularization
parameter with $k \geq m-1$ corresponding to interpolation, and $k=0$
corresponding to a constant. This method requires that the data abscissae
be distinct, and produces $f,u \in C^{\infty}[a,b]$ for $a = x_1$ and
$b = x_m$.
The polynomial is taken to be a linear combination of the monomial
basis functions $\phi_j(x) = x^j$ for $j = 0, \dots, k$, and the
coefficients are computed by solving the normal equations with a QR
factorization of the Vandermonde matrix.
We implemented only the simple global method, but a method that could be
more effective for dense data sets is locally weighted
polynomial regression, referred to as LOESS or LOWESS \cite{cleveland88}.
The idea is to fit a low-degree polynomial (degree 1 or 2) to the
nearest neighbors of each data point with weight proportional to
inverse distance from the abscissa of the neighbor to that of the point.
The number of points used in each fit
increases with the value of a smoothing parameter. Another method that
uses local polynomial regression (for uniformly distributed data points)
is the Savitsky-Golay smoothing filter \cite{savitsky64}.
\subsection{Tikhonov regularization}
Tikhonov regularization was introducted in \cite{tikhonov63} and first
applied to the numerical differentiation problem in \cite{cullum71}.
Refer also to \cite{tikhonov77}.
We include three methods, indexed by $k$ = 0, 1, and 2,
corresponding to the smoothness of the solution $u$. We assume that $g(a)$
is given and that $g$ is in the Sobolev space $H^{k+1}(a,b)$. We formulate
the problem as that of computing an approximate solution $u \in H^k(a,b)$
to the operator equation
\begin{equation}
A u(x) = \int_a^x u(t) \, dt = \hat{g}(x), \;\; x \in [a,b],
\end{equation}
where $\hat{g}(x) = g(x)-g(a)$. Once $u$ is computed, $f$ can be obtained
from $f(x) = \int_a^x u(t) \, dt + g(a)$.
We discretize the problem by representing $f$ as a vector $\mathbf{f}$ of function
values on a uniform grid that partitions $[a,b]$ into $n$ subintervals
of length $\Delta t = (b-a)/n$:
\begin{equation}
f_j = f(t_j), \quad t_j = a + (j-1)\Delta t, \quad
(j = 1, \dots, n+1).
\end{equation}
The function $u$ is represented by derivative values at the midpoints:
\[
u_j = f'(t_j+\Delta t/2), \quad (j = 1, \dots, n).
\]
The discretized system is
$A\mathbf{u} = \hat{\mathbf{y}}$, where $\hat{y}_i = y_i-g(a)$ and $A_{ij}$ is
the length of $[a,x_i] \cap [t_j,t_{j+1}]$:
\[
A_{ij} = \begin{cases}
0 & \text{if } x_i \leq t_j \\
x_i-t_j & \text{if } t_j < x_i < t_{j+1} \\
\Delta t & \text{if } t_{j+1} \leq x_i\,.
\end{cases}
\]
This linear system may be
underdetermined or overdetermined and is likely to be ill-conditioned.
We therefore use a least squares formulation with Tikhonov regularization.
We minimize the convex functional
\begin{equation}
E(\mathbf{u}) = \|A\mathbf{u}-\hat{\mathbf{y}}\|^2 + \alpha \|D\mathbf{u}\|^2,
\end{equation}
where $\alpha$ is a nonnegative regularization parameter, $\|\cdot\|$ denotes
the Euclidean norm, and $D$ is a differential operator of order $k$ defining
a discretization of the $H^k$ Sobolev norm:
\[
D^t = \begin{cases}
I & \text{if } k = 0 \\
(I \, D_1^t ) & \text{if } k = 1 \\
(I \, D_1^t \, D_2^t ) & \text{if } k = 2\,
\end{cases}
\]
where $I$ denotes the identity matrix, and $D_1$ and $D_2$ are first
and second difference operators. We use second-order central differencing
so that $D_1$ maps midpoint values to interior grid points, and $D_2$ maps
midpoint values to interior midpoint values. The regularization
(smoothing) parameter $\alpha$ defines a balance between fidelity to the
data (with high fidelity corresponding to low bias) on the one hand, and
the size of the solution norm, which is related to variance, on the other
hand. The optimal value depends on the choice of norm. Larger values of $k$
enforce more smoothness on the solution. Setting the gradient of $E$ to
zero, we obtain a linear system with an order-$n$ symmetric positive definite
matrix:
\[
(A^t A + \alpha D^t D)\mathbf{u} = A^t \hat{\mathbf{y}}.
\]
In the case that the error norm $\|\eta\|$ is known, a good value of
$\alpha$ is obtained by choosing it so that the residual norm
$\|A\mathbf{u}-\hat{\mathbf{y}}\|$ agrees with $\|\eta\|$. This is Morozov's
discrepancy principle \cite{morozov67}.
\subsection{Smoothing spline}
The cubic smoothing spline was introduced by Schoenberg \cite{schoenberg64}
and Reinsch \cite{reinsch67, reinsch71}. The method of generalized
cross validation for automatic selection of a smoothing parameter first
appeared in \cite{craven79}. The cubic spline was generalized to a spline
under tension by Schweikert \cite{schweikert66}, and methods for automatic
selection of tension factors were developed in \cite{renka87}. This
method is effective for relatively sparse data sets.
For this method we take $m = n$, and require that the abscissae are distinct
and include the endpoints:
\[
a = x_1 < x_2 < \dots < x_n = b.
\]
The smoothing function $f$ is a spline with knots at the abscissae.
Denote the knot values by $v_i = f(x_i)$, and
for each subinterval $[x_i,x_{i+1}]$, denote the subinterval length by
$\Delta x_i$, denote the slope of the piecewise linear interpolant by
$s_i = (v_{i+1}-v_i)/\Delta x_i$, and let $\tau_i$ be a nonnegative
tension factor associated with the subinterval. Then our measure of
smoothness for $f \in H^2[a,b]$ is
\[
q_1(f) = \int_a^b f''^2 + \sum_{i=1}^{n-1} \Big(\frac{\tau_i}
{\Delta x_i}\Big)^2 \int_{x_i}^{x_{i+1}} (f'-s_i)^2.
\]
For a given set of knot values $\{v_i\}$ the minimizer of $q_1$ is
in $C^2[a,b]$ and satisfies
\begin{equation}
f^{(4)} - (\tau_i/\Delta x_i)^2 f'' = 0 \label{smspline}
\end{equation}
on each subinterval so that $f$ is cubic in subintervals for which
$\tau_i = 0$, and $f$ approaches the linear interpolant of $(x_i,v_i)$
and $(x_{i+1},v_{i+1})$ as $\tau_i \to \infty$. Solutions to
\eqref{smspline} lie in the span of $\{1,x,e^{(\tau_i/\Delta x_i)x},
e^{-(\tau_i/\Delta x_i)x} \}$ so that $f$ is piecewise exponential.
Using a Hermite interpolatory representation of f, the four degrees of
freedom in each subinterval are taken to be the endpoint function values
$v_i$, $v_{i+1}$ and first derivative values $d_i = f'(x_i)$,
$d_{i+1} = f'(x_{i+1})$. Let $q_1$ now denote the quadratic functional
defined on the pair of $n$-vectors of knot function values $\mathbf{v}$
and derivatives $\mathbf{d}$.
We obtain a smoothing spline by minimizing $q_1$ subject to the constraint
\[
q_2(\mathbf{v}) = \sum_{i=1}^n \Big(\frac{y_i-v_i}
{\sigma_i}\Big)^2 \leq S,
\]
where $\sigma_i$ is the standard deviation in $y_i$ and $S$ is a
nonnegative number with nominal value in the confidence interval
$[n - \sqrt{2n}, n+\sqrt{2n}]$ for the weighted sum of squared deviations
from the data values $q_2(\mathbf{v})$. For $S$ sufficiently large,
$f$ is linear and $q_1 = 0$. For an active constraint the problem is
equivalent to minimizing
\[
E(\mathbf{v},\mathbf{d},\lambda) = q_1(\mathbf{v},\mathbf{d}) + \lambda
(q_2(\mathbf{v}) - S)
\]
for Lagrange multiplier $\lambda$ (whose reciprocal serves as a
regularization parameter). This problem is treated by finding a zero of
\[
\phi(\lambda) = \frac{1}{\sqrt{q_2(\mathbf{v})}} - \frac{1}{\sqrt{S}},
\]
where $\mathbf{v}$ and $\mathbf{d}$ satisfy the order-$2n$ symmetric positive
definite linear system obtained by setting the gradient of $E$ to zero
with $\lambda$ fixed.
Our test suite includes two methods in this category: a cubic spline
in which all tension factors are zero, and a tension spline with just
enough tension in each subinterval to preserve local monotonicity
and convexity of the data \cite{renka87, renka93}.
\subsection{Convolution smoothing with a Friedrichs mollifier}
\quad We compute a smooth approximation $f \in C^{\infty}[a,b]$ by
convolution of the piecewise linear interpolant $p$ of the data
values with the positive symmetric Friedrichs mollifier function
\cite{friedrichs44}
\begin{equation*}
\rho(x) = \begin{cases}
c e^{ 1/(x^2-1)} & \text{if } |x| < 1 \\
0 & \text{if } |x| \geq 1\,,
\end{cases}
\end{equation*}
where $c$ is chosen so that $\rho$ integrates to 1.
Then for $x \in [a,b]$,
\begin{equation*}
f(x) = \frac{1}{h} \int_{a-h}^{b+h} \rho\Big(\frac{x-s}{h}\Big)
p(s) \, ds
\end{equation*}
for small positive $h$. The derivative $g'$ is then approximated by
$u = f'$. Note that $h$ is a smoothing parameter: $f \to p$ in
$L^r(a,b)$, $1\le r<\infty$, as
$h \to 0$ (interpolation), and $f \to 0$ as
$h \to \infty$.
We represent the smoothed function $f$ by a discrete set of values
on a uniform grid with mesh width $\Delta t = (b-a)/(n-1)$:
\begin{equation*}
f_i = f(t_i), \quad t_i = a + (i-1) \Delta t \quad \text{for }
i = 1, \dots, n.
\end{equation*}
We use second-order central difference approximations to midpoint
first derivative values
\begin{equation*}
f'(t_i+\Delta t/2) = (f_{i+1}-f_i)/\Delta t \quad \text{for }
i = 1, \dots, n-1.
\end{equation*}
This method has the limitation that the abscissae must be distinct, and
the approximation is valid only on the subdomain $[a+h,b-h]$.
To our knowledge, this novel numerical application of the well-known
Friedrichs mollifier was first suggested by Aimin Yan as a way to handle
the numerical differentiation of a measurement dataset arising from a
function of two variables. This was needed for the solution of the inverse
groundwater problem in \cite{knowles04}, and elsewhere, and found to be
quite effective in those contexts.
\subsection{Knowles and Wallace variational method}
The method of Knowles and Wallace \cite{knowles95} was designed with
the goal of eliminating the difficult problem of determining an
effective regularization parameter when nothing is known about the
errors in the data.
Let $p(x) = g(x) + k$ for $x \in [a,b]$ and a constant $k$ large enough
that $p$ is bounded below by a positive constant and
$Q(x) = p''(x)/p(x)$ is bounded above by a curvature-limiting
constant $M < (\pi/(b-a))^2$ for $x \in [a,b]$. Then $u = g' = p'$ can be
obtained by solving $-p'' + Qp = 0$ with specified endpoint values
of $p$, where $Q$ is the unique global minimizer of the strictly convex
nonlinear energy functional
\begin{equation*}
E(q) = \int_a^b (p'-f')^2 + q (p-f)^2,
\end{equation*}
with $f$ as the solution to the two-point boundary value problem
\begin{gather*}
A_q f = -f'' + q f = 0 \quad \text{in } (a,b), \\
f(a) = p(a), \quad f(b) = p(b).
\end{gather*}
Note that $E(Q) = 0$ and $f = p$ so that $f-k = g$ at the solution. Since $g$
is specified only by discrete data points, possibly contaminated by
noise, using the data to discretize $E$ will not result in a zero at
the minimizer, but the computed function $f$ will be a smooth
approximation to $p$ and can be used to compute $u$. The idea is thus
to compute the best approximation to $p$ in the space of solutions
to the two-point boundary value problem.
An equivalent expression for the functional is
\begin{equation*}
E(q) = \int_a^b p'^2 + q p^2 - (f'^2 + q f^2) .
\end{equation*}
The gradient of $E$ is defined by the first Gateaux derivative
\begin{equation*}
E'(q)[h] = \int_a^b (p^2-f^2) h ,
\end{equation*}
and the Hessian is defined by the second derivative
\begin{equation*}
E''(q)[h][k] = 2 \int_a^b (f k) A_q^{-1}(f h) ,
\end{equation*}
where $A_q$ is restricted to functions satisfying homogeneous
Dirichlet boundary conditions.
In order to discretize the problem, we take $p$ to be the piecewise linear
interpolant of $\{(x_i,y_i+k)\}$, and we represent functions $p, f$, and $q$
as $n$-vectors of grid-point values on the uniform grid with mesh width
$\Delta t = (b-a)/(n-1)$:
\begin{equation*}
f_i = f(t_i), \quad t_i = a + (i-1)\Delta t \quad (i = 1, \dots, n).
\end{equation*}
We use second-order central difference approximations to midpoint
first derivative values
\begin{equation*}
(D_1 \mathbf{f})_i = f'(t_i+\Delta t/2) = (f_{i+1}-f_i)/\Delta t \quad
(i = 1, \dots, n-1).
\end{equation*}
We then have second-order approximations to $A_q \mathbf{f}$ at the interior
grid points
\begin{equation*}
(A_q \mathbf{f})_i = \frac{-f_{i-1}+2 f_i-f_{i+1}}{\Delta t^2} + q_i f_i
\end{equation*}
for $i = 2, \dots, n-1$ with $f_1 = p(a)$ and $f_n = p(b)$. The solution to
the two-point boundary value problem is $\mathbf{f} = A_q^{-1}\mathbf{c}$ for
order-($n$-2) matrix $A_q = \operatorname{diag}(\mathbf{q}) - D_2$ and
$\mathbf{c} = (p(a)/\Delta t^2) \mathbf{e}_1 + (p(b)/\Delta t^2) \mathbf{e}_{n-2}$,
where $D_2$ denotes the second difference operator, and $\mathbf{e}_1$ and
$\mathbf{e}_{n-2}$ are standard basis vectors. We approximate $E(\mathbf{q})$ with
a rectangle rule for the midpoint derivative values (first term)
and a trapezoidal rule for the interior grid-point values (second term):
\begin{equation*}
E(\mathbf{q}) = \frac{1}{\Delta t} \sum_{i=1}^{n-1} (w_{i+1}-w_i)^2 +
\Delta t \sum_{i=2}^{n-1} q_i w_i^2
\end{equation*}
for $\mathbf{w} = \mathbf{p}-\mathbf{f}$.
A Newton-like method for minimizing $E$ is not feasible because every
component of the Hessian would require solution of a linear system.
We therefore use a gradient descent iteration, but not with the ordinary
gradient. The standard method of steepest descent is notoriously
slow, but we obtain an effective method by using the Sobolev gradient
\cite{neuberger10}:
\begin{equation*}
\mathbf{q}_{k+1} = \mathbf{q}_{k} - \beta \nabla_S E({\bf q_k}),
\end{equation*}
where $\beta$ denotes a positive step size, and $\nabla_S E(\mathbf{q})$ is the
$H^1$ Sobolev gradient of $E$ at $\mathbf{q}$ defined by
\begin{equation*}
\nabla_S E(\mathbf{q}) = (I + D_1^t D_1)^{-1} \nabla E(\mathbf{q})
\end{equation*}
for Euclidean gradient $\nabla E(\mathbf{q})$ with components $p_i^2 - f_i^2$.
Note that $D_1$ is restricted to the interior grid points, and
$I + D_1^t D_1 = I-D_2$. The inverse of this matrix is a smoothing operator
which serves as a preconditioner for the descent iteration.
\subsection{Total variation regularization}
In some applications it may be necessary to allow for discontinuities in
the derivative $u$ corresponding to corners in $g$. To this end, we
assume only that $g \in L^2[a,b]$ and we take the measure of regularity
to be the total variation in $u$: $F(u) = \int_a^b |u'|$ for $u$ in the
space $BV[a,b]$ of functions of bounded variation.
The method of total variation regularization was introduced by Rudin,
Osher, and Fatemi \cite{rudin92} for the purpose of removing noise from
images without smearing edges. It was applied to the problem of
differentiating noisy data by Chartrand \cite{chartrand11} using an
iterative method of Vogel and Oman \cite{vogel96}. Here we use a
generalized Sobolev gradient method to minimize the energy functional.
As in the case of Tikhonov regularization we assume that $g(a)$ is given
and we seek an approximate solution $u$ to the operator equation
\begin{equation*}
A u(x) = \int_a^x u(t) \, dt = \hat{g}(x), \quad x \in [a,b],
\end{equation*}
where $\hat{g}(x) = g(x)-g(a)$. We again represent $u$ by a vector $\mathbf{u}$
of discrete values at the midpoints of a uniform grid:
\[
u_j = f'(a+(j-1)\Delta t +\Delta t/2), \quad (j = 1, \dots, n)
\]
for $\Delta t = (b-a)/n$. The discretized system consists of
$m$ equations in $n$ unknowns $A\mathbf{u} = \hat{\mathbf{y}}$,
where $A_{ij}$ is the
length of $[a,x_i] \cap [t_j,t_{j+1}]$ and $\hat{y}_i = y_i-g(a)$.
The energy functional is
\begin{equation}
E(\mathbf{u}) = \frac{1}{2} \|A\mathbf{u}-\hat{\mathbf{y}}\|^2 + \alpha
\sum_{j=1}^{n-1} \sqrt{ (u_{j+1}-u_j)^2 + \epsilon},
\end{equation}
where $\|\cdot\|$ denotes the Euclidean norm, and $\alpha$ is a nonnegative
regularization parameter. The regularization term is the discretization
of the BV seminorm $F(u) = \int_a^b |u'|$ using the trapezoidal rule and
natural end conditions: $u'(a) = u'(b) = 0$. Since the gradient of $F$ is
$\nabla F(u) = -(u'/|u'|)'$, the perturbation by a small positive number
$\epsilon$ is necessary for differentiability of $E$ where $u' = 0$.
The gradient of $E$ is
\[
\nabla E(\mathbf{u}) = A^t(A\mathbf{u}-\hat{\mathbf{y}}) - \alpha \mathbf{s}',
\]
where the sign vector $\mathbf{s}$ is the approximation to $u'/|u'|$:
\[
s_i = (u_{i+1}-u_i)/\sqrt{(u_{i+1}-u_i)^2 + \epsilon} \quad
(i = 1, \dots, n-1),
\]
and
\[
(s')_1 = s_1, \quad (s')_i = s_i-s_{i-1} \quad (i = 2, \dots, n-1), \quad
(s')_n = -s_{n-1}.
\]
The Hessian of $E$ at $\mathbf{u}$ is
\[
H(\mathbf{u}) = A^tA + \alpha D_u^t(I-\operatorname{diag}(\mathbf{s})^2) D_u,
\]
where $D_u$ is the discretization of the operator that maps $v$ to
$v'/\sqrt{|u'|}$, and $1-s_i^2 = \epsilon/((u_{i+1}-u_i)^2+\epsilon)$.
We approximate $H(\mathbf{u})$ by the preconditioner
\[
A_u = A^tA + \alpha \Delta t D_u^t D_u.
\]
The Sobolev gradient $A_u^{-1} \nabla E(\mathbf{u})$ is the gradient associated
with the weighted Sobolev inner product
\[
\langle \mathbf{v},\mathbf{w} \rangle_u
= (A\mathbf{v})^t (A \mathbf{w}) +
\alpha \Delta t (D_u\mathbf{v})^t (D_u \mathbf{w}).
\]
The steepest descent iteration is
\[
\mathbf{u}_{k+1} = \mathbf{u}_{k} - \beta A_u^{-1} \nabla E({\bf u_k})
\]
with constant step size $\beta$, and initial value $\mathbf{u}_0$ obtained
by differencing $y$ values. Since the gradient and inner product
at step $k$ depend on $\mathbf{u}_k$, this is a variable-metric method.
The gridpoint values of $\mathbf{f}$ are computed as
\[
f_j = \Delta t \sum_{i=1}^{j-1} u_i + g(a) \;\; (j = 1,\dots,n+1).
\]
\section{Test results}
Our first test function is $g(x) = \cos(x)$ on $[-.5,.5]$. We generated two
data sets, one with $m = 100$ points, and one with $m=10$ points, and
both with uniformly distributed
abscissae $x_i$ and data values $y_i = g(x_i) + \eta_i$, where $\eta_i$
is taken from a normal distribution with mean 0 and standard deviation
$\sigma$. We tested with $\sigma = 0.01$ on both data sets, and
$\sigma = 0.1$ on the dense data set only. For the
Tikhonov regularization method we used the known value of $\|\eta\|$
to compute the optimal parameter value $\alpha$. In the case of
smoothing splines we used the known value of $\sigma$ to define the
smoothing parameter. Table \ref{table1} displays the maximum relative error in
the computed derivative for most of the methods. We omitted the
tension spline because the results are almost identical to those of
the cubic spline, and we omitted the total variation regularization
method because it is not designed to fit data from a smooth function.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|l|r|r|r|} \hline
Method & $m=100, \sigma = .01$ & $m=100, \sigma = .1$
& $m=10, \sigma=.01$ \\ \hline
Degree-2 polynomial & .0287 & .3190 & .2786 \\
Tikhonov, $k$ = 0 & .7393 & .8297 & .7062 \\
Tikhonov, $k$ = 1 & .1803 & .3038 & .6420 \\
Tikhonov, $k$ = 2 & .0186 & .0301 & .4432 \\
Cubic spline & .1060 & 1.15 & .3004 \\
Convolution smoothing & .1059 & .8603 & .2098 \\
Variational method & .1669 & .7149 & .3419 \\ \hline
\end{tabular}
\end{center}
\caption{Relative errors in derivative approximations}
\label{table1}
\end{table}
The best method on the dense data set is clearly Tikhonov regularization
with $k=2$. The solution is graphed in Figure \ref{fig1}. Note that the
curvature is exagerated by the mismatched axis scaling. The large error
in the cubic spline with $\sigma = .1$ is the result of fitting with a
polynomial of degree 1 because the constraint was not active. The last
three methods outperform Tikhonov regularization on the sparser data set,
with convolution smoothing producing the smallest error. The fitting
function is graphed in Figure \ref{fig2}.
The polynomial of degree 2 does reasonably well on both data sets for this
particular test function.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1a}
\includegraphics[width=0.45\textwidth]{fig1b}
\end{center}
\caption{Tikhonov regularization, $k=2, g(x) = \cos(x)$}
\label{fig1}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig2a}
\includegraphics[width=0.45\textwidth]{fig2b}
\end{center}
\caption{Convolution smoothing, $h=.3, g(x) = \cos(x)$}
\label{fig2}
\end{figure}
To demonstrate the benefits of tension factors we chose
test function $g(x) = x^7$ on $[-1,1]$ and created a data set consisting
of $m=10$ points with a uniform random distribution of the abscissae
and data values with Gaussian noise, $\sigma = .01$. Figure \ref{fig3} depicts
derivative curves associated with zero tension on the left, and
shape-preserving tension factors on the right.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig3a}
\includegraphics[width=0.45\textwidth]{fig3b}
\end{center}
\caption{Smoothing splines with no tension (left), nonzero tension (right)}
\label{fig3}
\end{figure}
Figure \ref{fig4} demonstrates the effectiveness of the total variation
regularization method for test function $g(x) = |x-.5|$ on $[0,1]$.
The data set is again 100 uniformly distributed points with $\sigma=.01$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig4a}
\includegraphics[width=0.45\textwidth]{fig4b}
\end{center}
\caption{Total variation regularization: $g(x) = |x-.5|$}
\label{fig4}
\end{figure}
The variational method of Knowles and Wallace was compared with the
spectral method of Anderssen and Bloomfield \cite{anderssen73} on a
data set provided by Pallaghy and L\"{u}ttge \cite{pallaghy70} in
\cite{knowles95}. The data set consists of 101 points with uniformly
distributed abscissae. Values and central difference approximations
to derivative values are depicted in Figure \ref{fig5}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig5a}
\includegraphics[width=0.45\textwidth]{fig5b}
\end{center}
\caption{Pallaghy and L\"{u}ttge data and derivative approximations}
\label{fig5}
\end{figure}
Figures \ref{fig6}, \ref{fig7} and \ref{fig8}
depict derivatives computed by some of the methods
described in the previous section. Parameters were chosen experimentally
to balance smoothness against fidelity to the data. Only a few tests were
required to select reasonable values: tolerance .05 resulting in degree 14
for the polynomial on the left side of Figure \ref{fig6};
parameters $\alpha = 10^{-3}, \alpha=10^{-7}$, and $\alpha=10^{-10}$ for
Tikhonov regularization with $k = 0, k = 1$, and $k=2$, respectively;
standard deviation $\sigma = 3 \times 10^{-3}$ for the smoothing spline
on the left side of Figure \ref{fig8}; and $h=.06$ for the convolution method
on the right side. The exponential tension spline with optimal tension
factors produced a fit nearly identical to that of the cubic spline.
Refer to \cite{knowles95} for results with the variational method.
The total variation method was not tested on this data set.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig6a}
\includegraphics[width=0.45\textwidth]{fig6b}
\end{center}
\caption{Polynomial (left), Tikhonov regularization, $k=0$ (right)}
\label{fig6}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig7a}
\includegraphics[width=0.45\textwidth]{fig7b}
\end{center}
\caption{Tikhonov regularization with $k=1$ (left), $k=2$ (right)}
\label{fig7}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig8a}
\includegraphics[width=0.45\textwidth]{fig8b}
\end{center}
\caption{Smoothing spline (left), convolution (right)}
\label{fig8}
\end{figure}
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\end{document}