Received: 13 June 2016 Revised: 19 April 2017 Accepted: 26 May 2017 Published on: 23 August 2017

DOI: 10.1002/sam.11355

O R I G I N A L A R T I C L E

Dimension reduction in magnetohydrodynamics powergeneration models: Dimensional analysis and active subspaces

Andrew Glaws1 Paul G. Constantine1 John N. Shadid2,3 Timothy M. Wildey2

1Department of Computer Science, University of

Colorado, Boulder, Colorado2Center for Computing Research, Sandia National

Laboratories, Albuquerque, New Mexico3Department of Mathematics and Statistics,

University of New Mexico, Albuquerque, New

Mexico

CorrespondencePaul G. Constantine, Department of Computer

Science, University of Colorado, Boulder, CO

80309.

Email: paul.constantine@colorado.edu

Funding InformationUS Department of Energy Office of Science,

DE-SC-0011077 and DE-NA0003525. US

Department of Energy’s National Nuclear Security

Administration, DE-AC04-94AL85000.

Magnetohydrodynamics (MHD)—the study of electrically conducting fluids—can

be harnessed to produce efficient, low-emissions power generation. Today, computa-

tional modeling assists engineers in studying candidate designs for such generators.

However, these models are computationally expensive, so thoroughly studying the

effects of the model’s many input parameters on output predictions is typically infea-

sible. We study two approaches for reducing the input dimension of the models: (i)

classical dimensional analysis based on the inputs’ units and (ii) active subspaces,

which reveal low-dimensional subspaces in the space of inputs that affect the outputs

the most. We also review the mathematical connection between the two approaches

that leads to consistent application. We study both the simplified Hartmann problem,

which admits closed form expressions for the quantities of interest, and a large-scale

computational model with adjoint capabilities that enable the derivative compu-

tations needed to estimate the active subspaces. The dimension reduction yields

insights into the driving factors in the MHD power generation models, which may

aid generator designers who employ high-fidelity computational models.

KEYWORDS

active subspaces, dimension reduction, dimensional analysis, magnetohydrodynam-

ics, MHD generator

1 INTRODUCTION

Magnetohydrodynamics (MHD) is an area of physics con-

cerned with electrically conducting fluids [13]; its various

models are found in disparate fields from geophysics to

fusion energy. The US Department of Energy’s investment

in MHD for power generation with low emissions genera-

tors dates back to the 1960s. Interest waned in the mid-1990s

after a proof-of-concept program revealed several technical

challenges to scaling and integrating MHD-based compo-

nents into a practical power generator [39]. However, in

the last two decades, several of these challenges have been

addressed via other investments; a recent workshop at the

National Energy Technology Laboratory brought together

current MHD researchers from labs, academia, and indus-

try to assess the state of the art in MHD power genera-

tion. In particular, given two decades of supercomputing

advances, the workshop report calls for improved simula-

tion tools that exploit modern supercomputers to build better

MHD models and integrate them into full scale generator

designs [25].

Our recent work has developed scalable simulations for

resistive MHD models [23,29,30,35]. To incorporate such

simulations into generator design, a designer must under-

stand the sensitivities of model predictions to changes in

model input parameters. With this in mind, we have developed

adjoint capabilities in the MHD simulation codes that enable

computation of derivatives of output quantities of interest

with respect to input parameters [31].

Derivative-based sensitivity analysis can reveal a

low-dimensional parameterization of the map from model

inputs to model predictions by identifying combinations of

parameters whose perturbations change predictions the most;

less important combinations can be safely ignored when

assessing uncertainty in predictions or employing the model

for design. Such dimension reduction may enable exhaustive

parameter studies not otherwise feasible with the expensive

simulation model. In this paper, we exploit the adjoint-based

Stat Anal Data Min: The ASA Data Sci Journal, 2017;10:312–325 wileyonlinelibrary.com/sam © 2017 Wiley Periodicals, Inc. 312

GLAWS ET AL. 313

gradient capabilities to uncover active subspaces [4] in each

of two quantities of interest (the average flow velocity and

the induced magnetic field) from a laminar flow MHD model

as a proof-of-principle demonstration. An active subspace is

the span of a set of directions in the model’s input space; per-

turbing the inputs along these directions changes the quantity

of interest more, on average, than perturbing the inputs along

orthogonal directions.

Our recent work has revealed active subspaces in

physics-based simulations for solar cell models [8], inte-

grated hydrologic models [19,20], and multiphysics scramjet

models [9]. We distinguish the current application to MHD

by connecting the active subspace-based dimension reduc-

tion to dimensional analysis—a standard tool in science

and engineering for reducing the number of parameters

in a physical model by examining the physical quantities’

units. Recent work in the statistics literature has recognized

the importance of dimensional analysis for inference and

experimental design when data are derived from physical

systems [22,32,33]; moreover, Albrecht et al. [1] argue that

dimensional analysis is underutilized by statisticians.

The Buckingham Pi Theorem (see Barenblatt [2], Chap-

ter 1)—an essential concept in dimensional analysis—loosely

states that a physical system with m parameters whose derived

units are products of powers of k<m fundamental units (e.g.,

meters, seconds, etc.) can be written in terms of m− k unit-

less quantities. Dimensional analysis can reveal fundamental

insights into the physical model before running any simula-

tions or conducting any experiments—merely by examining

units. Our recent work has revealed a mathematical connec-

tion between standard dimensional analysis and dimension

reduction with active subspaces [10]; namely, the dimensional

analysis provides an upper bound on the number of important

input space directions without any computation. We exploit

this connection to gain insight into the driving factors of two

MHD models.

The remainder of the paper proceeds as follows. In

Section 2, we review dimensional analysis, active subspaces,

and the connections between dimensional analysis and active

subspaces. In Section 3, we apply dimensional analysis to

the governing equations of MHD to determine the number of

dimensionless quantities that affect the system. In Section 4,

we estimate active subspaces in two MHD models: (i) the

Hartmann problem, which is a simplified description of an

MHD duct flow that has direct relevance to MHD generators

and (ii) large-scale steady state laminar flow simulation of an

MHD generator in three spatial dimensions. In both cases, we

comment on the insights revealed by comparing dimensional

analysis to the gradient-based, computationally discovered

active subspace.

1.1 Notation

We have made every attempt to use the following nota-

tion conventions consistently throughout the manuscript; any

departures from these conventions are noted in the text explic-

itly. Lowercase italicized letters (e.g., a) indicate scalars.

Boldface lowercase letters (e.g., a) are vectors. Boldface

uppercase letters (e.g., A) are matrices. Functional depen-

dence is indicated using the standard parentheses notation; for

example, y= f (x) indicates that the dependent scalar variable

y is a function of the independent vector-valued variable x.

2 DIMENSION REDUCTION METHODS

Reducing a system’s dimension can both yield insight into its

fundamental characteristics and enable parameter studies that

are otherwise infeasible when the dimension is too large. The

type of dimension reduction in the present study applies to

systems of the form

y = f (x), y ∈ R, x ∈ Rm, (1)

where the function f maps Rm to R. The scalar y is the

system’s quantity of interest, and x is a vector of input

parameters. Loosely speaking, we seek a parameterization of

Equation 1 with fewer than m parameters.

We assume the physical model is accompanied by a

known probability density function 𝛾(x) that represents uncer-

tainty or variability in the model inputs; this is a common

assumption when quantifying uncertainty in computer sim-

ulations of physical systems [34,37]. Further, we assume

the components of x are independent, which distinguishes

the type of dimension reduction methods we consider. In

particular, there is no covariance in the components of x,

so covariance based dimension reduction—such as princi-

pal component analysis [21]—is not appropriate. Instead, we

seek to reduce the dimension in a way that takes advantage

of structure in f (x) from Equation 1. In statistical regression,

related dimension reduction concepts and techniques (i.e.,

input space dimension reduction informed by an input/output

relationship) go by the name sufficient dimension reduction[11], where one seeks a subspace in the space of m pre-

dictors such that the response’s statistics are unchanged. In

contrast to the regression problem, the system Equation 1

is deterministic so that it justifiably models a determinis-

tic computer simulation. In other words, y from Equation 1

is not corrupted by random noise as in statistical regression

[12,40]. This difference creates challenges for proper interpre-

tation of sufficient dimension reduction in such physics-based

systems—for example., there is no notion of sufficiency in the

deterministic system.

We consider two types of dimension reduction for

Equation 1. The first is dimensional analysis, which is a

mature tool for reducing the number of variables in a physi-

cal system by examining the quantities’ units. The second is

based on active subspaces in f (x), which are subspaces of Rm

constructed such that perturbing x along an active subspace

changes y more, on average, than perturbing x orthogonally to

the active subspace. The former is analytical while the latter

314 GLAWS ET AL.

TABLE 1 International system of units (SI). Table taken from NationalInstitute of Standards and Technology [24]

Base quantity Name Symbol

Length meter m

Mass kilogram kg

Time second s

Electric current ampere A

Thermodynamic temperature kelvin K

Amount of a substance mole mol

Luminous intensity candela cd

is computational. In other words, dimensional analysis fol-

lows from the quantities’ units and can typically be performed

without the aid of a computer in small systems. In contrast,

the active subspace is estimated by computing y and its gradi-

ent at several values of x. We follow the linear algebra-based

presentation of dimensional analysis from Calvetti and Som-

ersalo [3, Chapter 4], which enables a precise comparison

with active subspaces in Section 2.3.

2.1 Dimensional analysis and Buckingham PiTheorem

Dimensional analysis enables dimension reduction by exam-

ining the physical units of the output and inputs. The central

result in dimensional analysis—the Buckingham Pi Theorem

([2], Chapter 1)—states that the original model Equation 1

can be written in a unitless form as

Π = f (𝚷) (2)

where Π∈R and 𝚷∈Rn denote the unitless output and input,

respectively. The number of unitless inputs in Equation 2 can-

not be more than the number of inputs in Equation 1, and

often, it will be less (i.e., n≤m). Thus, dimensional anal-

ysis and Buckingham Pi facilitate dimension reduction of

the model. This section focuses on the development of the

unitless quantities, Π and 𝚷.

To apply dimensional analysis, we assume Equation 1 is a

physical law, meaning that the output and inputs are accom-

panied by physical units. These units are derived from k≤mbase units, denoted generically as L1, … , Lk, which are a

subset of the 7 international system (SI) units; Table 1 shows

the SI base units. Note that if k is larger than m, then there are

more fundamental units than needed to describe the system

[2].

The unit function of a quantity, denoted by square brack-

ets, returns the units of its argument. For example, if y is

a velocity, then [y]=m/s. If a quantity is unitless, then its

unit function returns 1. We can generically write the units of

x= [x1, … , xm]T and y as

[xj] =k∏

i=1

Ldi,j

i , [y] =k∏

i=1

Luii . (3)

In words, the units of each physical quantity can be written

as a product of powers of the k base units. To derive the unit-

less quantities from Equation 2, define the k×m matrix D and

the k-vector u as

D =

[d1,1 · · · d1,m⋮ ⋱ ⋮

dk,1 · · · dk,m

], u =

[u1

⋮uk

], (4)

which contain the powers from the quantities’ units in

Equation 3. Assume that rank(D)= r. Let v= [v1, … , vm]T

satisfy Dv=u, and let U∈Rm× n be a basis for the null space

of D, that is, DU= 0∈Rk× n, where n=m− r. If r =m, then

dimensional analysis does not result in dimension reduction.

Note that the elements of v and U are not unique. We can

construct the unitless quantity of interest Π as

Π = ym∏

i=1

x−vii . (5)

The unit function, defined above, then returns [Π]= 1 by

construction. We similarly construct unitless parameters Πj as

Πj =m∏

i=1

xui,j

i , j = 1, … , n, (6)

where ui, j is the (i, j) element of U.

The Buckingham Pi Theorem [2] states that a physical law

Equation 1 can be written in unitless form

Π = f (𝚷), 𝚷 =[Π1 · · · Πn

]T. (7)

The number of inputs in the unitless form of the physical law

Equation 2 is n<m, which has reduced dimension compared

to Equation 1. This dimension reduction may enable more

accurate semi-empirical modeling of the map f given exper-

imental data, since a model with fewer inputs typically has

fewer parameters to fit with a given data set. Additionally, the

unitless quantities allow one to devise scale-invariant experi-

ments, since scaling the units does not change the form of the

unitless physical law Equation 2. Albrecht et al. [1] exploit the

advantages of dimensional analysis for statistical inference

and experimental design.

2.2 Active subspaces

The space of x from Equation 1 is equipped with a known

probability density function 𝛾(x). Additionally, assume that

(i) f is square-integrable with respect to 𝛾 and (ii) f is dif-

ferentiable with square-integrable partial derivatives. Denote

the gradient vector of f as 𝛻f (x). Define the m×m symmetric

and positive semidefinite matrix C as

C = ∫ 𝛻f (x)𝛻f (x)T𝛾(x)dx. (8)

Since C is symmetric, it admits a real eigenvalue

decomposition,

C = W𝚲WT , (9)

GLAWS ET AL. 315

where the columns of W are the orthonormal eigenvectors,

and 𝚲 is a diagonal matrix of the associated eigenvalues in

descending order such that 𝜆1 ≥ 𝜆2 ≥ · · ·≥ 𝜆m ≥ 0.

The eigenpairs are functionals of f for the given 𝛾 . Assume

that 𝜆n >𝜆n+1 (i.e., 𝜆n is strictly greater than 𝜆n+1) for some

n<m. Then we can partition the eigenpairs as

𝚲 =[𝚲1

𝚲2

], W =

[W1 W2

], (10)

where 𝚲1 contains the first n eigenvalues of C, and W1’s

columns are the corresponding eigenvectors. The active sub-space of dimension n is the span of W1’s columns; the

active variables y∈Rn are the coordinates of x in the active

subspace. Note that the vector-valued y differs from the

scalar-valued quantity of interest y in Equation 1. The active

subspace’s orthogonal complement, called the inactive sub-space, is the span of W2’s columns; its coordinates are

denoted z∈Rm− n and called the inactive variables.

The following property justifies the labels; see Lemma 2.2

from Constantine et al. [7]:

𝜆1 + · · · + 𝜆n = ∫ 𝛻yf (x)T𝛻yf (x)𝛾(x)dx,

𝜆n+1 + · · · + 𝜆m = ∫ 𝛻zf (x)T𝛻zf (x)𝛾(x)dx, (11)

where 𝛻y and 𝛻z denote the gradient of f with respect to the

active and inactive variables, respectively. Since the eigenval-

ues are in descending order, and since f is such that 𝜆n >𝜆n+1,

Equation 11 says that perturbations in y change f more, on

average, than perturbations in z.

If the eigenvalues 𝜆n+ 1, … , 𝜆m associated with the inactive

subspace are sufficiently small, then f can be approximated

by a ridge function [27], which is a function that is constant

along a subspace of its domain,

f (x) ≈ g(WT1 x), (12)

where g maps Rn to R. If we are given N function evalua-

tions (xi, f (xi)) and an estimate of W1, constructing g with

the N pairs (WT1 xi, f (xi)) in n<m variables may be possi-

ble, whereas N may be too small to construct a function of

all m variables. Thus, the active subspace enables dimension

reduction in the space of x for response surface construction.

In practice, we estimate C from Equation 8 with numeri-

cal integration; high-accuracy Gauss quadrature [15] may be

appropriate when the dimension of x is small and the inte-

grands are sufficiently smooth. Constantine and Gleich [6]

analyze the following Monte Carlo method. Draw M indepen-

dent samples xi according to the given 𝛾(x) and compute

C ≈ C = 1

M

M∑i=1

𝛻f (xi)𝛻f (xi)T = W��WT. (13)

The eigenpairs ��, W of C estimate those of C. Using results

from nonasymptotic random matrix theory [17,38], Constan-

tine and Gleich [6] analyze how large the number M of

samples must be to ensure quality estimates; the analysis

supports a heuristic of M = 𝛼(𝛿)k log(m) samples to estimate

the first k eigenvalues within a relative error 𝛿, where 𝛼 is typ-

ically between 2 and 10 and m is the dimension of f (x). Let

𝜀 denote the distance between the true n-dimensional active

subspace and its Monte Carlo estimate defined as [18]

𝜀 = ||W1WT1 − W1W

T1 ||2, (14)

where W1 contains the first n columns of W. Using standard

perturbation theory for invariant subspaces [36], Corollary

3.7 from Constantine and Gleich [6] shows that, for suffi-

ciently large M,

𝜀 ≤ 4𝜆1𝛿

𝜆n − 𝜆n+1

. (15)

Equation 15 shows that a large eigenvalue gap 𝜆n − 𝜆n+1

implies that the active subspace can be accurately estimated

with Monte Carlo. Therefore, the practical heuristic is to

choose the dimension n of the active subspace according to

the largest gap in the eigenvalues. Additionally, Constantine

et al. [7] analyze the effect of using estimated eigenvectors in

the ridge function approximation Equation 12.

2.3 Connecting dimensional analysis to activesubspaces

Next we study the relationship between these two dimension

reduction techniques; this closely follows the development

in Constantine et al. [10]. Dimensional analysis produces

the unitless physical law Equation 2 by defining unitless

quantities Equations 5 and 6 as products of powers of the

dimensional quantities. On the other hand, the coordinates

of the active subspace can be written as linear combina-

tions of the system’s inputs, that is, y = WT1 x and z = WT

2 xfor x∈Rm. Thus, these two techniques are connected via a

logarithmic transformation of the input space. Combining

Equations 5, 6 and 2,

ym∏

i=1

x−vii = f

( m∏i=1

xui,1i , … ,

m∏i=1

xui,ni

)

= f

(exp

(log

( m∏i=1

xui,1i

)), … ,

exp

(log

( m∏i=1

xui,ni

)))

= f

(exp

( m∑i=1

ui,1 log(xi)

), … ,

exp

( m∑i=1

ui,n log(xi)

))= f (exp(uT

1log(x)), … , exp(uT

n log(x))), (16)

where log(x) returns an m-vector with the log of each

component. Then we can rewrite y as

y = exp(vT log(x)) ⋅ f (exp(uT1

log(x)), … ,

exp(uTn log(x))) = g(AT log(x)), (17)

316 GLAWS ET AL.

where

A =[v u1 · · · un

]∈ R

m×(n+1), (18)

and g ∶ Rn+1 → R. In other words, the unitless physical

law can be transformed into a ridge function of the logs of

the physical inputs. Compare Equation 17 to the form of the

ridge approximation Equation 12.

Since the physical law can be written as a ridge function,

its active subspace is related to the coefficient matrix A. Let

x = log(x), and assume ��(x) is a probability density function

on the space of x derived from 𝛾(x). By the chain rule,

𝛻xg(AT x) = A𝛻g(AT x), (19)

where 𝛻x denotes the gradient with respect to x, and 𝛻g is

the gradient of g with respect to its argument, AT x. Plugging

Equation 19 into Equation 8,

∫ 𝛻xg(AT x)𝛻xg(AT x)T ��(x)dx

= A(∫ 𝛻g(AT x)𝛻g(AT x)T ��(x)dx

)AT . (20)

The first column of A is not in the null space of D from

Equation 4, and its remaining columns are a basis for D’s null

space. Therefore, A has full column rank. Then Equation 20

shows that the active subspace for the physical law—as a func-

tion of the logs of its inputs—has dimension at least n+ 1,

and it is a subspace of the A’s column space.

The connection between the dimensional analysis and the

active subspace provides an upper bound on the dimension

of the active subspace. However, the eigenvalues of C from

Equation 8 rank the importance of each eigenvector-defined

direction. So the eigenpairs of C reveal more about the

input/output relationship than the dimensional analysis alone.

3 MAGNETOHYDRODYNAMICS

In this section, we perform dimensional analysis on the gov-

erning equations of MHD to study the number of unitless

quantities affecting the system. MHD models the behavior

of electrically conducting fluids, such as ionized liquids

or plasmas. The governing equations for MHD couple the

Navier-Stokes equations for fluid dynamics with Maxwell’s

equations for electromagnetism. Under simplifying assump-

tions, we can write the equations for steady-state MHD as

𝛻 ⋅[𝜌u ⊗ u + (p0 + p)I − 𝜇(𝛻u + 𝛻uT )

− 1

𝜇0

(B ⊗ B − 1

2||B||2I

)]= 0,

𝛻 ⋅[

u ⊗ B − B ⊗ u − 𝜂

𝜇0

(𝛻B − 𝛻BT )]= 0,

𝛻 ⋅ u = 0, 𝛻 ⋅ B = 0, (21)

where the unknown quantities are the fluid velocity u, the

magnetic field B, and the fluid pressure p.

It is worthwhile to examine the parameters in the MHD

model in detail. For an incompressible flow system solved

in terms of the primitive variables (u, p, B), five parameters

appear in the governing equations: (i) the fluid viscosity 𝜇,

(ii) the fluid density 𝜌, (iii) the applied pressure p0 (more

specifically, the applied pressure gradient 𝛻p0), (iv) the mag-

netic resistivity of the fluid 𝜂, and (v) the permeability of

a vacuum 𝜇0. Two additional parameters do not appear in

Equation 21 but are essential to describing the physics of the

system. The first is an external magnetic field B0 which is

applied to the fluid flow. This quantity appears in the bound-

ary conditions, which we do not discuss here. The second is a

length scale 𝓁 that determines the size of the channel through

which the fluid is flowing.

Altogether, there are seven model parameters; however, for

the active subspace analysis in Section 4, we fix two of these

parameters—𝜇0 and 𝓁—such that we only have five indepen-

dent variables. The quantity 𝜇0 is fixed since the permeability

of a vacuum is a universal constant of a similar nature to

the speed of light. Treating this value as variable would not

provide useful information. The length scale is considered

fixed in this study to represent a specific MHD generator

configuration. Despite fixing these parameters for the active

subspace analysis, they appear in the dimensional analysis.

Proper application of the Buckingham Pi Theorem requires

that we include all dimensional quantities relevant to describ-

ing the physics of the system, regardless of whether or not

they are treated as fixed or variable. Furthermore, fixing these

parameters for active subspace analysis does not alter the

upper bound on the dimensionality of the reduced parameter

space provided by dimensional analysis.

We perform dimensional analysis for the MHD problem

from two perspectives. Both approaches are valid and

produce equivalent results; however, they each highlight dif-

ferent aspects of the problem. The first may be considered

a classical perspective on dimensional analysis. It does not

assume y= f (x) from Equation 1, that is, an input/output

system as described in Section 2. Instead, we apply the Buck-

ingham Pi Theorem directly to governing system of PDEs in

Equation 21 by constructing the dimension matrix using all

relevant dimensional quantities. Such analysis produces the

traditional unitless quantities (e.g., the Reynolds number and

Hartmann number) that appear in MHD. The second per-

spective is motivated by the connection between dimensional

analysis and active subspaces explained in Section 2.3. For

this analysis, we treat the seven model parameter as inputs

for f . We assume the output is some functional of the solu-

tion fields from Equation 21. We then follow the procedure

discussed in Section 2.1.

3.1 Classical dimensional analysis for MHD

Classical dimensional analysis is performed directly on the

governing equations, which allows us to reformulate them

in terms of unitless (or nondimensional) quantities. Perform-

ing such analysis requires that we consider all dimensional

GLAWS ET AL. 317

quantities that are fundamental to the model. These include

the seven model parameters as well as a velocity—which we

denote by v—due to the dependent variable u in Equation 21.

We need not include additional pressure and magnetic

field terms since these appear in the parameters. The nec-

essary base units are L= length, T= time, M=mass, and

C= electric current; see Table 1. The model’s dimensional

quantities, with their units, are

• length, 𝓁, with [𝓁]=L,

• velocity, v, with [v] = L

T,

• fluid viscosity, 𝜇, with [𝜇] = M

LT,

• fluid density, 𝜌, with [𝜌] = M

L3,

• pressure gradient, 𝛻p0, with [𝛻p0] = M

L2T2,

• fluid magnetic resistivity, 𝜂, with [𝜂] = ML3

T3C2,

• magnetic field, B0, with [B0] = M

T2C,

• and the vacuum permeability, 𝜇0, with [𝜇0] = ML

T2C2.

The associated matrix D (see Equation 4) is

D =

𝓁 v 𝜇 𝜌 𝛻p0 𝜂 B0 𝜇0

LTMC

⎡⎢⎢⎢⎣1 1 −1 −3 −2 3 0 10 −1 −1 0 −2 −3 −2 −20 0 1 1 1 1 1 10 0 0 0 0 −2 −1 −2

⎤⎥⎥⎥⎦. (22)

A basis for the null space of D is

⎧⎪⎪⎪⎨⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎢⎢⎣

11−110000

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎢⎣

10

−1∕200

−1∕210

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎢⎣

11000−101

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1−20−11000

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎬⎪⎪⎪⎭. (23)

Since the dimension of the null space of D is 4, the Buck-

ingham Pi Theorem states that the system depends on four

unitless quantities; see Equation 6. In this case,

Π1 = 𝓁1v1𝜇−1𝜌1𝛻p00𝜂0B0𝜇0

0= 𝓁v𝜌

𝜇,

Π2 = 𝓁1v0𝜇−1∕2𝜌0𝛻p00𝜂−1∕2B1

0𝜇0

0= 𝓁B0

𝜇1∕2𝜂1∕2,

Π3 = 𝓁1v1𝜇0𝜌0𝛻p00𝜂−1B0

0𝜇1

0= 𝓁v𝜇0

𝜂,

Π4 = 𝓁1v−2𝜇0𝜌−1𝛻p10𝜂0B0

0𝜇0

0=

𝓁𝛻p0

v2𝜌. (24)

An alternative way to determine the unitless quantities for

Equation 21 is to nondimensionalize the equations. To do so,

we express the spatial variables and the outputs as products of

a unitless quantity (denoted by *) and a characteristic quantity

(denoted with a subscript c),

x = x∗𝓁c, u = u∗vc, B = B∗Bc, p0 = p∗0pc, and p = p∗pc,

(25)

where the characteristic quantities have units matching their

corresponding dimensional quantities.

Plug Equation 25 in Equation 21 and simplify to obtain

𝛻∗ ⋅[u∗ ⊗ u∗ + (p∗

0+ p∗)I − 1

Re(𝛻∗u∗ + 𝛻∗u∗T )

− 1

Rm

Ha2

Re

(B∗ ⊗ B∗ − 1

2||B∗||2I

)]= 0,

𝛻∗ ⋅[

u∗ ⊗ B∗ − B∗ ⊗ u∗ − 1

Rm(𝛻∗B∗ − 𝛻∗B∗T )

]= 0,

𝛻∗ ⋅ u∗ = 0, 𝛻∗ ⋅ B∗ = 0. (26)

The unitless spatial variables appear in the derivatives

𝛻∗ =[

𝜕

𝜕x∗1

𝜕

𝜕x∗2

𝜕

𝜕x∗3

]T. (27)

Equation 26 depends on four unitless quantities that are

well-known from fluid dynamics [27] and electromagnetics

[14]:

• the Reynolds number, Re = 𝓁cvc𝜌

𝜇,

• the Hartmann number, Ha = 𝓁cBc

𝜇1∕2𝜂1∕2,

• the magnetic Reynolds number, Rm = 𝓁cvc𝜇0

𝜂,

• and a dimensionless pressure gradient, 𝛻∗ ⋅ (p∗0I) = 𝛻∗p∗

0=

𝓁c𝛻p0

v2c𝜌

.

Scaling the MHD equations is consistent with the unitless

quantities derived from the basis Equation 23. In other words,

the unitless quantities from the classical Buckingham Pi anal-

ysis match those in Equation 26.

3.2 Active subspaces-motivated dimensional analysisfor MHD

In this section, we perform dimensional analysis motivated

by the connection between active subspaces and dimensional

analysis discussed in Section 2.3. This analysis assumes a

problem of the form y= f (x). We consider the seven model

parameters to be the inputs x. The scalar-valued output is

a functional of the unknown solution fields (e.g., velocity,

magnetic field) in the governing equations. In Section 4,

we consider two outputs: the average flow velocity and the

magnitude of the induced magnetic field. For the dimen-

sional analysis in this section, we have a single D matrix (see

Equation 4) associated with the set of inputs and one distinct

u vector for each output.

The matrix D from Equation 4 is

D =

𝓁 𝜇 𝜌 𝛻p0 𝜂 B0 𝜇0

LTMC

⎡⎢⎢⎢⎣1 −1 −3 −2 3 0 10 −1 0 −2 −3 −2 −20 1 1 1 1 1 10 0 0 0 −2 −1 −2

⎤⎥⎥⎥⎦. (28)

318 GLAWS ET AL.

A basis for the null space of D is⎧⎪⎪⎪⎨⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎢⎣

10010−21

⎤⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎣

1−1∕2

00

−1∕210

⎤⎥⎥⎥⎥⎥⎥⎦,

⎡⎢⎢⎢⎢⎢⎢⎣

01−10−101

⎤⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎬⎪⎪⎪⎭. (29)

If we consider the average velocity output, then the u vector

from Equation 4 is

u =[1 −1 0 0

]T. (30)

Solving Dv=u yields

v =[0 0 −1∕3 1∕3 1∕3 0 −1∕3

]T. (31)

The matrix A from Equation 18 is

A =

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 00 0 −1∕2 1

−1∕3 0 0 −11∕3 1 0 01∕3 0 −1∕2 −1

0 −2 1 0−1∕3 1 0 1

⎤⎥⎥⎥⎥⎥⎥⎦. (32)

This matrix has full column rank. Therefore, the average flow

velocity depends on at most four linear combinations of the

log-transformed input parameters, as shown in Section 2.3.

Next, we consider the total induced magnetic field. The uvector from Equation 4 for this output is

u =[0 −2 1 −1

]T, (33)

and the solution to Dv=u is

v =[0 0 1∕6 1∕3 1∕3 0 1∕6

]T. (34)

The matrix A is

A =

⎡⎢⎢⎢⎢⎢⎢⎣

0 1 1 00 0 −1∕2 1

1∕6 0 0 −11∕3 1 0 01∕3 0 −1∕2 −1

0 −2 1 01∕6 1 0 1

⎤⎥⎥⎥⎥⎥⎥⎦. (35)

Once again, A has full column rank implying that the induced

magnetic field depends on four or fewer linear combinations

of the log-transformed input parameters. In the next section,

we analyze the active subspaces of the particular outputs,

average velocity and total induced magnetic field, as functions

of the model parameters; we numerically verify the theoreti-

cal results about the number of linear combinations of inputs

that affect the outputs.

4 ACTIVE SUBSPACES FOR MHD

The dimensional analysis in the previous section, coupled

with the analysis from Section 2.3, indicates that both the

average flow velocity and the induced magnetic field can be

written as a ridge function of four linear combinations of the

FIGURE 1 Depiction of the Hartmann problem. A magnetic fluid flows

between two parallel plates in the presence of a perpendicular magnetic

field. The field acts as a damping force on the fluid while the flow induces a

horizontal magnetic field

log transformed inputs. Therefore, we expect numerical tests

to reveal an active subspace of dimension 4 or less. We study

two MHD models with active subspaces: (i) the Hartmann

problem that models a simplified duct flow and (ii) a numer-

ical model of an idealized MHD generator in three spatial

dimensions.

4.1 Hartmann problem

The Hartmann problem is a standard test problem in MHD.

It models laminar flow between two parallel plates. In gen-

eral, we assume these plates are separated by distance 2𝓁. We

define 𝓁 = 1 m for the numerical investigations in this section.

The fluid is assumed to be a conducting fluid and a uniform

magnetic field is applied perpendicular to the flow direction.

As the fluid flow bends the magnetic field, an induced field

in the horizontal direction is generated and a corresponding

magnetic stress is developed that resists, or damps, the flow

nonuniformly in the boundary layer and the core of the flow.

This can be seen in Figure 1.

The advantage of working with the Hartmann problem

is that it admits closed form analytical expressions for the

quantities of interest in terms of their input parameters. This

enables thorough numerical studies where errors can be com-

puted exactly. We consider two quantities of interest: (i)

average flow velocity across the channel uavg and (ii) the

induced magnetic field Bind. Derivations of these quantities of

interest for the Hartmann problem from the governing MHD

Equation 21 are not presented here; they may be in found in

Cowling and Lindsay [13]. As functions of the model inputs

(see Section 3.1),

uavg = −𝜕p0

𝜕x𝜂

B20

(1 − B0𝓁√

𝜂𝜇coth

(B0𝓁√𝜂𝜇

)), (36)

and

Bind =𝜕p0

𝜕x𝓁𝜇0

2B0

(1 − 2

√𝜂𝜇

B0𝓁tanh

(B0𝓁

2√𝜂𝜇

)). (37)

The expressions for the Hartmann problem quantities of inter-

est have five input parameters: fluid viscosity 𝜇, fluid density

𝜌, applied pressure gradient 𝜕p0/𝜕x (where the derivative is

GLAWS ET AL. 319

TABLE 2 Indices and intervals for the parameters x of the Hartmannproblem. These intervals represent the expected operating conditionsfor an MHD generator modeled with the Hartmann problem

Index Name Notation Interval

1 Fluid viscosity log(𝜇) [log(0.05), log(0.2)]

2 Fluid density log(𝜌) [log(1), log(5)]

3 Applied pressure gradient log(

𝜕p0

𝜕x

)[log(0.5), log(3)]

4 Resistivity log(𝜂) [log(0.5), log(3)]

5 Applied magnetic field log(B0) [log(0.1), log(1)]

with respect to the flow field’s spatial coordinate), resistivity

𝜂, and applied magnetic field B0. Note that fluid density 𝜌

does not appear in either Equation 36 or Equation 37. We treat

𝜌 as an input parameter for two reasons. First, 𝜌 appears in

the governing MHD Equation 21; its absence from the closed

form solutions, Equations 36 and 37, is not apparent without

knowing the closed form solutions. Second, the problem pro-

vides an interesting test for whether or not the active subspace

analysis can identify the lack of dependence on 𝜌.

Recall from Section 2.3 that we consider the quantities of

interest as functions of the log-transformed inputs. Let

x =[log(𝜇) log(𝜌) log

(𝜕p0

𝜕x

)log(𝜂) log(B0)

]T, (38)

be the vector of inputs for the active subspace analysis from

Section 2.2. To estimate the active subspace for each quan-

tity of interest, we set the input density 𝛾(x) to be a uniform

density on a five-dimensional hyperrectangle. The ranges of

each of x’s components are in Table 2; they are chosen to

represent the expected operating conditions of an MHD gen-

erator modeled with the Hartmann problem. We estimate

C from Equation 8 with a tensor product Gauss-Legendre

quadrature rule [15] with 11 points in each dimension—a total

of 115 = 161 051 points. This is sufficient for ten digits of

accuracy in the eigenvalue estimates.

Figure 2 shows the results of analyzing the active subspace

of the Hartmann problem’s average flow velocity uavg from

Equation 36. Figure 2A shows that all but 2 eigenvalues

are zero (to machine precision). This implies that the active

subspace of dimension n= 2 is sufficient to describe the rela-

tionship between the log-transformed inputs and the quantity

of interest. Figure 2B shows the components of the first two

eigenvectors of C’s quadrature estimate; the index on the hor-

izontal axis maps to the specific input as in Table 2. A large

eigenvector component reveals that the corresponding param-

eter is important in defining the active subspace. Notice that

both eigenvector components corresponding to log(𝜌) (the

second input) are zero; this is consistent with the definition of

uavg in Equation 36, which does not depend on fluid density 𝜌.

Figure 2C a summary plot of 1000 samples of uavg—taken

from the quadrature evaluations used to estimate C—versus

the corresponding samples of the active variable. Such plots

are commonly used in regression graphics [11]. The plot

shows a strong relationship between the first active variable

and uavg, so a ridge function of the form Equation 12 with

one linear combination would be a good approximation;

this is validated by the three-order-of-magnitude gap

between the first and second eigenvalues. Figure 2D shows a

two-dimensional summary plot with the same data, where the

color is the value of uavg, the horizontal axis is the first active

variable (defined by the first eigenvector of C), and the ver-

tical axis is the second active variable (defined by the second

eigenvector of C). Since the eigenvalues with index greater

than 2 are zero, the two-dimensional summary plot reveals

the complete relationship between the log-transformed inputs

and uavg.

Figure 3 shows the same plots as Figure 2 for the induced

magnetic field Bind from Equation 37 as the quantity of inter-

est. Compared to uavg, the one-dimensional summary plot for

Bind, Figure 3C, shows greater departure from a univariate

relationship when the first active variable wT1x is less than 0.

This structure is confirmed in the two-dimensional summary

plot, Figure 3D, which shows the truly two-dimensional struc-

ture in map from inputs to Bind. The eigenvectors in Figure 3B

differ substantially from the eigenvectors associated with uavg

in Figure 2B; these differences reflect the different depen-

dence between inputs and outputs, which are verified in the

analytical expressions from Equations 36 and 37. Notably, the

second component of the eigenvectors is zero for both out-

puts, which indicates that neither output depends on the fluid

density 𝜌 (the second input parameter).

The dimensional analysis from Section 3 showed that the

quantities of interest should depend on four unitless quanti-

ties. Equation 26 expresses the governing equations in terms

of the Reynolds number, the Hartmann number, the magnetic

Reynolds number, and a dimensionless pressure gradient.

With scaling, we can write unitless forms of the quantities of

interest, uavg from Equation 36 and Bind from Equation 37, in

terms of unitless parameters:

u∗avg = −

𝜕p∗0

𝜕x∗Re

Ha2(1 − Ha coth(Ha)), (39)

and

B∗ind

=𝜕p∗

0

𝜕x∗ReHa

Rm

(1 − 2

Hatanh

(Ha2

)). (40)

Notice several of the unitless quantities appear only as a

product in Equations 39 and 40. This product of unitless

quantities defines a new unitless quantity. Thus, u∗avg and B∗

ind

depend only on two unitless quantities. This explains why the

eigenvalues of C are zero after the second.

4.2 MHD generator problem

This model is a steady-state MHD duct flow configuration

representing an idealized MHD generator; the governing

equations for the flow and magnetic field are the resis-

tive MHD equation—a slightly modified version∗ of the

∗The minor modification to the governing equations does not change the

dimensional analysis.

320 GLAWS ET AL.

(A)

(C) (D)

(B)

FIGURE 2 These figures represent the active subspace-based dimension reduction for the Hartmann problem’s average flow velocity uavg from Equation 36.

(A) The eigenvalues of C, and (B) the components of C’s first two eigenvectors. (C) and (D) One- and two-dimensional summary plots of the quantity of

interest. They reveal the low-dimensional relationship between the two active variables and uavg

governing equations presented in Section 3; see Shadid et al.

[30] for details. The MHD generator induces an electrical

current by supplying a set flow-rate of a conducting fluid

through an externally supplied vertical magnetic field. The

bending of the magnetic field lines produces a horizontal

electrical current. The geometric domain for this problem is a

square cross-sectional duct of dimensions 15 m× 1 m× 1 m.

The simple geometry problem facilitates scalability studies

as different mesh sizes can be easily generated. The velocity

boundary conditions are set with Dirichlet inlet velocity of [1,

0, 0] m/s, no slip on the top, bottom and sides of the channel,

and natural boundary conditions on the outflow. The magnetic

field boundary conditions on the top and bottom are specified

as a set magnetic field configuration (0,Bgeny , 0), where

Bgeny = 1

2B0

[tanh

(x − xon

𝛿

)− tanh

(x − xoff

𝛿

)]. (41)

The values xon and xoff indicate the locations in the x direc-

tion where the magnetic field is active. The inlet, outlet, and

sides are perfect conductors with B ⋅ n = 0 and E × n = 0,

that is, the current and magnetic fluxes are zero at these

boundaries. Homogeneous Dirichlet boundary conditions are

used on all surfaces for the Lagrange multiplier enforcing the

solenoidal constraint 𝛻 ⋅B= 0; see Shadid et al. [30]. This

problem has similar characteristics to the Hartmann problem

with (i) viscous boundary layers, (ii) Hartmann layers occur-

ring at the boundaries, and (iii) a flow field that is strongly

modified by the magnetic field in the section of the duct

where it is active. Figure 4 shows a solution for this problem

for Reynolds number Re= 2500, magnetic Reynolds number

Rem = 10, and Hartmann number Ha= 5. The image shows

x velocity iso-surface colored by y velocity, where the modi-

fication of the inlet constant profile and the parabolic profile

at the region where the magnetic field is active are evident.

Vectors (colored by magnitude) show the vertical magnetic

field (applied and induced) and horizontal induced current

from the bending of the magnetic field lines.

The fixed physical parameters for the MHD generator are

𝜇0 = 1, xon = 4.0, xoff = 6.0, and 𝛿 = 0.1. The variable input

parameters are the same as in the Hartmann problem. How-

ever, the generator uses different input ranges, which can

be found in Table 3. The probability density function on the

space of inputs is a uniform density on the hyperrectangle of

log-transformed parameters defined by the ranges in Table 3.

The quantities of interest are the average flow velocity uavg

and the induced magnetic field Bind, as in the Hartmann

problem.

Given values for the input parameters, the MHD generator’s

solution fields are computed with the Sandia National Lab-

oratory’s Drekar multiphysics solver package [26,31]. The

GLAWS ET AL. 321

(A)

(C) (D)

(B)

FIGURE 3 These figures represent the active subspace-based dimension reduction for the Hartmann problem’s induced magnetic field Bind from

Equation 37. (A) The eigenvalues of C, and (B) shows the components of C’s first two eigenvectors. (C) and (D) One- and two-dimensional summary plots of

the quantity of interest. They reveal the low-dimensional relationship between the two active variables and Bind

FIGURE 4 Visualization of flow field from an idealized 3D MHD generator model. The image shows x velocity iso-surface colored by the y velocity.

Vectors (colored by magnitude) show vertical magnetic field (applied and induced) and horizontal induced current

package has adjoint capabilities, which enables computation

of the derivatives of the quantities of interest with respect

to the input parameters [31]. Each MHD generator model

run uses 5.3 CPU-hours (ten min on 32 cores), so estimat-

ing C from Equation 8 with a tensor product Gauss-Legendre

quadrature rule is not possible. Instead, we use a Monte Carlo

method to estimate C using M = 483 independent samples

from the uniform density on the log-transformed parameters.

For this study, our budget was roughly 500 simulations, which

is common for simulation models whose size and complexity

are comparable to the Drekar-based MHD model. For details

on the accuracy of the Monte Carlo method for estimating

active subspaces, see Constantine and Gleich [6].

Figure 5A shows the eigenvalue estimates computed with

Monte Carlo for the uavg quantity of interest. The dashed

lines show upper and lower bounds on the eigenvalue esti-

mates computed with a nonparametric bootstrap with 500

bootstrap replicates from the set of 483 gradient samples. We

322 GLAWS ET AL.

TABLE 3 Indices and intervals for the parameters x of the MHDgenerator problem. These intervals represent the expected operatingconditions for the idealized MHD generator

Index Name Notation Interval

1 Fluid viscosity log(𝜇) [log(0.001), log(0.01)]

2 Fluid density log(𝜌) [log(0.1), log(10)]

3 Applied pressure gradient log(

𝜕p0

𝜕x

)[log(0.1), log(0.5)]

4 Resistivity log(𝜂) [log(0.1), log(10)]

5 Applied magnetic field log(B0) [log(0.1), log(1)]

emphasize that since there is no randomness in the map from

inputs to outputs (i.e., the computer simulation is determin-

istic), the bootstrap is a heuristic to estimate the variability

due to the Monte Carlo sampling. Estimates of standard error

from sample variances are not appropriate, since the eigen-

values are nonlinear functions of the gradient samples. For an

example of a similar bootstrap computation for eigenvalues,

see (Chapter 7.2 of Efron and Tibshirani [16]).

The fifth eigenvalue is 0.00024% of the sum of the five

eigenvalues, which is consistent with the dimensional anal-

ysis from Section 3—that is, there should be no more than

four linear combinations of the model parameters that affect

the quantity of interest. And this restriction is reflected in the

small fifth eigenvalue.

The first two eigenvectors of C’s Monte Carlo estimate

(for uavg) are shown in Figure 6B. The magnitudes of the

eigenvector components can be used to determine which

physical parameters influence the active subspace—that is,

they provide sensitivity information. (See Constantine and

Diaz [5] for how to construct sensitivity metrics from active

subspaces.) The fluid viscosity 𝜇 and the pressure gradient

𝜕p0/𝜕x are the most important parameters for the average

(A)

(C) (D)

(B)

FIGURE 5 These figures represent the active subspace-based dimension reduction for the MHD generator problem’s average flow velocity uavg. (A) The

eigenvalues of C, and (B) the components of C’s first two eigenvectors. (C) and (D) One- and two-dimensional summary plots of the quantity of interest.

They reveal the low-dimensional relationship between the two active variables and uavg

GLAWS ET AL. 323

(A) (B)

(C) (D)

FIGURE 6 These figures represent the active subspace-based dimension reduction for the MHD generator’s induced magnetic field Bind. (A) The

eigenvalues of C, and (B) the components of C’s first two eigenvectors. (C) and (D) One- and two-dimensional summary plots of the quantity of interest.

They reveal the low-dimensional relationship between the two active variables and Bind

fluid velocity. This insight agrees with physical intuition,

and it is consistent with the same metrics from the Hartmann

problem; see Figure 2B.

Figure 5C,D shows the one- and two-dimensional summary

plots for uavg as a function of the first two active variables

using all 483 samples. Similar to Figure 2C,D, we see a nearly

one-dimensional relationship between the log-transformed

input parameters and the average velocity, where the one

dimension is the first active variable.

Figure 6A shows the eigenvalues for C’s Monte Carlo esti-

mate, with bootstrap ranges, for the induced magnetic field

quantity of interest Bind. In this case, the fifth eigenvalue is

0.000001% of the sum of the eigenvalues, which is consistent

with the dimensional analysis from Section 2.3 that shows

that any quantity of interest will depend on at most four linear

combinations of the log-transformed input parameters. The

first eigenvector in Figure 6B shows that Bind depends on all

input parameters except the fluid density 𝜌. This is remarkably

similar to the dependence seen in the Hartmann problem; see

Figure 3B.

The one- and two-dimensional summary plots for Bind are

in Figure 6C,D. There appears to be a region in the parame-

ter space—when the first active variable is positive—where

the relationship between the inputs and Bind is well char-

acterized by one linear combination of the log-transformed

inputs. However, the one-dimensional character of that rela-

tionship degrades as the first active variable decreases. (Note

that the first eigenvector is only unique up to a sign, so

this relationship could be inverted.) The relative signs of the

eigenvector components yield insight into the how the out-

put varies as each input is varied. For example, the fourth and

fifth components of the first eigenvector have opposite signs.

324 GLAWS ET AL.

Thus, the corresponding model parameters—resistivity 𝜂 and

applied magnetic field B0—affect Bind in opposite directions,

on average.

5 CONCLUSIONS

We have reviewed two methods for dimension reduction in

physical systems: (i) dimensional analysis that uses the phys-

ical quantities’ units and (ii) active subspaces that use the

gradient of the output with respect to the inputs to identify

an important low-dimensional subspace of the input space.

We also reviewed the connection between these two methods

via a log transform of the input parameters—namely, that the

dimensional analysis provides an upper bound on the num-

ber of linear combinations of log-transformed parameters that

control any quantity of interest.

We applied these techniques to two quantities of

interest—average flow velocity and induced magnetic

field—from two magnetohydrodynamics models that apply

to power generation: (i) the Hartmann problem that admits

closed form expressions for the quantities of interest and

(ii) a large-scale computational model of coupled fluid flow,

magnetic fields, and electric current in a three-dimensional

duct. The computational model has adjoint capabilities that

enable gradient evaluations.

The insights from the active subspace are consistent with

the dimensional analysis. In particular, there are at most

four linear combinations of log-transformed parameters that

affect the quantity of interest—which is a reduction from

an ambient dimension of 5 to an intrinsic dimension of

4. The Hartmann problem has a further reduction to an

intrinsic dimension of 2—that is, two linear combinations

of log-transformed parameters are sufficient to character-

ize the quantities of interest. The eigenvalues of the matrix

C that defines the active subspace rank the importance

of the linear combinations, which offers more insight into

the input/output relationships than the dimensional analysis

alone.

The eigenvector components that define the active subspace

reveal the sensitivities of the quantities of interest with respect

to the input parameters. These metrics are consistent between

the Hartmann problem and the large-scale generator model.

In particular, (i) the average flow velocity depends mainly on

the fluid viscosity and the applied pressure gradient, and (ii)

the induced magnetic field depends strongly on all parameters

except the fluid density.

The summary plots with the first and second eigenvec-

tors show the low-dimensional relationship between the

log-transformed inputs and the quantities of interest. In par-

ticular, the average flow velocity can be well approximated by

a function of one linear combination of the log-transformed

inputs in both the Hartmann problem and large-scale

generator. The same is true for the induced magnetic field

from the Hartmann problem. However, the induced magnetic

field in the large-scale generator admits a one-dimensional

characterization in a subset of the parameter domain. By

combining the information in the one-dimensional summary

plot with the sensitivities reveals by the first eigenvector’s

components, we conclude that in regions of (i) low fluid

viscosity, (ii) high applied pressure gradient, (iii) low resis-

tivity, and (iv) high applied magnetic field, the relationship

between the log-transformed inputs and the induced magnetic

field requires more than two linear combinations of input

parameters for accurate approximation.

These insights offer directions for further development

in MHD models for effective power generation. Addition-

ally, the revealed relationships between the inputs and out-

puts will assist in (i) quantifying parametric uncertainties

and (ii) enabling computational design for MHD power

generation.

ACKNOWLEDGMENTS

Glaws’ work is supported by the Ben L. Fryrear Ph.D. Fel-

lowship in Computational Science at the Colorado School of

Mines. Constantine’s work is supported by the US Depart-

ment of Energy Office of Science, Office of Advanced

Scientific Computing Research, Applied Mathematics pro-

gram under Award Number DE-SC-0011077. This work

was partially supported by the DOE Office of Science

Advanced Scientific Computing Research Applied Mathe-

matics Program at Sandia National Laboratories under con-

tract DE-NA0003525. Sandia National Laboratories is a

multimission laboratory managed and operated by National

Technology and Engineering Solutions of Sandia, LLC, a

wholly owned subsidiary of Honeywell International, Inc., for

the US Department of Energy’s National Nuclear Security

Administration under contract DE-NA0003525.

ORCIDPaul G. Constantine http://orcid.org/

0000-0003-3726-6307

REFERENCES1. M. C. Albrecht, C. J. Nachtsheim, T. A. Albrecht, and R. D. Cook, Exper-

imental design for engineering dimensional analysis, Technometrics 55

(2013), pp. 257–270.

2. G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics,

Cambridge Univ. Press, Cambridge, 1996.

3. D. Calvetti and E. Somersalo, Computational mathematical modeling: anintegrated approach across scales, SIAM, Philadelphia, 2013.

4. P. G. Constantine, Active subspaces: emerging ideas for dimension reductionin parameter studies, SIAM, Philadelphia, 2015.

5. P. G. Constantine and P. Diaz, Global sensitivity metrics from active sub-spaces, Reliab. Eng. Syst. Saf. 162 (2017), pp. 1–13.

6. P. Constantine and D. Gleich, Computing active subspaces with MonteCarlo, 2015. arXiv:1408.0545v2.

7. P. Constantine, E. Dow, and Q. Wang, Active subspace methods in theoryand practice: applications to kriging surfaces, SIAM J. Sci. Comput. 36

(2014), pp. A1500–A1524.

8. P. G. Constantine, B. Zaharatos, and M. Campanelli, Discovering an activesubspace in a single-diode solar cell model, Stat. Anal. Data Min. ASA Data

Sci. J. 8 (2015), pp. 264–273.

GLAWS ET AL. 325

9. P. G. Constantine, M. Emory, J. Larsson, and G. Iaccarino, Exploiting activesubspaces to quantify uncertainty in the numerical simulation of the HyShotII scramjet, J. Comput. Phys. 302 (2015), pp. 1–20.

10. P. Constantine, Z. del Rosario, and G. Iaccarino, Many physical laws areridge functions, 2016. arXiv:1605.07974.

11. R. D. Cook, Regression graphics: ideas for studying regressions throughgraphics, 2nd ed., John Wiley & Sons, New York, 1998.

12. R. D. Cook and S. Weisberg, Applied regression including computing andgraphics, John Wiley & Sons, New York, 1999.

13. T. G. Cowling and R. B. Lindsay, Magnetohydrodynamics, Phys. Today 10

(1957), p. 40.

14. P. A. Davidson, An introduction to magnetohydrodynamics, Cambridge

Univ. Press, Cambridge, 2001.

15. P. J. Davis and P. Rabinowitz, Methods of numerical integration, 2nd ed.,

Academic Press, Inc., New York, 1984.

16. B. Efron and R. J. Tibshirani, An introduction to the bootstrap, CRC Press,

1994.

17. A. Gittens and J. A. Tropp, Tail bounds for all eigenvalues of a sum ofrandom matrices, 2011 arXiv:1104.4513.

18. G. H. Golub and C. F. V. Loan, Matrix computations, 4th ed., The Johns

Hopkins University Press, Baltimore, 2013.

19. J. L. Jefferson, J. M. Gilbert, P. G. Constantine, and R. M. Maxwell, Activesubspaces for sensitivity analysis and dimension reduction of an integratedhydrologic model, Comput. Geosci. 83 (2015), pp. 127–138.

20. J. L. Jefferson, R. M. Maxwell, and P. G. Constantine, Exploring the sensi-tivity of photosynthesis and stomatal resistance parameters in a land surfacemodel, J Hydrometeorol. 18 (2017), pp. 897–915.

21. I. T. Jolliffe, Principal component analysis, Springer Science+Business

Media, LLC, 1986.

22. D. K. J. Lin and W. Shen, Comment: some statistical concerns on dimen-sional analysis, Technometrics 55 (2013), pp. 281–285.

23. P. T. Lin, J. N. Shadid, R. S. Tuminaro, M. Sala, G. L. Hennigan, and R.

P. Pawlowski, A parallel fully coupled algebraic multilevel preconditionerapplied to multiphysics PDE applications: drift-diffusion, flow/transport/reaction, resistive MHD, Int. J. Numer. Methods Fluids 64 (2010), pp.

1148–1179.

24. National Institute of Standards and Technology, International system of units(SI). June 11, 2016. Available at http://physics.nist.gov/cuu/Units/units.html

25. NETL, Magnetohydrodynamic power generation workshop summary report.Technical report, National Energy Technology Laboratory, 2014.

26. R. P. Pawlowski, J. N. Shadid, T. M. Smith, E. C. Cyr, and P. D. Weber,

Drekar::CFD—a turbulent fluid-flow and conjugate heat transfer code.Theory manual (version 1.0). Technical Report SAND2012-2697, Sandia

National Laboratories, 2012.

27. A. Pinkus, Ridge functions, Cambridge Univ. Press, Cambridge, 2015.

28. R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, and E. M. Gates, Fluid flow:a first course in fluid mechanics, 4th ed., Prentice Hall, New Jersey, 1999.

29. J. N. Shadid, R. P. Pawlowski, J. W. Banks, L. Chacón, P. T. Lin, and R.

S. Tuminaro, Towards a scalable fully-implicit fully-coupled resistive MHDformulation with stabilized FE methods, J. Comput. Phys. 229 (2010), pp.

7649–7671.

30. J. N. Shadid, R. P. Pawlowski, E. C. Cyr, R. S. Tuminaro, L. Chacón, and

P. D. Weber, Scalable implicit incompressible resistive MHD with stabilizedFE and fully-coupled Newton-Krylov-AMG, Comput. Methods Appl. Mech.

Eng. 304 (2016), pp. 1–25.

31. J. N. Shadid, T. M. Smith, E. C. Cyr, T. M. Wildey, and R. P. Pawlowski,

Stabilized FE simulation of prototype thermal-hydraulics problems withintegrated adjoint-based capabilities, J Comput. Phys. 321 (2016), pp.

321–341.

32. W. Shen, Dimensional analysis in statistics: theories, methodologies and

applications, Ph.D. Thesis, The Pennsylvania State Univ., 2015.

33. W. Shen, T. Davis, D. K. J. Lin, and C. J. Nachtsheim, Dimensionalanalysis and its applications in statistics, J. Qual. Technol. 46 (2014),

pp. 185–198.

34. R. C. Smith, Uncertainty quantification: theory, implementation, and appli-cations, SIAM, Philadelphia, 2013.

35. D. Sondak, J. N. Shadid, A. A. Oberai, R. P. Pawlowski, E. C. Cyr, and T.

M. Smith, A new class of finite element variational multiscale turbulencemodels for incompressible magnetohydrodynamics, J. Comput. Phys. 295

(2015), pp. 596–616.

36. G. W. Stewart, Error and perturbation bounds for subspaces associated withcertain eigenvalue problems, SIAM Rev. 15 (1973), pp. 727–764.

37. T. Sullivan, Introduction to uncertainty quantification, Springer, New York,

2015.

38. J. A. Tropp, User-friendly tail bounds for sums of random matrices, Found.

Comput. Math. 12 (2012), pp. 389–434.

39. USGAO, Fossil fuels: the Department of Energy’s MagnetohydrodynamicsDevelopment Program. Technical Report RCED-93-174, U.S. Government

Accountability Office, 1993.

40. S. Weisberg, Applied linear regression, 3rd ed., John Wiley & Sons, New

York, 2005.

How to cite this article: Glaws A, Constantine

PG, Shadid J and Wildey TM. Dimension reduction

in magnetohydrodynamics power generation models:

Dimensional analysis and active subspaces, Statisti-cal Analysis and Data Mining: The ASA Data Sci-ence Journal 2017;10:312–325. https://doi/10.1002/

sam.11355.