Application of Polynomial Chaos Expansion for Climate ...1188950/FULLTEXT01.pdf · Polynomial Chaos Expansions (PCEs), ett alternativ till Monte Carlo-metoder. Med hjälp av PCEs

IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Application of Polynomial Chaos Expansion for Climate Economy Assessment

ROBIN NYDESTEDT

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Application of Polynomial Chaos Expansion for Climate Economy Assessment ROBIN NYDESTEDT Degree Projects in Optimization and Systems Theory (30 ECTS credits) Degree Programme in Engineering Physics KTH Royal Institute of Technology year 2018 Supervisor at Karlsruhe Institute of Technology: Dr. Timm Faulwasser Supervisor at KTH: Prof. Xiaoming Hu Examiner at KTH: Prof. Xiaoming Hu

TRITA-SCI-GRU 2018:031 MAT-E 2018:10 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

In climate economics integrated assessment models (IAMs) are used to predict economic impactsresulting from climate change. These IAMs attempt to model complex interactions between humanand geophysical systems to provide quanti�cations of economic impact, typically using the Social Costof Carbon (SCC) which represents the economic cost of a one ton increase in carbon dioxide. Anotherdi�culty that arises in modeling a climate economics system is that both the geophysical and economicsubmodules are inherently stochastic. Even in frequently cited IAMs, such as DICE and PAGE, thereexists a lot of variation in the predictions of the SCC. These di�erences stem both from the modelsof the climate and economic modules used, as well as from the choice of probability distributionsused for the random variables. Seeing as IAMs often take the form of optimization problems thesenondeterministic elements potentially result in heavy computational costs. In this thesis a new IAM,FAIR/DICE, is introduced. FAIR/DICE is a discrete time hybrid of DICE and FAIR providing apotential improvement to DICE as the climate and carbon modules in FAIR take into account feedbackcoming from the climate module to the carbon module. Additionally uncertainty propagation inFAIR/DICE is analyzed using Polynomial Chaos Expansions (PCEs) which is an alternative to MonteCarlo sampling where the stochastic variables are projected onto stochastic polynomial spaces. PCEsprovide better computational e�ciency compared to Monte Carlo sampling at the expense of storagerequirements as a lot of computations can be stored from the �rst simulation of the system, andconveniently statistics can be computed from the PCE coe�cients without the need for sampling. APCE overloading of FAIR/DICE is investigated where the equilibrium climate sensitivity, modeledas a four parameter Beta distribution, introduces an uncertainty to the dynamical system. Finally,results in the mean and variance obtained from the PCEs are compared to a Monte Carlo referenceand avenues into future work are suggested.

2

Sammanfattning

Inom klimatekonomi används integrated assessment models (IAMs) för att förutspå hur klimat-förändringar påverkar ekonomin. Dessa IAMs modellerar komplexa interaktioner mellan geofysiskaoch mänskliga system för att kunna kvanti�era till exempel kostnaden för den ökade koldioxidhal-ten på planeten, i.e. Social Cost of Carbon (SCC). Detta representerar den ekonomiska kostnadensom motsvaras av utsläppet av ett ton koldioxid. Faktumet att både de geofysiska och ekonomiskasubmodulerna är stokastiska gör att SCC-uppskattningar varierar mycket även inom väletableradeIAMs som PAGE och DICE. Variationen grundar sig i skillnader inom modellerna men också frånatt val av sannolikhetsfördelningar för de stokastiska variablerna skiljer sig. Eftersom IAMs ofta ärformulerade som optimeringsproblem leder dessutom osäkerheterna till höga beräkningskostnader. Idenna uppsats introduceras en ny IAM, FAIR/DICE, som är en diskret tids hybrid av DICE ochFAIR. Den utgör en potentiell förbättring av DICE eftersom klimat- och kolmodulerna i FAIR ävenbehandlar återkoppling från klimatmodulen till kolmodulen. FAIR/DICE är analyserad med hjälp avPolynomial Chaos Expansions (PCEs), ett alternativ till Monte Carlo-metoder. Med hjälp av PCEskan de osäkerheter projiceras på stokastiska polynomrum vilket har fördelen att beräkningskost-nader reduceras men nackdelen att lagringskraven ökar. Detta eftersom många av beräkningarnakan sparas från första simuleringen av systemet, dessutom kan statistik extraheras direkt från PCEkoe�cienterna utan behov av sampling. FAIR/DICE systemet projiceras med hjälp av PCEs där enosäkerhet är introducerad via equilibrium climate sensitivity (ECS), vilket i sig är ett värde på hurkänsligt klimatet är för koldioxidförändringar. ECS modelleras med hjälp av en fyra-parameters Betasannolikhetsfördelning. Avslutningsvis jämförs resultat i medelvärde och varians mellan PCE imple-mentationen av FAIR/DICE och en Monte Carlo-baserad referens, därefter ges förslag på framtidautvecklingsområden.

3

Contents

1 Introduction 1

2 Modeling Climate Economy 2

2.1 DICE2013-R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 FAIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 FAIR/DICE Hybrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Using Integrated Assessment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Polynomial Chaos Expansion for Uncertainty Quanti�cation 18

3.1 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Hilbert Spaces of Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Polynomial Chaos Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Polynomial Chaos Expansion of the FAIR/DICE Dynamics . . . . . . . . . . . . . . . . . 27

4 Simulating Stochastic FAIR/DICE 36

5 Results 38

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Conclusion 45

1

1 Introduction

Since the millenium the interest in climate change has been steadily increasing. Globally new climateaccords are being signed, such as the Paris Agreement in 2016, where nations agree on making an e�ortto reduce carbon emissions. To shape global policy it is important to have projections supporting theirimplementations, in other words the planetary harm caused by climate change needs to be quanti�ed.What does this harm e�ectively entail for us, or for future generations?

Due to the growing interest in the topic the interdisciplinary �eld of climate economics has emerged toprovide a di�erent perspective, one combining both economics and climate physics. For a climate physi-cist the validity of the geophysical models is paramount, whereas for an economist said models providea tool for better �nancial predictions. There is after all a consensus that humans are causing climatechange [2], thus humans can also, if not prevent, at least reduce the rate of change.

The question asked by climate economics is whether there is an optimal investment policy in reducingclimate change. Taking into account technological progress in renewables energy or carbon reduction aswell as current available funds there is a need for a long term strategy of investments. Climate economicsprovides a �nancial incentive for expending available funds in carbon reduction by looking at social wel-fare.

Any investment policy should ultimately provide a net positive gain not only for the environment, butalso for its inhabitants. This is done by maximizing social welfare under the constraints provided by eco-nomics and physics over a chosen time span, typically around 300 years. This leads to another question,how much can these constraints actually be relied upon? This report aims to investigate this question,or more speci�cally, analyze uncertainty in a climate economics model we have developed.

The model in question, FAIR/DICE, is a merge of a recently proposed geophysical model FAIR [9] andthe economic module of a climate economics model, DICE [12]. DICE in particular has been analyzed foruncertainty propogation in for example [6]. This analysis is done using what is referred to as polynomialchaos expansions, which allows for projections of uncertainties onto stochastic polynomial spaces. It isnot a method typical of climate economics, where traditional Monte Carlo sampling is more common. Butit does provide an e�ective way to handle uncertainties in large systems which are prevalent in climateeconomics.

2 2 MODELING CLIMATE ECONOMY

2 Modeling Climate Economy

Climate economy models are typically referred to as Integrated Assessment Models, or IAI's, due to theirinterdisciplinary nature. Speci�cally they integrate several di�erent �elds into a larger one, that is thenused for assessment of for example economic damage due to climate damage. This is all contained in aninterdisciplinary numerical model.

As climate economy IAI's are used to inform policy change decisions it is imperative that a value isassigned to damage done by increases carbon emissions. The value that is typically used is the socialcost of carbon, or the SCC, which gives a measure of the cost of emissions expressed in decrease inconsumption.

De�nition 1 (Social Cost of Carbon). The SCC in a particular year is the decrease in aggregate con-sumption in that year that would change the current value of the social welfare by the same amount asone unit increase in carbon emissions in that year. The unit of the SCC is typically [$/tCO2] [16].

The social welfare will be described more in depth in the next section but for now it su�ces to know it isa general measure quality of life. One way to interpret the social cost of carbon is to consider the socialwelfare pathway obtained by not changing the current emissions policies. How does the value of socialwelfare change in that year if instead one GtCO2 is emitted today? That di�erence is the social cost ofcarbon.

In essence the SCC is a measure of long term damage caused expressed in consumption, alternatively thevalue of avoided damages due to a small emissions reduction. The SCC can then be used as a guide forpolicy making, such as to determine carbon taxes. Clearly however, de�ning the SCC is of no use withoutmodels for consumptions and emissions which leads to the next section, where the IAI DICE2013-R willbe introduced.

2.1 DICE2013-R

The Dynamic Integrated Climate-Economy Model, or DICE model, is an IAI used to estimate the SCCdeveloped by W. Nordhaus in the early 1990s. He has since continued its development and at the timeof writing this thesis the latest release is DICE2013, though there is a public beta DICE2016 available.The umbrella term "DICE" will, unless otherwise speci�ed, refer to the version DICE2013-R, a precursorto DICE2016, as it is the version used in this report.

Moving on from version descriptions, DICE is in essence is a dynamic optimization model that estimatesthe optimal path of greenhouse gas reductions. However, the question of what constitutes an "optimalpath" bears asking. In DICE the "optimal path" refers to the optimal path with respect to social welfare,here a discounted sum of the population weighted per capita consumption over the relevant time horizon.

For DICE Nordhaus chose to view climate change economics through the lens of neoclassical economics,whereby investments in for example capital and technology will increase future consumption at the ex-pense of reduced consumption today. Greenhouse gas emissions are then viewed as a negative capitalthat may be reduced with investments, namely emissions reductions. In short, investments in counteringglobal warming today will yield bene�ts in the future.

From a systems and control standpoint DICE can be formulated as a discrete optimal control problem(OCP) with �ve year time steps. The objective function is the social welfare, whilst the controls arethe savings and mitigation rates describing the share of fraction of net economic output investment andemissions mitigated respectively. The SCC is determined by solving the OCP and attaining the Lagrangemultipliers of emission and consumption, which will be detailed after the complete DICE model is intro-duced.

2.1 DICE2013-R 3

The carbon, climate and economic dynamics can be neatly condensed into �gure (1) (from Faulwasseret al. 2016). The speci�c inputs and outputs will be detailed in the next sections, though the reader isrecommended to refer back to this �gure as it is easy to get lost in the details of the speci�c modules.

Figure 1: DICE carbon, climate and economics cycle. [8]

2.1.1 Objective function

The objective function is the discounted sum of the population weighted utility of per capita consumption.It is important to note that consumption is a generalized consumption, including not only goods andservices as is traditional in economics but also leisure, health status and environmental services. Themathematical de�nition of the objective function over a time horizon [0, N ] with time steps k is as follows

W =

N∑k=1

U [c(k), L(k)]R(k). (1)

Here c(k) is per capita consumption, L(k) is population, and R(k) is the discount factor.

The utility, U [c(k), L(k)], is chosen in DICE to be represented by a constant elasticity function

U [c(k), L(k)] =L(k)c(k)1−α

1− α

where α is the marginal utility of consumption which may be interpreted as aversion to generationalinequality. Finally, the discount factor R(k) is

R(k) = (1 + ρ)−k.

This essentially reduces the utility provided by future generations, governed by the pure rate of socialtime preference ρ. The pure rate of social time preference is a parameter surrounded by controversy inthe world of economics, as it quanti�es the value of future generations compared to our own. In DICE itis chosen such that it represents the policy factors as they currently exist.

2.1.2 Economic model

Even though the economic module of DICE is based on well researched economics, there is a very impor-tant distinction worth pointing out. Climate economics requires longer time horizons than are typicallyused in economics, it is thus hard to quantify the accuracy of the DICE predictions. In fact the economicdynamics of DICE are simpli�ed relative to other economic models, for example highlighted by the fact


that DICE assumes a single commodity.

The reader is referred to [12] for a more extensive discussion of derivations of the model. The shortversion is that Nordhaus gathered data on a regional basis which he aggregated to a global total forDICE, including everything from population growth to capital accumulation. It is worth to note thatthese projections typically see changes between DICE versions to better match other estimates at thetime of the versions release.

To begin with, the gross output Y (k), in 2005 $US trillions per year, is introduced

Y (k) = A(k)K(k)γL(k)1−γ .

This is the Cobb-Douglas production function which incorporates the total factor productivity A(k)capturing Hicks-neutral technological progress, capital stock K(k) and labour supply L(k). De�ning theexogenous L(k) and A(k) �rst

L(k) = L(k − 1)[1 + gL(k)]

with the population growth rate gL(k) de�ned as

gL(k) =gL(k − 1)

1 + δL.

The constant δL is set such that the population will approach a limit of 10.5billion in 2100, and is equalto the 2013 U.N. predictions for 2050.

Accounting for technological progress, the the total factor productivity is

A(k) = A(k − 1)[1 + gA(k)],

with the productivity growth rate

gA(k) =gA(k − 1)

1 + δA.

Here A(k) is set such that DICE models the gross world product in 2010. Having de�ned the grosseconomic output it is necessary to estimate how much of it is degraded. This is done via the damagefunction Ω(k) ∈ [0, 1] and CO2 abatement cost function Λ(k) ∈ [0, 1]

Ω(k) =1

1 + 0.00267(TAT (k))2.

The damage function is interpreted as one minus the fraction of net output lost due to climate change.The global mean atmospheric temperature TAT (k will be introduced properly in a later section, for nowit su�ces to know that the damages function dependent on it. Note however that value of TAT (k) isrelative to the atmospheric temperature in 1900. As a starting point the damage function is derived frommonetized damages, though Nordhaus increased it to account for non-monetized impacts.

Representing the ratio of abatement cost to output is the abatement cost function

Λ(k) = θ1(k)µ(k)θ2 .

Here the �rst control is introduced, µ(k) ∈ [0, 1], denoting the fraction of total CO2 emissions abated.θ1(k) is a reduction factor accounting for decreasing costs in emissions mitigation. θ2 is a constant ad-justing the cost of the control µ(k). Finally, the abatement cost function has been adjusted to include abackstop technology, i.e. some technology that replaces all fossil fuels. This technology is initially set ata high price which reduces over time. This is introduced by adjusting the time path the cost abatementparameters such that the backstop price for a given year matches the abatement cost for µ(k) = 1.

2.1 DICE2013-R 5

Now, the net economic output of climate damages and mitigation costs may be introduced

Q(k) = Ω(k)[1− Λ(k)]Y (k) = C(k) + I(k)

where C(k) is the consumption and I(k) investments. This leads to the second control variable used inthe OCP formulation

s(k) =I(k)

Q(k).

This is the savings rate, denoting the fraction of net economic output invested. Next up is the capital,de�ned as

K(k) = I(k)− δKK(k − 1), (2)

where the capital follows a perpetual inventory method with an exponential deprecation rate with thedeprecation constant δK .

The �nal pieces of the economic module relates to emissions, which have both endogenous and exogenousparts. In previous versions of DICE greenhouse gases and carbon were looked at separately, but inDICE2013-R everything is converted to its carbon equivalent. The endogenous industrial emissions, inGtCO2 per year, are given by

EInd(k) = σ(k)[1− µ(k)]Y (k).

Here the carbon intensity σ(k) relates the gross economic output to emissions, given in units tCO2 per$1000 of gross domestic product, and the term in the square brackets represents the emission reduction.The carbon intensity σ(k) is modeled as exogenous and is built upon estimates of emissions. It is de�nedas

σ(k) = σ(k − 1)[1 + gσ(k)],

where

gσ(k) =gσ(k − 1)

1 + δσ.

It is set such that it matches the carbon intensity of 2010 and gσ(2015) = −1.0% per year. The lastequation introduced in this section represents the limit of fossil fuels, set at 6000 tonnes of carbon content,2.12 is a conversion factor.

2.12 · 6000 ≥N∑k=1

EInd(k)

2.1.3 Geophysical model

The geophysical model of DICE consists of two integrated submodels, namely the carbon and tempera-ture cycles. The industrial emissions outputted from the economy model acts an an input for the carboncycle where any net increase in emissions gives an increase in carbon. Said increase gives rise to radiativeforcing in the atmosphere causing the atmospheric temperature to rise, which feeds back into the damagefunction of the economy model.

The equations linking the three submodels have not changed drastically with DICE versions, insteadupdates have been in the particular dynamics in any submodel. Similar to the economic model the geo-physical model is simpli�ed, yielding transparency of the optimization. In fact the equations governingboth the climate and carbon cycles are mostly linear, with the radiative forcing equation serving as anexception.

Before introducing the dynamics, it is important to point out that in DICE only the industrial CO2 isa�ected by the emissions mitigation rate. Other greenhouse gas emissions or other sources of CO2 areregarded as exogenous. The reasoning for this is that the non-controlled contributors to global warmingare likely controlled in di�erent ways. Let us introduce the carbon cycle, see �gure (2).

As the above �gure shows, DICE employs a reservoir based model for the carbon (as well as temperature,which will be shown later). The model is then calibrated to existing carbon-cycle models and historical


Figure 2: DICE carbon cycle[8]

data.

To make the equations more concise, let the following be de�ned as

M(k) = [MAT (k), MUP (k), MLO(k)]T

Φ =

φ11 φ21 0φ12 φ22 φ320 φ32 φ33

.Here MAT (k), MUP (k), MLO(k) denotes the carbons (in GtC) in the atmosphere, upper and lower oceanrespectively. The constant Φ holds the �ow parameters governing the �ow between the carbon reservoirs.The �nal piece of the carbon cycle is the emissions, in the economic section the industrial emissions wereintroduced but here the other emissions need to be taken into account. Let the total emissions be E(k),then

E(k) = EInd(k) + ELand(k),

Here ELand(k) is exogenous, representing the other carbon dioxide sources described previously. It'sset to match current land-use changes, which is 3GtCO2 per year. The carbon cycle in actuality is avery complex, and the simpli�cations done with this linear model has for example over predicting at-mospheric absorption when comparing to historical data. The reader is referred to [12] for further details.

The dynamics for the carbon cycle are now presented. Note that e1 denotes the �rst unit vector, i.e.e1 = [1, 0, 0]

T . The dynamics are thus

M(k) = ΦM(k − 1) + e1E(k), (3)

Recall that the carbon tied in with the climate through radiative forcing, whose equation is in fact thesame used in FAIR as will be seen later. It is given by

F (k) = ηlog2

(MAT (k)

MAT (1750)

)+ FEX(k).

Here η is the forcings due to a doubling of CO2, which is set to 3.8Wm−2. The term FEX(k) is the

exogenous forcings. With higher radiative forcing the atmospheric temperature gets warmer which inturn also warms the deep ocean. The climate cycle may be visualized in �gure (3).

The similarities to the carbon cycle are evident, as both are linear discrete time system, with one externalinput from the industrial emissions in the carbon cycle and radiative forcing in the climate cycle.

2.1 DICE2013-R 7

Figure 3: DICE climate cycle[8]

Like for the climate cycle, a constant �ow matrix Ψ and the temperature vector T(k) are introduced

T(k) = [TAT (k), TLO(k)]T

Ψ =

[ψ11 ψ12ψ21 ψ22

].

The climate dynamics are thenT(k) = ΨT(k − 1) + e1F (k). (4)

2.1.4 Complete DICE model

Putting (1) - (4) together yields the complete DICE model is

maxs,µ

W =

N∑k=1

U [c(k), L(k)]R(k)

s.t T(k) = ΨT(k − 1) + e1F (k)M(k) = ΦM(k − 1) + e1E(k)K(k) = I(k)− δKK(k − 1).

(5)

The SCC is then determined by using Lagrange multipliers. Given a social welfare pathway W (t), anemissions pathway E(t) and �nally consumption pathway C(t) the SCC at time t is given by

SCC(t) = −∂W (t)∂E(t)

∂W (t)∂C(t)

= −∂C(t)∂E(t)

.

2.1.5 MATLAB Implementation

In this report a MATLAB implementation of DICE is used. Here the economic module in particular di-vides the equations up di�erently compared to the original DICE2013 manual, for a detailed descriptionof this implementation please see[14], only the endogenous equations will be covered here. The end resultis of course equivalent and has been compared to Nordhaus' original GAMS implementation, also usinga 5 year time step with N = 60 number of steps.


The climate and capital states are given as follows[TAT(k + 1)TLO(k + 1)

]=

[ψ11 ψ12ψ21 ψ22

] [TAT(k)TLO(k)

]+

[ξ10

]F (k) (6) MAT(k + 1)MUP(k + 1)

MLO(k + 1)

= φ11 φ12 0φ21 φ22 φ23

0 φ32 φ33

MAT(k)MUP(k)MLO(k)

+ ξ20

0

E(k) (7)K(k + 1) = (1− δ)∆K(k) + ∆

(1− a2 TAT(k)a3 − θ1(k)µ(k)θ2

)A(k)K(k)γ

(L(k)1000

)1−γs(k),

With the emissions and radiative forcing being governed by

E(k) = σ(k)(1− µ(k))A(k)K(k)γ(L(k)1000

)1−γ+ ELand(k) (8)

F (k) = η log2

(ζ11MAT(k) + ζ12MUP(k) + ξ2E(k)

MAT,1750

)+ FEX(k). (9)

The auxiliary states of the net economic output and gross economic output are

NEO(k) =(1− a2 TAT(k)a3 − θ1(k)µ(k)θ2

)(10)

GEO(k) = ATFP (k)K(k)γ

(L(k)

1000

)1−γ. (11)

Finally, the utility and consumption

U(C(k), L(k)) = L(k)

(

1000C(k)L(k)

)1−α− 1

1− α− 1

(12)

C(k) =(1− a2 TAT(k)a3 − θ1(k)µ(k)θ2

)A(k)K(k)γ

(L(k)1000

)1−γ(1− s(k)). (13)

Which ultimately leads to the optimization problem, the scales here are utility multipliers and o�sets.

maxs,µ

∆ ∗ scale1 ∗N∑i=1

U(C(k), L(k))

(1 + ρ)5(k−1)− scale2 (14)

subject to (6)− (8)µ(1) = µ0µ(k) ≥ 0, k = 2, . . . , Nµ(k) ≤ 1, k = 2, . . . , N

0 ≤ s(k) ≤ 1, k = 1, . . . , N.

(15)

2.2 FAIR 9

2.2 FAIR

Unlike DICE, the Finite Amplitude Impulse-Response model, FAIR, is not an IAI but it is a simpli�edgeophysical model of the carbon and temperature cycle. By expanding on the Impulse-Response modelfrom the Intergovernmental Panel on Climate Changes Fifth Assessment Report (AR5-IR) FAIR man-ages to match results from Comprehensive Earth system Models (ESMs) under a number of idealizedexperiments and future emission scenarios [9].

The bene�t of FAIR, in contrast to ESMs, is the lower computational costs. While the explicit simu-lations employed by ESMs accurately predict the evolution of atmospheric CO2 concentrations and theassociated climate response they are computationally expensive [4]. This makes ESMs less suitable forIAIs as they lack a set emissions pathway, and often require sampling which ultimately leads to excessivecomputational times.

Though there are many simpli�ed carbon-climate models available, many have not been explicitly eval-uated in terms of their response to a pulse emission of CO2. This is particularly disconcerting for IAIsas the SCC is often computed using such a pulse. DICE is included in the list of such models, thoughDICE typically uses Lagrange multipliers rather than a CO2 pulse to compute the SCC.

Additionally FAIR captures the increase in airborne fraction, i.e. the fraction of emitted CO2 that re-mains in the atmosphere after some period in scenarios with extensive global warming. This is importantas this airborne fraction causes the CO2 concentrations to be approximately linear with respect to CO2caused warming which leads to a simple equivalence between carbon and warming. Hence warming re-strictions may be described in terms of carbon restrictions, or a carbon budget.

In short, FAIR is an extension of AR5-IR which improves response to CO2 pulse emissions to bettermimic more complex models. Unlike DICE where carbon is allotted to the atmospheric, upper and lowerocean reservoirs the four carbon reservoirs in FAIR do not have direct physical interpretations. Thespeci�c physical processes relating to, for example atmospheric carbon, are only de�ned implicitly withinthe four pools. The carbon equations are given as

dRi(t)

dt= aiE(t)−

Ri(t)

α(t)τi, i = 1, . . . , 4. (16)

Unlike DICE FAIR operates in ppm, thus here E(t) is emissions in ppm per year, ai is the fraction ofemissions entering reservoir i and τi is the decay time constant for the respective reservoir. The scalingterm α(t) that is the extends AR5-IR to FAIR. With the introduction of α(t) FAIR mimics the CO2impulse response of ESMs.

The scaling term is governed by an algebraic expression, whose introduction is simpli�ed by �rst de�ningglobal mean atmospheric temperature (TAT (t) in DICE)

T (t) = T1(t) + T2(t).

The speci�cs of the two temperature reservoirs will be introduced later, for now it su�ces to know whatT (t) represents. Secondly, atmospheric CO2 concentrations

C(t) = C0 +∑i

Ri(t),

and �nally the accumulated carbon, representing emitted carbon no longer in the atmosphere

Cacc(t) =

∫ tt0

E(t) dt− (C(t)− C(t0)). (17)

Note that C(t) is not to be confused with the consumption in DICE, despite using the same variable name.


Having de�ned the above, the algebraic equation governing the scaling factor α(t) is introduced

4∑i=1

α(t)aiτi

[1− exp

(−100α(t)τi

)]= r0 + rCCacc + rTT (t). (18)

The left hand side of (18) is the 100-year integrated impulse response function, iRF100, representing theaverage airborne fraction over a period of time. r0, in years, is the pre-industrial iIRF100, rC , in yearsper ppm, the increase iIRF100 increase by cumulative carbon uptake and, rT , in years per K, the samefor temperature increases.

By assuming (18) is a function of accumulated carbon, Cacc(t), and the temperature T (t), a linear rela-tionship approximately represents the behaviour of ESMs. In the original paper [? ] α(t), is assumed tobe independent of time, such that (18) only holds for an in�nitesimal length of time which is adequatefor the discretized implementation used in the paper. Here it is instead modeled as a continuous timedependent state such that (18) holds for all t.

The values for the parameters are chosen such that FAIR mimics the behaviour in the CMIP5 ESM. Atable of parameter values are provided at the end of the section.

Having de�ned the non-linear and slightly cumbersome algebraic constraint for the carbon dynamics, thetemperature dynamics provides a stark contrast in its simplicity

dTj(t)

dt=qjF (t)− Tj(t)

djj = 1, 2. (19)

Recall that the atmospheric temperature T (t) =∑i Ti(t). Here the two temperature reservoirs repre-

sents contributions to T (t) from the upper and deep oceans. F (t) is the radiative forcing, its de�nitionequivalent to its de�nition in DICE except for a slightly lower value for η (3.74Wm−2 instead of 3.8Wm−1)

F (t) = ηlog2

(C(t)

C0

)+ FEX(t), (20)

dj are the carbon-cycle response timescales chosen to match means in [5]. The thermal equilibrationconstants qi, given in Km

2/W, are calibrated to have the equilibrium climate sensitivity and transientclimate response satisfy ECS = 2.75K and TCR = 1.6K. The ECS is related as per follows

ECS = η(q1 + q2). (21)

This is of particular interest to us as the uncertainty distribution of the ECS is well documented as willbe discussed in later sections. The ECS is de�ned as the change in temperature caused by a sustaineddoubling of atmospheric CO2 or its equivalent. For the de�nition of the TCR see [? ], it is omitted hereas it is not used directly in this report.

2.2.1 The complete FAIR model

Due to the scaling factor α, FAIR turns into a Di�erential Algebraic Equation (DAE), provided below.The DAE is given as, using (16) � (19)

dRi(t)

dt= aiE(t)−

Ri(t)

α(t)τi, i = 1, . . . , 4

dTj(t)

dt=qjF (t)− Tj(t)

djj = 1, 2

dCacc(t)

dt= E(t)−

4∑i=1

dRi(t)

dt

4∑i=1

α(t)aiτi

[1−exp

(−100α(t)τi

)]= r0 + rCCacc + rTT (t).

(22)

2.3 FAIR/DICE Hybrid 11

2.3 FAIR/DICE Hybrid

In the FAIR/DICE hybrid developed within this project the carbon and climate dynamics of DICE arereplaced with the dynamics of FAIR. As DICE is a complete IAI whereas FAIR is a carbon climate modelit is reasonable to adapt FAIR to an existing DICE implementation.

The �rst issue to tackle by incorporating the FAIR dynamics to the DICE implementation is the dis-cretization of FAIR. Similar to the original GAMS implementation by Nordhaus the DICE implementa-tion uses a discretized time horizon with �ve year time steps. The DICE dynamics are adapted usingForward Euler, though unfortunately its continuous counterpart proved unreliable for the FAIR dynamics.

FAIR is instead discretized by solving the di�erential algebraic equation at each time step and then usingthe endpoint of the solution for the subsequent state values. A caveat with this method is that FAIRuses continuous inputs in emissions, E(t), and radiative forcing, F (t) whereas the discrete DICE modelclearly only has discrete representations.

To solve this conundrum external inputs to FAIR are approximated to be piecewise constant over anytime step interval, the details of the values chosen are described further down. To formalize this notion,let T = [tk, tk+1] be any given time step in DICE. Additionally let x(t) be a state vector containing allstates from FAIR except the algebraically constrained α(t), where ẋ(t) = f(x, t). Finally, let x0 = x(tk),ie. the initial value in T. Then the discretization of a state xi(t) is done as follows

1. ∀j 6= i assume that xj(t) = xj(tk), ∀t ∈ T.

2. Let the auxiliary states be constant, e.g assume E(t) = E(tk), ∀t ∈ T

3. Analytically solve the IVP ẋj(t) = fj(x, t), xj(0) = x0j . Let the solution be x′j .

4. The value for xj(tk+1) is given by x′j(tk+1 − tk).

For clarity we de�ne the length of the time step, ∆ = tk+1 − tk. After computing x(tk+1) then α(tk+1)is found by simply solving for the root of the algebraic constraint (18).

2.3.1 Temperature

Following the procedure outlined above for the temperature yields a non-homogeneous linear system. Forj ∈ {1, 2} we have

Ṫj = −Tjdj

+qjF

dj.

Note that for brevity Newtons derivative notation is used and the time argument t is omitted. The sameapplies to the state index j as the solution of �nding T1(tk+1) is equivalent to �nding T2(tk+1).

Recall that the radiative forcing F (t) = F is constant in our time step, T. The homogeneous andparticular solutions to this ODE are then given by

T ′H = Ae− t′d A ∈ R

T ′P = qF.

Here t′ ∈ [0, tk+1 − tk] and A is a yet to be determined constant. This yields

T ′ = Ae−t′d + qF.

A may now be determined using the initial value

T ′(0) = A+ qF = T (tk) =⇒ A = T (tk)− qF.

Finally, Tk+1 is given by

T (tk+1) = T′(∆) = (T (tk)− qF )e−

∆d + qF.


2.3.2 Carbon

Similar to the temperature, a non-homogeneous linear system is obtained for the carbon concentrationafter approximating the non-carbon states to be piecewise constant in the interval

Ṙi = aiE −Riατ

.

The algebraic constraint term α is troublesome however, as it appears in an exponential term in the exactsolution. This exponential term will prove troublesome when the PCE's are introduced which motivatesanother approach to the carbon discretization. Using the same method as in the temperature, the primedsystem is

R′(t′) = (R(tk)− aατE)e−t′ατ + aατE

=⇒ Ṙ′(t′) = aατE −R(tk)ατ

e−t′ατ .

Now de�ne the auxiliary variable

z(t′) = e−t′ατ

=⇒ ż(t′) = − 1ατ

e−t′ατ .

We can now add the carbon dynamics to the following system, where the exponential term is implicitlyde�ned in the dynamics of z(t′)

Ṙ′(t′) =aατE −R(tk)

ατz(t′), R′(0) = R(tk)

ż(t′) = − 1ατ

z(t′), z(0) = 1.

Using Backward Euler yields

R′(∆) = R(tk) + ∆aταE −R(tk)

ατz(∆)

z(tk+1 − tk) = z(0)−∆

ατz(∆).

Substituting zk+1 in the Rk+1 equation yields

R(tk+1) = R′(∆) = R(tk) + ∆

aταE −R(tk)ατ + ∆

z(0)

z(∆) = z(0)− ∆ατ

z(∆).

Where we used that

z(∆) =z(0)

1 + ∆ατ.

Furthermore, as z(0) = 1 the equation becomes

R(tk+1) = R(tk) + ∆aταE −R(tk)

ατ + ∆.

Thus we have for the accumulated carbon, Cacc the solution is rather trivial, as both E and Ri are keptconstant in T. Thus we have

Ċacc = E −∑i

Ṙi

=⇒ Cacc(tk+1) = Cacc(tk) + ∆(E −∑i

Ṙi(tk)).


2.3.3 FAIR Discretization

The �nal discretized FAIR dynamics are thus

T (tk+1) = (T (tk)− qF (R(tk)))e−∆d + qF (R(tk))

R(tk+1) = R(tk) + ∆aταE −R(tk)

ατ + ∆.

Cacc(tk+1) = Cacc(tk) + ∆(E(tk)−

4∑i=1

Ṙi(tk)).

(23)

With α(tk+1) determined by �nding the root of the algebraic constraint, i.e. solving

4∑i=1

α(tk+1)aiτi

[1− exp

(−100

α(tk+1)τi

)]= r0 + rCCacc(tk+1) + rTT (tk+1) (18)

with respect to α(tk+1).

Figure 4: Comparison of continuous (red) and discrete (blue) FAIR showing that the discretizationprovides a good approximation..


2.3.4 FAIR Initial values

The �nal piece of the FAIR adaptation comes from the initial values. This is necessitated as the carbonand temperature reservoirs do not have explicit analogues in DICE. To solve this conundrum a simulationof the FAIR dynamics from pre-industrial times to the starting date of DICE is employed, i.e. 1850-2010.Using an emissions pathway deduced from historical data [1] the simulation then yields the fraction ofcarbon and temperature that goes into each reservoir. The data unfortunately does not include land-usechange and forestry emissions, instead it is assumed that this will have a minimal e�ect on the proportionsallotted to each reservoir. The initial values from FAIR may then easily be solved for using the initialvalues in DICE.

To begin with the temperature, the initial values are

T (2010) = 0.8

(0.10390.8961

).

Using DICEs TAT (2010) = 0.8, and that∑2j=1 Tj = T = TAT . The vector holds the proportion coe�-

cients from the simulation of FAIR.

Similarly, for the carbon

R(2010) = (391− C0)

0.51390.35550.11390.0166

.

391 comes from the measured 2010 value of atmospheric carbon. The di�erence is obtained fromC = C0 +

∑iR which relates atmospheric carbon to the carbon reservoirs.

Finally, the initial value for the accumulated carbon is

Cacc(2010) = 168.4561−∑i

R(2010).

This initial value for Cacc is a lower estimate as the �rst term comes from integrating the emissionspathway which excluded forestry and land-use change emissions.


2.3.5 Complete FAIR/DICE model

The complete FAIR/DICE model OCP is thus, using (5), (8), (13), (23). Here tk ∈ Z[t0, tf ] andk = 0, . . . , N .

maxs,µ

∆ ∗ scale1 ∗tf∑

tk=t0

U(C(tk), L(tk))

(1 + ρ)∆(tk−1)

Tj(tk+1) = (Tj(tk)− qjF (tk))e− ∆dj + qjF (tk), j = 1, 2

Ri(tk+1) = Ri(tk) + ∆aiτiαE(tk)−Ri(tk)

ατi + ∆, i = 1, 2, 3, 4

Cacc(tk+1) = Cacc(tk) + ∆

(E(tk)−

4∑i=1

(aiE(tk)−

Ri(tk)

α(tk)τi

))

F (tk) = ηlog2

(C(tk)

C0

)+ FEX(tk)

4∑i=1

α(tk+1)aiτi

[1− exp

(−100

α(tk+1)τi

)]= r0 + rCCacc(tk+1) + rTT (tk+1)

E(tk) = σ(tk)(1− µ(tk))A(tk)K(tk)γ(L(tk)1000

)1−γ+ ELand(tk)

K(tk + 1) = (1− δ)∆K(tk) + ∆

1− a2 2∑j=1

Tj(tk)

a3 − θ1(tk)µ(tk)θ2

µ(1) = µ0

µ(tk) ≥ 0µ(tk) ≤ 1

0 ≤s(tk) ≤ 1.

Where the utility and consumption are

U(C(tk), L(tk)) = L(tk)

(

1000C(tk)L(tk)

)1−α− 1

1− α− 1

C(tk) =(1− a2 TAT(tk)a3 − θ1(tk)µ(tk)θ2

)A(tk)K(tk)

γ(L(tk)1000

)1−γ(1− s(tk)).

Note that the exogeneous states are omitted, they are detailed in section 2.1.


In �gure (5) an SCC, control and atmospheric temperature TAT = T1 + T2 comparison for FAIR/DICEand DICE is provided. Interestingly FAIR/DICE has a noticeably higher SCC compared to DICE,in part likely attributed to how the control µ rises to its max value quicker. It is also worth notingthat FAIR/DICE, unlike FAIR, has an increasing atmospheric temperature even towards the end of theinterval.

Figure 5: Comparison of FAIR/DICE and DICE SCC and temperature.

2.4 Using Integrated Assessment Models 17

2.4 Using Integrated Assessment Models

Before proceeding some notes regarding the use of IAI's is important. In this report deterministic trajec-tories for the controls are obtained after which the system is forward simulated. These trajectories areobtained by solving the open-loop optimization problem, which are susceptible to turnpike behaviour. Inessence this means that the solution will diverge at the ends of the interval, regardless of the length ofthe interval.

Intuitively this makes sense, as when optimizing over a �nite time span there is no regard to e�ects out-side of it. If the climate damage is catastrophic two years after or not does not a�ect the optimal controltrajectories, thus in the case of these climate economy models emissions tend to increase rapidly near the�nal years of the domain. Note for example the carbon reservoirs in 4 that all seemingly diverge towardsthe end of the time horizon. If the time domain was increased to say, 2350, the divergent behaviour wouldbe translated further in time.

Any policy decision using IAI's should clearly not follow the trajectories in the �nal years. One remedyto this is to use a Model Predictive Control implementation, where the optimal control problem is solvedfor a shorter time and then solved again at the end of the sub interval. For examples of this see [8].

18 3 POLYNOMIAL CHAOS EXPANSION FOR UNCERTAINTY QUANTIFICATION

3 Polynomial Chaos Expansion for Uncertainty Quanti�cation

Uncertainty quanti�cation is well researched in control, indeed it is of fundamental importance in anydiscipline of engineering or applied mathematics. As is often quoted, the statistician G. Box stated in oneof his papers that "all models are wrong, some are useful", indicating that models are but a representationof reality. This is especially true in the �eld of climate physics, and as a consequence climate economics.Even the most advanced models running on top of the line supercomputers cannot provide anything butestimates of how the climate will evolve.

While DICE and FAIR have both been calibrated to accurately model historical data there are a range ofuncertainties that need to be accounted for when attempting to predict how the complex climate systemwill evolve past the present day. Even if the equations provide a solid basis for climate predictions everyparameter presents a potential uncertainty, which in other words presents a potential inaccuracy in theprediction.

The main purpose of climate economics is to provide scienti�cally grounded guidelines to be used forpolicy enactment. It is thus of vital importance that uncertainties do not go unaccounted for. Even byfocusing on the physical systems and disregarding human factors such as the pure rate of social time pref-erence it is evident that the stochastics are non trivial. For example looking at the equilibrium climatesensitivity it can be seen that there is not even a strict consensus for the characteristics of the probabilitydistribution, as seen in [13].

Alas it is not new that stochastics are important in any modeling of physical processes. One very commonmethod to deal with uncertainty quanti�cation in essentially every �eld ranging from control to physicsis to use Monte Carlo sampling. By sampling the relevant stochastic variables beforehand simulationsmay be run repeatedly in a deterministic fashion, and relevant statistics is extracted from the data.While functional Monte Carlo sampling has one obvious downside, for large complex systems it is verycomputationally costly to run thousands of simulations.

An alternative to this approach was presented by N. Wiener in 1938 [11]. Using orthogonal polynomials,speci�cally Hermite polynomials, he showed that Gaussian stochastic processes may be represented aspolynomial functions of the stochastic variables. This has since been expanded for a wide range of dis-tributions and polynomial spaces [17]. Collectively it is known as Polynomial Chaos Expansions (PCE),or Generalized Polynomial Chaos. The advantage of this method is that it provides an explicit polyno-mial representation of the stochastics, allowing for much cheaper computational costs. However, beforeformally introducing PCEs the next few sections will introduce the necessary mathematical preliminariesbeginning with probability theory.

3.1 Probability theory

The modern notion of probability theory was not introduced until the 1930s, when it was formalized byKolmogorov. The �eld is as di�cult conceptually as it is to tackle rigorously. The aim of this sectionis to provide a brief introduction to the relevant concepts of probability theory, and it is by no meansextensive. The de�nitions in this section are provided in T. Koski's lecture notes on probability theory[7].

The formalism of probability theory is centered around probability spaces, loosely speaking sets of ran-domly occurring events.

De�nition 2 (Probability Space). A probability space (Ω, F, P ) consists of three parts:

1. A sample space Ω, containing all outcomes.

2. A set of subsets of Ω, F ⊂ Ω, containing zero or more outcomes.

3. A function P : F → [0, 1], assigning probabilities to every outcome.

3.1 Probability theory 19

Here the set F is a sigma algebra. Omitting rigorous mathematical treatment, a sigma algebra is a setthat includes the empty subset, is closed under complement and under countable unions and intersec-tions. Roughly speaking this prevents issues when integrating over sets that cannot easily be integratedover, such sets may appear when Ω is uncountably in�nite. Issues stemming from the set of outcomes Fnot being a sigma algebra typically only appear in purely mathematical constructs however, in fact aninterpretation of F su�cient to follow this report is that it's the set of relevant events.

Given Ω and F it is possible to de�ne a measure over the space, the analogue to the length of a real lineor area of a surface. In this case the measure would give a value of how likely an event or collection ofevents is to occur, in other words P is our probability measure. A formal introduction of measure theoryis outside the scope of this work and the reader is referred to any preferred textbook on probability theory.

De�nition 3 (Random Variable). A random variable X is a real valued function X : Ω→ R such thatfor every set A ∈ B where B contains all the sets of form (a, b) ∈ R then

X−1(A) = {ω : X(ω) ∈ A} ∈ F.

Though the de�nition is anything but intuitive, in essence it allows the random variable to be a measur-able function, hence for our purposes making it tractable.

For the remainder of this section, it is assumed that random variables are de�ned in a suitable probabil-ity space as described above. There are a few additional concepts that necessitate introduction, namelythe mean, variance and of course the probability density function (PDF) and its associated cumulativedensity function (CDF).

De�nition 4 (Probability Density Function). The probability density function of a random variable Xon a probability space (Ω, F, P ) is a function p(x) such that for any measurable space A ⊂ F it holdsthat

P (X ∈ A) =∫X−1A

dP =

∫A

p(x)dx.

De�nition 5 (Cumulative Density Function). Using the same random variable X as de�ned above, thecumulative density function Fc : A→ [0, 1] such that

Fc(X) = P (X ≤ x).

Having de�ned the PDF the de�nitions of the mean and variance naturally follows.

De�nition 6 (Expected Value). The mean, or expected value, of a random variable X with the PDFp(x) over the measurable space A ⊂ F is

E[X] =

∫Ω

X(ω)dP (ω) =

∫A

xp(x)dx.

De�nition 7 (Variance). The variance of the random variable X is

V ar[X] = E[(X − E[X])2] =∫A

(x− E[X])2p(x) dx.

Typically when working with variance it is easier to employ Steiner's formula

V ar[X] = E[X2]− E[X]2.

To round o� this section on probability theory the n'th moment of a random variable warrants a de�nition

De�nition 8 (Moment). The n'th moment of the random variable X is

E[Xn] =

∫A

xnp(x)dx.

The reader may note expected value of X is simply the �rst moment of X.


3.2 Hilbert Spaces of Random Variables

Like the previous section, this serves as a brief introduction to vector spaces, or Hilbert spaces in par-ticular, and the reader is referred to any common textbook on the subject for a more in depth analysis.Hilbert spaces are generalizations of Euclidian spaces, in essence it is a space that has a "length" de�nedthrough an inner product. It is de�ned as follows.

De�nition 9 (Hilbert Space). A Hilbert space is an inner product space with respect to a length function,the inner product. The space is also a complete metric space. The inner product satis�es the followingproperties for any x and y in the space

The inner product of x and y is its own complex conjugate.

〈y, x〉 = 〈x, y〉.

The inner product is linear in its �rst argument, for any a ∈ C and b ∈ C and x1 and x2 in theHilbert space

〈ax1 + bx2, y〉 = a〈x1, y〉+ b〈x2, y〉.

The inner product with itself is positive de�nite

〈x, x〉 ≥ 0.

Additionally the norm of a Hilbert space is the real valued function

||x|| =√〈x, x〉.

Intuitively a complete metric space refers to a space with no points missing, hence "complete". Whilethis de�nition is familiar to anyone with a basic understanding of linear algebra, it does not clarify howit may be used for random variables. There is in fact not even a need for an extension, by letting theinner product be the expected value all the criteria hold, though now a space of stochastic rather thanreal or complex variables. The proofs are omitted, but as the expected value is an integral none of theproperties are particularly hard to show.

De�nition 10 (Inner Product of a Hilbert Space of Stochastic Variables). The inner product of X andY in a Hilbert space of random variables is given by

〈X,Y 〉 = E[XY ].

One of the useful properties of a Hilbert space is that they are all in an L2 space, which is the set ofsquare integrable functions. Thus, for the expected value we have that for a stochastic variable X

E[X2]

3.3 Orthogonal Polynomials 21

Before continuing to which speci�c Hilbert bases will be used, it is important to de�ne how in�nite basismay be approximated to a �nite basis which is necessary for numerical application. The question thatneeds to be answered is "given a complete and potentially in�nite Hilbert space how do we optimallyapproximate it to a �nite subspace of our choosing?". Given a Hilbert space H with element x ∈ H andwith a subspace U ⊂ H, we need to determine which x̄ ∈ U best approximates x, i.e.

x̄ = argminx̂

||x− x̂||

s.t x ∈ Hx̂ ∈ U.

The answer to this lies in the Hilbert Projection Theorem [15]

Theorem 1 (Hilbert Projection Theorem). Let H be a Hilbert space and U a closed subspace of H.Corresponding to any vector x ∈ H there is a unique vector u0 ∈ U such that

||x− u0|| ≤ ||x− u|| ∀u ∈ U.

Furthermore, a necessary and su�cient condition for u0 ∈ U to be a minimizer is that (x − u0) isorthogonal to M .

To apply this theorem to a subspace a subspace U spanned by an n-dimensional orthogonal basis {Φl}nl=0.The projection operator T : H → U is de�ned as

Tx =

n∑l=0

〈x,Φl〉〈Φl,Φl〉

Φl =

n∑l=0

clΦl.

Readers familiar with Fourier analysis will recognize the Fourier coe�cient cl. The derivation of thisprojection is omitted and the reader is referred to literature on linear algebra or, of course, Fourieranalysis. It is su�cient to know that using such a projection will yield the point satisfying the propertiesof the Hilbert Projection Theorem.

3.3 Orthogonal Polynomials

While there exists a vast number of di�erent bases which may span a given Hilbert space it is importantto choose ones that are tractable and ideally facilitates numerical computations. One such set of bases isthe set of polynomial bases, which are easy to handle both analytically and numerically. There is afterall a reason Taylor expansions are so commonly used.

De�nition 13 (Orthogonal Polynomials). A set of polynomials, P with support S, is said to be orthog-onal if for any polynomials Pi(x) ∈ P and Pj(x) ∈ P the following holds∫

SPi(x)Pj(x)w(x) dx = Ciδij Ci ∈ R.

Where w(x) is any weight function such that the above property holds for all polynomials in P. For sucha polynomial set and weight function an inner product in P may then be de�ned as

〈Pi(x), Pj(x)〉 =∫SPi(x)Pj(x)w(x) dx.

A general property of orthogonal polynomials is that they are uniquely de�ned by a recurrence relation

−xPn(x) = AnPn+1(x)− (An + Cn)Pn(x) + CnPn−1(x), n ≥ 1.

Here the real numbers An and Cn satisfy that An 6= 0, Cn 6= 0 and CnAn−1 > 0. Using this, an arbitrarydimension of a polynomial basis may be computed.


To tie this in with the previous chapter, imagine if w(x) is also a probability density function for somerandom variables Xm and Xn with support S where they may be represented by a linear combinationof polynomials in P. Additionally, let P be an orthogonal set with polynomials up to degree P where Pmay be in�nite. More speci�cally

Xm =

P∑i=0

diPi(Xm) di ∈ {0, 1}

Xn =

P∑i=0

fiPi(Xn) fi ∈ {0, 1}.

Then the properties of the expected value must hold even within the polynomial basis

〈Xm, Xn〉 = E[XmXn] =∫S

( P∑i=0

diPi(x)

)( P∑i=0

fiPi(x)

)w(x) dx

=

∫S

P∑i=0

P∑j=0

didjPi(x)Pj(x)w(x) dx =

{C ∈ R if for any i: di = fi 6= 00 otherwise.

More concisely stated〈Xm, Xn〉 = δmnCm.

By letting Xm = Pi(X) and Xn = Pj(X) for a random variable X with the same PDF w(x) and supportS we have that

〈Pi(X), Pj(X)〉 = δijCi.This provides the relation between the stochastic variables and the polynomial bases; by �nding a suitablebasis with the relevant weight function the theory of Hilbert spaces may be applied.

3.4 Polynomial Chaos Expansion

With the tools available from the previous sections PCE's may �nally be de�ned. To begin with a simpleexample, consider a discrete one dimensional random variable x(k, ξ), where k ∈ N denotes the discretetime and ξ ∈ R is a stochastic germ in a Hilbert space H spanned by the polynomial basis {Φi(ξ)}∞i=0,where Φi : R→ R. Then the polynomial chaos expansion of x(k, ξ) may be written as

x(k, ξ) = Tx(k, ξ) =

∞∑p=0

x̂p(k)Φp(ξ).

Recall that T is the projection operator, explicitly the Fourier coe�cient is given by

x̂p(k) =〈x(k, ξ),Φp(ξ)〉〈Φp(ξ),Φp(ξ)〉

where the inner product of H over support S is de�ned as

〈Φi(ξ),Φj(ξ)〉 =∫S

Φi(ξ)Φj(ξ)w(ξ) dξ.

The PCE of x(k, ξ) is exact, unfortunately it is not feasible to use for numerical computations due to thein�nite span. Instead, for some P ∈ Z[0,∞), the truncated PCE of x(k, ξ) is

x(k, ξ) ≈P∑p=0

x̂p(k)Φp(ξ).

It is important to note that PCE's are not restricted to one dimensional stochastic variables. The generalprinciples remain the same for multivariate PCE's, they are however outside the scope of this report.

Preliminary PCE's may seem like nothing but an exercise in academia, and indeed it is not until relativelyrecently PCE's has made a surge in the world of system and control. The power of PCE's is made moreapparent when applied to a dynamical system.

3.4 Polynomial Chaos Expansion 23

Example 3.1. Consider the state vector x(k, ξ) ∈ R where k ∈ Z[0, N ]. Additionally, let x(k, ξ) begoverned by the dynamics

x(k + 1, ξ) = ax(k, ξ) a ∈ Rx(0, ξ) = x0 x0 ∈ R.

Furthermore let x(k, ξ) be in the Hilbert spaceH with the polynomial basis {Φi(ξ)}i as de�ned previously.Then the PCE of x(k, ξ) is obtained by projecting the dynamics onto the polynomial basis. To obtainthe PCE coe�cient of x(k + 1, ξ), �rst take the inner product with the p'th polynomial

〈x(k + 1, ξ),Φp(ξ)〉 = 〈ax(k, ξ),Φp(ξ)〉〈P∑

m=0

x̂m(k + 1)Φm(ξ),Φp(ξ)

〉=

〈a

P∑m=0

x̂m(k)Φm(ξ),Φp(ξ)

〉.

Using the linearity and orthogonality of inner products, an explicit equation for the PCE coe�cients isobtained. First applied on the left hand side〈

P∑m=0

x̂m(k + 1)Φm(ξ),Φp(ξ)

〉=

P∑m=0

x̂m(k + 1)〈Φm(ξ),Φp(ξ)〉

= x̂p(k + 1)〈Φp(ξ),Φp(ξ)〉.

Using the same procedure for the right hand side yields〈a

P∑m=0

x̂m(k)Φm(ξ),Φp(ξ)

〉= ax̂p(k)〈Φp(ξ),Φp(ξ)〉.

The PCE of the dynamics is thus, for p = 0, . . . , P ∈ Z[0,∞)

x̂p(k + 1)〈Φp(ξ),Φp(ξ)〉 = ax̂p(k)〈Φp(ξ),Φp(ξ)〉x̂p(k + 1) = ax̂p(k).

Similarly for the initial value, the left hand side gives

〈x(0, ξ),Φp(ξ)〉 = x̂p(0)〈Φp(ξ),Φp(ξ)〉.

In the right hand side we use that Φ0 = 1

〈x0,Φp(ξ)〉 = x0〈1,Φp(ξ)〉 = x0〈Φ0(ξ),Φp(ξ)〉.

Putting it together, note that C00 = 〈Φ0(ξ),Φ0(ξ)〉

x̂p(0)〈Φp(ξ),Φp(ξ)〉 = x̂0〈Φ0(ξ),Φp(ξ)〉 = x0δ0pC00.

As the inner products are known and may be computed beforehand, there is now a very simple systemof equations which may be solved to determine the PCE coe�cients for x(k, ξ)

x̂p(k + 1) = ax̂p(k)

x̂p(0) = x0δ0pC00.

After which x(k, ξ) is given as

x(k, ξ) =

P∑m=0

x̂m(k)Φm(ξ).

The key point is that there is now a polynomial representation of x(k, ξ) with respect to ξ which is thencomputationally e�cient to sample over as there is no need to simulate the entire system from k = 0 tok = N for each sample.


Clearly the di�erences in computational times would be minor in this example, but for more complexsystems where the simulations are more time consuming PCE's may provide an e�ciency increase. Anobvious example is for large optimization problems that are time consuming to solve, with PCE's theoptimization will only have to be solved once by for example optimizing over the expected value of theobjective function.

The crux however, is that even though PCE's provide a faster way to sample is that by using PCE'ssampling is no longer needed. In fact statistics can be extracted directly from the PCE coe�cients.

Consider a stochastic variable x with the PCE

x =

P∑p=0

x̂pΦp(ξ).

The mean and variance can are then obtained as follows

E[x] = x̂0

V ar[x] =

P∑p=1

x̂2p〈Φp(ξ),Φp(ξ)〉.

The expected value is already computed when the PCE is obtained, and as the inner products maybe saved it does not require much computational power to compute the variance. Most certainly moree�cient than sampling the dynamics or optimization problem from scratch, as would be done with MonteCarlo. There are some trade-o�s however, which will be better illustrated after the following example.

Example 3.2. Consider the one dimensional stochastic variables x1(ξ), . . . , xn(ξ) in the same Hilbertspace as outlined previously, with the polynomial basis of order P . Let the stochastic variable z(ξ) bede�ned as

z(ξ) =

n∏i=1

xi(ξ).

The PCE of each xi is known, then The PCE coe�cients of z(ξ) are given by

ẑp(ξ) =

P∑m1=0

...

P∑mn=0

x̂m1 . . . x̂mn〈Φm1(ξ) . . .Φmn(ξ),Φp(ξ)〉.

A derivation of this example will be provided in a later section, what is being demonstrated is the sizeof the inner product. Recall that it is evaluated as an integral, for high n it is potentially expensive tocompute. Fortunately it only needs to be computed once.

One way to see PCE's is thus as a method to decrease computation times at the expense of increasingstorage requirements. It should be noted however that there is overhead in deriving the system of equa-tions used to determine the PCE's, which is not present when using Monte Carlo sampling. Additionallywhen the expressions to be overloaded aren't polynomial more advanced methods have to be employed,which will be discussed in a later section. To round o� a more elaborate example is provided showinghow PCE's may be used in practice.

Example 3.3. Consider the FAIR temperature dynamics under a deterministic pathway for the radiativeforcing

Ti(k + 1) = Ti(k)e− 5di − qiF (k)e−

5di + qiF (k). (24)

Where k ∈ {1, 2, . . . , N + 1} denotes the time step and i ∈ {1, 2} denotes the state. Given that theequilibrium climate sensitivity is related to Ti via qi through the expression ECS = η(q1 + q2) PCE's canbe used to determine how uncertainties in the ECS e�ect changes in temperature.

3.4 Polynomial Chaos Expansion 25

Assume that the ECS follows a four parameter beta distribution, such that ECS ∼ B(α, β, a′, b′). Here αand β are shape parameters, whereas a′ and b′ are the lower and upper bounds respectively. Additionallyassume that η is a known constant and let q = q1 + q2, such that ECS = ηq. Then a scaling of thebounds provides the distribution for q

q ∼ B(α, β,

a′

η,b′

η

)= B(α, β, a, b)

Where a = a′

η and b =b′

η . As we would like to work in a PCE basis with support [0, 1] a transformationfrom the four parameter beta distribution to the more common two parameter beta distribution is nec-essary. Let the stochastic germ ξ ∼ B(α, β) on support [0, 1]. It can be shown that ξ has the exact PCErepresentation in the Jacobi polynomial basis [18]

ξ = ξ0J0(ξ) + ξ1J1(ξ) = ξ0 + ξ1J1(ξ) ξ0 ∈ R, ξ1 ∈ R

ξ0 =α

α+ β, ξ1 =

1

α+ β.

This is useful as q may be written as a function of ξ through the following linear transformation

q(ξ) = ξ(b− a) + a.

Now let ci denote the proportion of q allotted to each qi taken from the deterministic values of qi, i.e.ci =

qi,deterq1,deter+q2,deter

Introducing the uncertainty to the FAIR dynamics (24) provides

Ti(k + 1, ξ) = Ti(k, ξ)e− 5di − ciq(ξ)F (k)e−

5di + ciq(ξ)F (k, ξ).

Now substituting q with its transformation and re-arranging the terms

Ti(k + 1, ξ) = Ti(k, ξ)e− 5di + ciF (k)(1− e−

5di )q(ξ)

= Ti(k, ξ)e− 5di + ciF (k)(1− e−

5di )((b− a)ξ + a)

= Ti(k, ξ)e− 5di + ciF (k)(1− e−

5di )(b− a)ξ + ciF (k)(1− e−

5di )

= Ti(k)Ai,0 + ξAi,1(k) +Ai,2(k)

For simplicity, Ai,j are introduced

Ai,0 ≡ e−5di

Ai,1(k) ≡ ciF (k)(1− e−5di )(b− a)

Ai,2(k) ≡ ciF (k)(1− e−5di ).

Because T1 and T2 have the same linear dynamics with only constants di�ering, the index i is omitted inthe rest of the section for brevity. The dynamics are thus

T (k + 1, ξ) = T (k, ξ)A0 + ξA1(k) +A2(k)

Let T̂p(k) denote the p'th PCE coe�cient for T (k)

T̂p(k) =〈T (k), Jp(k)〉〈Jp(k), Jp(k)〉

Then T (k) may be represented by the truncated PCE

T (k, ξ) ≈P∑p=0

T̂p(k)Jp(ξ) P ∈ Z[0,∞)

Projecting the dynamics onto the p'th Jacobi polynomial gives

〈T (k + 1, ξ), Jp(ξ)〉 = 〈T (k, ξ)A0 + ξA1(k) +A2(k), Jp(ξ)〉


Using that the inner product is a linear operator and that the the polynomial basis is orthogonal, i.e.〈Jm(ξ), Jn(ξ)〉 = δmn〈Jm(ξ), Jm(ξ)〉 the right hand side may be simpli�ed to

〈T (k + 1, ξ), Jp(ξ)〉 = A0〈T (k, ξ), Jp(ξ)〉+A1(k)〈ξ, Jp(ξ)〉+A2(k)δ0,p〈Jp(ξ), Jp(ξ)〉

Using the PCE representation of T (k, ξ) the left hand side is rewritten as

〈T (k + 1, ξ), Jp(ξ)〉 =〈 P∑m=0

T̂m(k + 1)Jm(ξ), Jp(ξ)

〉= T̂p(k + 1)〈Jp(ξ), Jp(ξ)〉

Clearly the same argument applies to the temperature term on the right hand side. Now, recall that ξhad an exact PCE representation, ξ = ξ0 + ξ1J1(ξ). The projection is thus done in analogy with T (k, ξ),which leads to the expression for T̂p(k + 1)

T̂p(k + 1)〈Jp(ξ), Jp(ξ)〉 = 〈Jp(ξ), Jp(ξ)〉(A0T̂p(k) +A1(k)(ξ0δ0p + ξ1δ1p) +A2(k)δ0p

)Dividing both sides by 〈Jp(ξ), Jp(ξ)〉 and then simplifying yields

T̂p(k + 1) = A0T̂p(k) +A1(k)(ξ0δ0p + ξ1δ1p) +A2(k)δ0p

T̂p(k + 1) = A0T̂p(k) + δ0p(A0(k)ξ0 +A2(k)) + δ1pξ1A1(k)

Finally, with the state index i included the system of equations for the PCE coe�cients is

T̂i,p(k + 1) = Ai,0T̂i,p(k) + δ0p(Ai,0(k)ξ0 +Ai,2(k)) + δ1pξ1Ai,1(k).

This example is meant to showcase some of the common manipulations done to determine the PCEcoe�cients. As will be seen in the next section the computation is much shorter if the dependence on ξis treated as implicit in the PCE of qi(ξ).

3.4.1 Order of operations

Figure 6: Order of operations, in this report the discretization is done �rst.

When overloading the continuous FAIR dynamics a decision was made to �rst discretize the system andthen project the discretized system onto the polynomial stochastic space. The question that needs to beasked is how the results would be a�ected if the order was reversed. In other words, what if the continousdynamics were overloaded before being discretized?

3.5 Polynomial Chaos Expansion of the FAIR/DICE Dynamics 27

Consider the following system:

ẋ(t, ξ) = f(x(t, ξ))

Where x ∈ Rn and f : Rn 7→ Rn and ξ is some known stochastic germ. These dynamics may be projecteddirectly onto a stochastic space {Φi}∞i=0, as per follows:

〈ẋ(t, ξ),Φp(ξ)〉 = 〈f(x(t, ξ)),Φp(ξ)〉

Which would yields PCE coe�cients for the time derivative:

˙̂xp(t) = f̂p(t)

It may be easier to discretize this system, but more importantly approximations done via for exampleTaylor expansions could be a�ected. While not attempted in this report it is important to point out thatthere is alternative route that warrants further analysis.

3.5 Polynomial Chaos Expansion of the FAIR/DICE Dynamics

Now the PCE overloading of the FAIR/DICE dynamics will be presented. With the exception of theequilibrium climate sensitivity (ECS) where the basis has a direct e�ect on the expressions the derivedexpressions are general to any other orthogonal polynomial basis. The uncertainty handled in this reportenters the system in the ECS and then propagates through all the endogenous states, it is thus naturalin beginning with a presentation of the ECS PCE.

The ECS in FAIR is given byECS = η(q1 + q2).

As was mentioned in the FAIR model section the stochastic characteristics of the ECS are well docu-mented, there is however no consensus on how it is distributed. In �gure (3.5) a few examples of di�erentdistributions taken from the literature are shown.

[13]

The shape of the distribution is similar to a skewed four parameter beta distribution which can be repre-sented by Jacobi polynomials. The derivations in the following sections are general to an arbitrary betadistribution.

3.5.1 Equilibrium Climate Sensitivity

Before beginning the derivations, note that the time steps in this section are denoted by k = 1, . . . , N+1.The expression to overload is

ECS = η(q1 + q2).


Where η is the forcing due to a doubling of CO2, and q1 and q2 are thermal equilibrium of deep oceanand thermal equilibration adjustment of upper ocean constants respectively. Because η exhibits littlevariation in the literature any uncertainty in the ECS likely comes from qi. In other words, assuming theECS is beta distributed, we have for a stochastic germ ξ ∈ β(α, β)

ECS(ξ) = η(q1(ξ) + q2(ξ)).

The reader is reminded that the basis used is the Jacobi polynomials, here denoted {Φi}∞i=0. Like in theintroductory example, de�ne

Q(ξ) = q1(ξ) + q2(ξ)

=⇒ ECS(ξ) = ηQ(ξ).

Capital Q(ξ) is used here to prevent ambiguity as the reservoir index i is often omitted in the followingsections. As Q(ξ) is just the ECS times a constant it will also be beta distributed, i.e.

Q(ξ) ∈ B(α, β, a, b),

for some shape parameters α and β, in support [a, b]. Explicitly we have that

Q(ξ) = ξ(b− a) + a,

where the truncated PCE of Q(ξ) is

Q(ξ) =

P∑m=0

Q̂mΦm(ξ).

In this basis ξ has the exact representation

ξ =α

α+ βΦ0(ξ) +

1

α+ βΦ1(ξ) = ξ0Φ0(ξ) + ξ1Φ1(ξ).

Projecting onto the basis yields

〈Q(ξ),Φp(ξ)〉 = Q̂p〈Φp(ξ),Φp(ξ)〉 = 〈ξ(b− a) + a,Φp(ξ)〉.

The PCE coe�cient Q̂p may then be extracted

Q̂p〈Φp(ξ),Φp(ξ)〉 = (b− a)〈ξ,Φp(ξ)〉+ 〈a,Φp(ξ)〉= (b− a)〈ξ0Φ0(ξ) + ξ1Φ1(ξ),Φp(ξ)〉+ δ0pa〈Φ0(ξ),Φ0(ξ)〉= δ0p(b− a)ξ0〈Φ0(ξ),Φ0(ξ)〉+ δ1p(b− a)ξ1〈Φ1(ξ),Φ1(ξ)〉+ δ0pa〈Φ0(ξ),Φ0(ξ)〉.

Dividing by 〈Φp(ξ),Φp(ξ)〉 gives

Q̂p = δ0p(b− a)ξ0 + δ1p(b− a)ξ1 + δ0pa= δ0p(a+ ξ0(b− a)) + δ1pξ1(b− a).

Thus Q(ξ) has the exact PCE representation

Q(ξ) = Q̂0Φ0(ξ) + Q̂1Φ1(ξ),

where

Q̂0 = ξ0(b− a) + aQ̂1 = ξ1(b− a).

That concludes the PCE overloading of the ECS. Next up is the temperature.


3.5.2 Temperature

Before beginning this section, it is convenient to clarify that a summation without the indices speci�edrefers to a summation from 0 to the chosen order of the basis P . Allowing for slight abuse of notation

∑m

:=

P∑m=0

.

The temperature dynamics are

T (k + 1, ξ) = T (k, ξ)e−5d − q(ξ)F (k, ξ)e− 5d + q(ξ)F (k, ξ). (25)

With the reservoir index i omitted. Here q(ξ) refers to ci(q1(ξ) + q2(ξ) for the respective i, where ci isthe fraction of the deterministic values, i.e. ci = qi/(q1 + q2). With respect to stochastic variables thereare two linear terms and one nonlinear. The system of equations for the temperature becomes

〈T (k + 1, ξ),Φp(ξ)〉 = 〈T (k, ξ)e−5d − q(ξ)F (k, ξ)e− 5d + q(ξ)F (k, ξ),Φp(ξ)〉

=⇒ T̂p(k + 1) = e−5d T̂p(k) + (1− e−

5d )∑m

∑m′

q̂mF̂m′(k)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉〈Φp(ξ),Φp(ξ)〉

.

Where the �nal term comes from that a product of two PCE's, for example X and Y , is expanded to

〈XY,Φp(ξ)〉 =∑m

∑m′

X̂mŶm〈Φm(ξ)Φm′(ξ),Φp(ξ)〉.

3.5.3 Carbon

The carbon dynamics are, with the index i = 1 . . . 4 omitted

R(k + 1, ξ) = R(k, ξ) + 5aτα(k, ξ)E(k, ξ)−R(k, ξ)

α(k, ξ)τ + 5. (26)

Doing a PCE overloading of this expression is di�cult due to the denominator, but by multiplying bothsides by the expression becomes tractable

R(k + 1, ξ)(α(k, ξ)τ + 5) = R(k, ξ)(α(k, ξ)τ + 5) + 5aτα(k, ξ)E(k, ξ)− 5R(k, ξ)= τR(k, ξ)α(k, ξ) + 5aτα(k, ξ)E(k, ξ).

The dynamics are now given in the implicit expression above, which simpli�es the PCE projections.Beginning with the left hand side by expanding the PCE's and projecting

〈R(k + 1, ξ)(α(k, ξ)τ + 5),Φp(ξ)〉 =〈∑m

∑m′

τR̂m(k + 1)α̂m′(k)Φm(ξ)Φm′(ξ) + 5∑n

R̂n(k)Φn(ξ),Φp(ξ)

〉=∑

m

∑m′

τR̂m(k + 1)α̂m′(k) 〈Φm(ξ)Φm′(ξ),Φp(ξ)〉+ 5∑n

R̂n(k + 1) 〈Φn(ξ),Φp(ξ)〉 .

Applying the same procedure to the right hand side yields

〈τR(k, ξ)α(k, ξ) + 5aτα(k, ξ)E(k, ξ),Φp(ξ)〉 =〈∑m

∑m′

τR̂m(k)α̂m′(k)Φm(ξ)Φm′(ξ) + 5∑n

∑n′

aτÊn(k)α̂n′(k)Φn(ξ)Φn′(ξ),Φp(ξ)

〉=∑

m

∑m′

τR̂m(k)α̂m′(k) 〈Φm(ξ)Φm′(ξ),Φp(ξ)〉+ 5∑n

∑n′

aτÊn(k)α̂n′(k) 〈Φn(ξ)Φn′(ξ),Φp(ξ)〉 .


This may be simpli�ed by combining the series, i.e. letting n = m and n′ = m′∑m

∑m′

(τR̂m(k)α̂m′(k) + 5aτÊm(k)α̂m′(k)

)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉 =∑

m

∑m′

τα̂m′(k)(R̂m(k) + 5aÊm(k)

)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉 .

Putting everything together we observe∑m

∑m′

τR̂m(k + 1)α̂m′(k) 〈Φm(ξ)Φm′(ξ),Φp(ξ)〉+ 5∑n

R̂n(k + 1) 〈Φn(ξ),Φp(ξ)〉 =∑m

∑m′

τα̂m′(k)(R̂m(k) + 5aÊm(k)

)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉 .

By rearranging and combining sums it can be simpli�ed to the following system of equations∑m

∑m′

(τR̂m(k + 1)α̂m′(k)− τα̂m′(k)(R̂m(k) + 5aÊm(k))

)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉 =

−5∑n

R̂n(k + 1)〈Φn(ξ),Φp(ξ)〉.

3.5.4 Accumulated Carbon

The accumulated carbon dynamics are given as

Cacc(k + 1, ξ) = Cacc(k, ξ) + 5

(E(k, ξ)−

∑i

Ṙi(k, ξ)

)

= Cacc(k, ξ) + 5

(E(k, ξ)−

∑i

(aiE(k, ξ)−

Ri(k, ξ)

α(k, ξ)τi

))

= Cacc(k, ξ) + 5∑i

Ri(k, ξ)

α(k, ξ)τi.

Seeing as∑i ai = 1. The dynamics are then implicitly given as

α(k, ξ)Cacc(k + 1, ξ) = α(k, ξ)Cacc(k, ξ) + 5∑i

Ri(k, ξ)

τi.

Moving the cross terms to the left hand side and PCE overloading yields∑m

∑m′

α̂m(k)(Ĉacc,m′(k + 1)− Ĉacc,m′(k)

)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉 =

∑i

∑m

5

τiR̂i,m(k)〈Φm(ξ),Φp(ξ)〉.

3.5.5 Algebraic Constraint

The algebraic constraint governing α(k, ξ) is

α(k, ξ)

4∑i=1

(aiτi(1− e

−100α(k,ξ)τi )

)− (r0 + rCCacc(k, ξ) + rTT (k, ξ)) = 0. (27)

Unlike the previously derived PCE's there is an exponential term here. While projecting arbitrarypolynomials onto a polynomial space is simple �nding the PCE coe�cients of an exponential function isnot obvious. One option, which will be used here, is to use a Taylor's expansion around the mean whichdoes a "good enough" approximation given that the distribution is not "too skewed" [3].


Here a Taylor expansion is employed, for simplicity de�ne

H(α(k, ξ)) =

4∑i=1

aiτi(1− e−100

α(k,ξ)τi ).

The Taylor expansion of H(α(k, ξ)) with respect to α(k, ξ) around the mean, i.e. the point α̂0(k) is

H(α(k, ξ)) =

N∑n=0

(α(k, ξ)− α̂0(k))n

n!H(n)(α̂0(k))

= H(α̂0(k)) +

N∑n=1

n∑l=0

(n

l

)(−1)lH(n)(α̂0(k))

n!α(k, ξ)n−lα̂0(k)

l.

Using the binomial theorem this can be rearranged to

〈α(k, ξ)H(α(k, ξ)),Φp(ξ)〉 = H(α̂0(k))α̂p(k)〈Φp(ξ),Φp(ξ)〉+N∑n=1

n∑l=0

∑(M1,M2,...,Mn−l+1)

h(n, l, k)α̂M1(k) . . . α̂Mn−l+1(k)〈ΦM1(ξ) . . .ΦMn−l+1(ξ),Φp(ξ)〉,(28)

with h(n, l, k) de�ned as before

h(n, l, k) =

(n

l

)(−1)lH(n)(α̂0(k))

n!α̂0(k)

l.

As the remaining terms are standard the derivation is not done explicitly, we have the nonlinear systemof equations with

〈(27),Φp(ξ)〉 = 0 = (28)− 〈Φp(ξ),Φp(ξ)〉[δ0pr0 + rCĈacc,p(k) + rT T̂p(k)

].

Note that there are P + 1 equations that can be solved for α̂0(k), . . . , α̂P (k).

3.5.6 Radiative Forcing

The radiative forcing is given by

F (k, ξ) =η

log(2)log

(C(k, ξ)

C0

)+ FEX(k).

The exogenous forcings are given by

FEX(k) = 0.25 +

{0.025(k − 1), k ∈ [1, 18]0.45, k ≥ 19.

The atmospheric carbon C(k, ξ) =∑4i=1Ri(k, ξ) +C0. The reader is again reminded that C0 here is the

pre-industrial atmospheric carbon. The PCE for C(k, ξ) is

C(k, ξ) =

P∑m=0

( 4∑i=1

R̂i,m(k) + C0δ0m

)Φm(ξ).

=⇒ Ĉp(k) =4∑i=1

R̂i,m(k) + C0δ0m

The logarithmic term poses numerical stability issues when a Taylor expansion is applied, instead discretedynamics for the radiative forcing are derived using Backward Euler. First, the derivative is

Ḟ (k, ξ) =ηĊ(k, ξ)

log(2)C(k, ξ)+ ḞEX(k).


The carbon and exogenous forcing time derivatives are given as

ḞEX(k) =

{0.025 k ∈ [1, 18]0 k ≥ 19

Ċ(k, ξ) =

4∑i=1

Ṙi(k, ξ) =

4∑i=1

aiE(k)−Ri(k)

α(k)τi.

Backward Euler is then readily applied to the radiative forcing

F (k + 1, ξ) = F (k, ξ) + 5Ḟ (k + 1, ξ)

F (k + 1, ξ) = F (k, ξ) + 5

(ηĊ(k + 1, ξ)

log(2)C(k + 1, ξ)+ ḞEX(k + 1)

)

F (k + 1, ξ) = F (k, ξ) + 5

η∑4i=1 aiE(k + 1, ξ)− Ri(k+1,ξ)α(k+1,ξ)τilog(2)C(k + 1, ξ)

+ ḞEX(k + 1)

.After multiplication with C(k + 1, ξ) thus yields

C(k + 1, ξ)F (k + 1, ξ) =

C(k + 1, ξ)F (k, ξ) + 5

η∑4i=1 aiE(k + 1, ξ)− Ri(k+1,ξ)α(k+1,ξ)τilog(2)

+ C(k + 1, ξ)ḞEX(k + 1)

.Rearranging gives

C(k + 1, ξ)(F (k + 1, ξ)− F (k, ξ)− ḞEX(k + 1)) = 5

η∑4i=1 aiE(k + 1, ξ)− Ri(k+1,ξ)α(k+1,ξ)τilog(2)

.Finally, multiplying by α(k + 1, ξ)

LHS = α(k + 1, ξ)C(k + 1, ξ)(F (k + 1, ξ)− F (k, ξ)− ḞEX(k + 1))

RHS =5η

log(2)

4∑i=1

(aiE(k + 1, ξ)α(k + 1, ξ)−

Ri(k + 1, ξ)

τi

).

The PCE projection will thus only involve polynomial products. The LHS is

〈LHS,Φp(ξ)〉 =∑M1,M2,M3

α̂M1(k + 1)ĈM2(k + 1)(F̂M3(k + 1)− F̂M3(k))〈ΦM1(ξ)ΦM2(ξ)ΦM3(ξ),Φp(ξ)〉−∑M1,M2

α̂M1(k + 1)ĈM2(k + 1)ḞEX(k + 1)〈ΦM1(ξ)ΦM2(ξ),Φp(ξ)〉.

And the RHS is obtained by

〈RHS,Φp(ξ)〉 =5η

log(2)

4∑i=1

∑M1,M2

aiÊM1(k + 1)α̂M2(k + 1)〈ΦM1(ξ)ΦM2(ξ),Φp(ξ)〉−

5η

log(2)

4∑i=1

∑M1

Ri(k + 1)

τi〈ΦM1(ξ),Φp(ξ)〉.


3.5.7 Damages

Conveniently �nding the PCE for the damages function is simple after having obtained the PCE coe�-cients for the temperature reservoirs. The damages function is given as

Damages(T (ξ)) = 1− 0.00267(T (ξ))2.

Where T = T1 +T2 is the atmospheric temperature. The PCE coe�cients for T is simply the sum of thecoe�cients for T1 and T2∑

m

T̂m(k, ξ)Φm(ξ) =∑m

T̂1,m(k, ξ)Φm(ξ) +∑m

T̂2,m(k, ξ)Φm(ξ)

=∑m

(T̂1,m(k, ξ) + T̂2,m(k, ξ))Φm(ξ).

Similar to the previous section m ∈ Z[0, P ] for some P ∈ Z[0,∞), this will hold throughout this section.Following the same procedure as previously to determine the PCE coe�cients

Damages(T (ξ)) ≈∑m

D̂m(k)Φm(ξ)

∑m

D̂m(k)Φm(ξ) = 1− 0.00267(∑m

T̂m(k)Φm(ξ))2.

Projecting both sides of the equality onto the p'th Jacobi polynomial

D̂p(k)〈Φp(ξ),Φp(ξ)〉 = δ0p〈Φp(ξ),Φp(ξ)〉 − 0.00267〈(∑

m

T̂m(k)Φm(ξ)

)2,Φp(ξ)

〉.

Dividing by the square inner product term yields the �nal expression for D̂p(k)

D̂p(k) = δ0p − 0.00267∑m

∑m′

T̂m(k)T̂m′(k)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉〈Φp(ξ),Φp(ξ)〉

.

3.5.8 Capital

The discrete dynamics of K(k, ξ) are given as

K(k + 1, ξ) = (1− δ)5K(k, ξ) + 5NEO(k, ξ)s(k).

Similar to the temperature this leads to a rather pleasant expression for the p'th PCE coe�cient as it'slinear with respect to stochastic terms

K̂p(k + 1) = (1− δ)5K̂p(k) + 5s(k)N̂p(k).

3.5.9 Gross Economic Output

The gross economic output is modeled as

GEO(k, ξ) = ATFP (k)K(k, ξ)γ

(L(k)

1000

)1−γ.

The expression is simpli�ed by de�ning

AL(k) ≡ ATFP (k)(L(k)

1000

)1−γ.

HenceGEO(k, ξ) = AL(k)K(k, ξ)γ .


Taking the usual projection onto the p'th Jacobi polynomial, and letting Ĝp(k) denote the GEO PCEcoe�cients gives

〈GEO(k, ξ),Φp(ξ)〉 = AL(k)〈(∑

m

K̂m(k)Φm(ξ)

)γ,Φp(ξ)

〉. (29)

This poses some di�culties as γ is non-integer. Fortunately, γ = 0.3 is a rational number. It follows thatK(k, ξ)0.3 = K(k, ξ)3/10. By de�ning the auxiliary random variables

c(k, ξ) = K(k, ξ)1/10

y(k, ξ) = c(k, ξ)3.

We have thaty(k, ξ) = K(k, ξ)3/10 = K(k, ξ)0.3.

The strategy here is to determine the PCE of c(k, ξ), then y(k, ξ) which will simpli�es the computationsin (29). To determine c(k, ξ), its governing equation is rewritten as

c(k, ξ)10 = K(k, ξ).

To reduce the size of the system of equations for determining the PCE coe�cients of c(k, ξ), de�ne

d(k, ξ) = c(k, ξ)5.

The procedure is then to �rst solve for the d(k, ξ) PCE coe�cients∑m1,m2

d̂m1(k)d̂m2(k)〈Φm1(ξ)Φm2(ξ),Φp(ξ)〉 = K̂p(k)〈Φp(ξ),Φp(ξ)〉.

And then the same for c(k, ξ) and y(k, ξ)

d̂p(k)〈Φp(ξ),Φp(ξ)〉 =∑

m1,...,m5

ĉm1 . . . ĉm5〈Φm1(ξ) . . .Φm5(ξ),Φp(ξ)〉

ŷp(k)〈Φp(ξ),Φp(ξ)〉 =∑m1

∑m2

∑m3

ĉm1(k)ĉm2(k)ĉm3(k)〈Φm1(ξ)Φm2(ξ)Φm3(ξ),Φp(ξ)〉.

Finally, the PCE coe�cients of the gross economic output, Ĝp(k), is

Ĝp(k) = AL(k)ŷp(k).

3.5.10 Net Economic Output

The net economic output is given as

NEO(k, ξ) = (Damages(k, ξ)− θ1(k)µ(k)θ2)GEO(k, ξ).

Recall that the m'th PCE coe�cient for the damages and GEO are D̂m(k) and Ĝm(p) respectively. Onlythe stochastic cross term requires any treatment, again projecting onto the p'th polynomial

Damages(k, ξ)GEO(k, ξ) =∑m

∑m′

D̂m(k)Ĝm′(k)Φm(ξ)Φm′(ξ)

〈∑m

∑m′

D̂m(k)Ĝm′(k)Φm(ξ)Φm′(ξ),Φp(ξ)

〉=∑m

∑m′

D̂m(k)Ĝm′(k)〈Φm(ξ)Φm′(ξ),Φp(ξ)〉.

The PCE coe�cient for the net economic output, N̂p(k) is then

N̂p(k) =∑m

∑m′

(D̂m(k)Ĝm′(k)

〈Φm(ξ)Φm′(ξ),Φp(ξ)〉〈Φp(ξ),Φp(ξ)〉

)− θ1(k)µ(k)θ2Ĝp(k).


3.5.11 Emissions

The emissions are de�ned as

E(k, ξ) = σ(k)(1− µ(k))GEO(k, ξ) + 0.8ELand(k).

Once again using the linearity in the stochastic terms

Êp(k) = σ(k)(1− µ(k))Ĝp(k) + 0.8ELand(k)δ0p.

3.5.12 Consumption

Given thatC(k, ξ) = (1− s(k))NEO(k, ξ).

In what amounts to a near copy paste of the emissions PCE the PCE coe�cient for C(k, ξ) is

Ĉp(k) = (1− s(k))N̂p(k).

3.5.13 Social Welfare

The objective function is given by

J(k + 1, ξ) = J(k, ξ)− L(k)1 + ρ5(k−1)

(

1000C(k,ξ)L(k)

)1−α− 1

1− α− 1

.As α = 1.45 is non-integer it is rounded to its one decimal, α ≈ 1.5. Let Q(k,C(k, ξ) be de�ned as

Q(k,C(k, ξ)) =

(1000C(k, ξ)

L(k)

)−1/2=

(L(k)

1000C(k, ξ)

)1/2.

Squaring both sides yields

Q(k,C(k, ξ))2 =L(k)

1000C(k, ξ).

Which after multiplication with C(k, ξ) is

C(k, ξ)Q(k,C(k, ξ))2 =L(k)

1000.

The PCE is then∑m1,m2,m3

Ĉm1(k)Q̂m2(k)Q̂m3(k)〈Φm1(ξ)Φm2(ξ)Φm3(ξ),Φp(ξ)〉 = δ0,pL(k)

1000〈Φp(ξ),Φp(ξ)〉.

The PCE projection of J yields

Ĵp(k + 1, ξ) = Ĵp(k, ξ)−L(k)

1 + ρ5(k−1)

[Q̂p(k)− δ0p

1− α− δ0p

].

The method used here is referred to as immersion, the non linearity in the decimal power is traded foran increase in system order.

36 4 SIMULATING STOCHASTIC FAIR/DICE

4 Simulating Stochastic FAIR/DICE

Having introduced PCE's and derived the analytical PCE equations for FAIR/DICE what remains isrunning the simulations. The end goal of overloading the dynamics here is of course to eventually over-load the optimization problem as whole. This has been done in for example optimal power �ow [10].

The full optimization problem overload is not done in this report, instead the system is simulated usingoptimal controls obtained by solving the deterministic optimization problem. The purpose of these sim-ulations is twofold. Firstly, they will show how the uncertainty propagates through the system dynamicsand secondly, the PCE equations derived can be directly implemented into an overload of the entireoptimization problem.

In other words the following optimization problem is solved, using the default values of the ECS:

maxs,µ

Social Welfare

s.t Dynamics global of temperature and carbon cycles - FAIR

Dynamics of global economy - DICE

Coupling between economy and climate

From which the controls are extracted, see the �gure below:

Figure 7: FAIR/DICE deterministic optimal controls for the emissions control rate µ and savings rate s.

37

Using these controls the FAIR/DICE system is then forward simulated by simply evaluating the k+ 1'thstate from the PCE equations. After which the mean and variance can be compared to traditional MonteCarlo sampling.

38 5 RESULTS

5 Results

First probability distributions of the FAIR/DICE dynamics are presented and compared to the MonteCarlo sampling. A few plots are presented in a "top down" view, for example the algebraic constraint,as the 3D view is not very informative by eye. In the following plots the PCE generated probabilitydistributions and Monte Carlo generated ones are provided. At the end of the section sampling ofPCE overloaded FAIR and continuous FAIR is compared, where the continuous DAE is solved usingMATLAB's sti� solver ode23s. The sampling size throughout is 20000.

Figure 8: Pr

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Application of Polynomial Chaos Expansion for Climate ...1188950/FULLTEXT01.pdf · Polynomial Chaos Expansions (PCEs), ett alternativ till Monte Carlo-metoder. Med hjälp av PCEs