the heat equation with unknown inclusions Numerical solution of an inverse boundary value problem for Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions

Haibing Wang

^∗

, Yi Li

SchoolofMathematics,SoutheastUniversity,Nanjing 210096,PRChina

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received1August2017

Receivedinrevisedform3February2018 Accepted4May2018

Availableonline9May2018

Keywords:

Inverseproblem Unknowninclusions Heatequation Layerpotential

Weconsidertheproblemofreconstructingunknowninclusionsinsideathermalconductor fromboundarymeasurements,whicharisesfromactivethermographyandisformulated as an inverse boundary value problem for the heat equation. In our previous works, weproposed asampling-type methodfor reconstructingthe boundaryofthe unknown inclusionand gaveitsrigorousmathematical justification.Inthispaper,wecontinueour previous works and provide afurther investigation ofthe reconstruction method from both the theoretical and numerical points of view. First, weanalyze the solvability of the Neumann-to-Dirichlet map gap equationand establish arelation of its solution to the Greenfunctionofan interior transmissionproblemfor the inclusion.Thisnaturally provides away of computingthis Greenfunction from the Neumann-to-Dirichlet map.

Ournewfindingsrevealtheessence ofthereconstruction method.Aconvergenceresult for noisy measurement data is also proved. Second, basedon the heatlayer potential argument,weperformanumericalimplementationofthereconstructionmethodforthe homogeneousinclusioncase.Numericalresultsare presentedtoshowthe efficiencyand stabilityoftheproposedmethod.

©^{2018ElsevierInc.Allrightsreserved.}

1. Introduction

Consider the heat conduction in a two-layered medium (Fig. 1.1). Denote by D0 and D the outer and inner layers, respectively. Set

=^D∪^D0. Suppose that the thermal conductivities of D₀ and D are 1 and k, respectively. We also assumethattheboundaries

∂

D0 and

∂

D ofD0 andD,respectively,areofclass C².Forsimplicityofnotations,throughout thispaperwedenote X×

(

0

,

T

)

and

∂

X×

(

0

,

T

)

by X_T and

(∂

X

)

T,respectively,where X isaboundeddomaininR^{2 and}

∂

X denotesits boundary.Injecting aheatflux g on

∂

oversome timeinterval

(

0

,

T

)

,the temperaturedistribution u in

T canbemodeledbythefollowinginitial-boundaryvalueproblem:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(∂

t

− ∇ ·

^k

∇)

^u

=

^{0 inD}T

, (∂

_t

− )

u

=

^{0 in}

( \

^D

)

_T

,

u

|

−

−

^u

|

+

=

^{0 on}

(∂

D

)

T

,

k

∂

_νu

|

₋

− ∂

_νu

|

₊

=

^{0 on}

(∂

D

)

T

,

∂

νu

=

^{g on}

(∂)

T

,

u

=

^{0 att}

=

⁰

,

(1.1)

*

Correspondingauthor.

E-mailaddress:hbwang@seu.edu.cn(H. Wang).

https://doi.org/10.1016/j.jcp.2018.05.008

0021-9991/©^{2018ElsevierInc.Allrightsreserved.}

Fig. 1.1.Configuration of the medium.

where

ν

on

∂

D (or

∂

)istheunitnormalvectordirectedintotheexteriorof D (or

).Herethesubscripts“+^”and“−^” indicatethetracetakenfromtheexteriorandinteriorofD,respectively.

The abovemodelhasmanyimportantapplications insciencesandengineering. Inactive thermography, D isregarded as an inclusion and D₀ is the background medium. In thiscase, the forward problem is to determine the temperature distributionin

T foranyinjectedheatfluxgon

(∂)

T,whiletheinverseproblemistoreconstructtheunknowninclusion D fromboundary measurements. Instead of recovering the thermal conductivity, we are more interested in finding the location,sizeandshapeoftheinclusionasadefectinsidetheconductor.Weprovedin[17] thatforanyg∈^H−12,−14

((∂)

T

)

thereexistsauniquesolutionuto(1.1) in H˜^1,12

(

T

)

.DefinetheNeumann-to-Dirichletmapby

D

:

H^−12,−14

((∂)

T

) →

H^12,14

((∂)

T

),

g

→

u

|

(∂)T

,

whichisan idealizedmeasurementdataforactive thermography.Thenourinverseproblemistoreconstruct D from

D, wherekisunknown.Theuniquenessandstabilityestimateareestablishedin[7,8].Asforreconstructionmethods,werefer to[6,9,13–16,21] andthereferencestherein,wherethedynamicalprobemethodandtheenclosuremethodaredeveloped.

Recently, the authors establisheda linear sampling-typemethod fortheheat equation in [12,17,18]. However, numerical studiesofthesereconstructionmethodsforparabolicinverseboundaryvalueproblemsareratherlimited[19].Somerelated worksonotherkindsofparabolicinverseboundaryvalueproblemscanbefoundin[2,3,5,10,11,20].

Inthiswork,basedontheheat layerpotentialtheory,weinvestigateboththeforwardandinverseproblemsfromthe numerical pointofview.Especially, thesampling-typereconstruction methodestablished in[17] for ourinverseproblem will be numerically implemented. Roughly speaking,this reconstruction methodis based on the characterization of the solutiontotheso-calledNeumann-to-Dirichletmapgapequation

(

D

−

_∅

)

g

=

^G_(y,s)

(

x

,

t

),

(1.2)

where

_∅ is theNeumann-to-Dirichlet map when D= ∅^{, and G}₍y,s)

(

x

,

t

)

:=^G

(

x

,

t;^y

,

s

)

isthe Green function forthe heat operator

∂

t−

in

T withhomogeneous Neumannboundaryconditionon

(∂ )

T.Intermsofthischaracterization, thenormofthesolutionto(1.2) servesasanindicatorfunctionandtheboundaryofD canbereconstructedapproximately by computingthevaluesoftheindicator functionataset ofsamplingpoints. Althoughthesampling-typereconstruction methodforinversescatteringproblemshasbeenextensivelystudied;see[1] andthereferencestherein,veryfewnumerical resultsforparabolicinverseboundaryvalueproblemsarereported.Werecentlystudiedin[19] thenumericalimplementa- tionofthesamplingmethodforidentifyingunknowncavitiesinthethermalconductor,buttherigidinclusioncasehasnot yetconsidered.

In this paper, we continue our previous worksand investigatethe numerical realizationof the samplingmethod for parabolicinverseboundaryvalueproblemswithunknowninclusions.Firstofall,wesupplementthetheoreticalanalysisof ourreconstructionmethodbyanalyzingthesolvabilityoftheequation(1.2) andshowingtherelationofitssolutiontothe Greenfunctionofan associatedinteriortransmissionproblem.Thesenewfindingsrevealtheessence ofthesampling-type reconstruction method. In addition, a convergence resultfor noisy measurement data is proved. Then, we simulate the measurementdata

D bysolvingtheforwardproblem(1.1),andcomputetheNeumann-to-Dirichletmap

_∅andtheGreen function G_(y,s)

(

x

,

t

)

by solving the problem(1.1) with D= ∅^{. Byexpressingthe solutionasa} single-layer heat potential, theinitial-boundary valueproblem(1.1) istransformedintoasystemofboundaryintegral equations.Anumericalscheme for solvingthe resultingintegral equationsisintroduced. Finally,we solvethe discretizedNeumann-to-Dirichlet map gap equation usingthe Tikhonov regularizationtechnique. We show the performance ofthe reconstruction method fromthe following twoaspects.1.We testthemethodforinclusionsofdifferentshapesandthermalconductivities.2.We testthe method with shorttime measurements, namely, usingmeasured dataonly in a very short time interval. Our numerical resultsillustratetheefficiencyofthereconstructionmethod.

Therestofthispaperisorganizedasfollows.InSection2,werevisitthesampling-typereconstructionmethodandshow some newtheoretical results.InSection 3,weintroduceanumericalschemeforsolvingtheforwardproblemtosimulate themeasurementforourinverseproblem.TwonumericalexamplesareprovidedinSection4.Finally,inSection5,wegive someconcludingremarks.

2. Reconstructionmethodfortheinverseproblem

In this section, we revisit the sampling-type reconstruction method for the heat equation with unknown inclusions studiedin[17],andgiveafurtherinvestigationonthismethodbyshowingsomenewtheoreticalresults.Explicitly,wewill analyzethesolvabilityoftheNeumann-to-Dirichletmap gapequationandexplainwhatisthesolutionifitdoesexist.The relationof its solutionto theGreen function ofa relatedinteriortransmissionproblemis established,which revealsthe essenceofthereconstructionmethod.Fornoisymeasurementdata,aconvergenceresultisalsoproved.

Tobeginwith,letusintroducetheanisotropicSobolevspaces.For p

,

q≥^{0 wedefine}

H^p,q

(R

²

× R) :=

^L2

(R;

^Hp

(R

²

)) ∩

^Hq

(R;

^L2

(R

²

)).

Forp

,

q≤^{0 wedefinethespaceHp,q bydualityHp,q}

(R

²×R):=

H^−p,−q

(R

²×R)

.ByH^p,q

(

XT

)

wedenotethespaceof restrictionsofelements inH^p,q

(

R²×R

)

to XT.ThespaceH^p,q

((∂

X

)

T

)

isdefinedanalogously.Wealsodefinethefunction spaces

H

˜

^1,12

(

XT

) :=

u

∈

H^1,12

(

X

× (−∞,

T

))

_u

(

x

,

t

) =

0 fort

<

0

,

H

ˆ

^1,12

(

XT

) :=

u

∈

^H1,12

(

X

× (

0

, +∞ ))

u

(

x

,

t

) =

^{0 fort}

>

T

,

H

˜

^1,12

(

X_T

; ∂

t

− ∇ · γ ∇) :=

u

∈ ˜

^H1,12

(

X_T

) (∂

t

− ∇ · γ ∇)

^u

∈

^L2

(

X_T

)

,

where

γ

=¹+

(

k−¹

) χ

D,and

χ

D isthecharacteristicfunctionofD.

Inthispaper,weworkintheaboveSobolevspaces,andsolutionstoinitial-boundaryvalueproblemsshouldbeunder- stoodintheweaksense.Forexample,thesolutionto(1.1) shouldbeinterpretedasfollows.Forgiveng∈^H−12,−14

((∂)

T

)

, weseeku∈ ˜^H1,12

(

T;

∂

t− ∇ ·

γ

∇

)

suchthat

T 0

γ _∇

u

· ∇

^v

+ ∂

u

∂

tv

dxdt

=

^g

,

v

foranyv∈ ˆ^H1,12

(

T

)

,where istheDirichlettracemapon

(∂)

T and·,·^{shouldbeunderstoodinthesenseofduality.}

Forlateruse,wedenoteby

G^a

(

x

,

t

;

^y

,

s

) :=

⎧ ⎨

⎩

1

4a

π (

t

−

^s

)

^exp

− |

^x

−

^y

|

² 4a

(

t

−

^s

)

,

t

>

s

,

0

,

t

≤

^s

(2.1)

thefundamentalsolutionoftheheatoperator

∂

t−^a

,wherea isaconstant.WesometimeswriteitasG^a_(y,s)

(

x

,

t

)

. Notethat

_∅isdefinedby

_∅g:=^v|(∂)T withv satisfying

⎧ ⎪

⎨

⎪ ⎩

(∂

t

− )

v

=

^{0 in}

T

,

∂

νv

=

^{g on}

(∂)

T

,

v

=

^{0 att}

=

⁰

.

(2.2)

Definetheoperator

F

:=

D

−

_∅

.

Then our reconstruction scheme is based on the characterization of the solution to the Neumann-to-Dirichlet map gap equation

(

F g

)(

x

,

t

) =

^G_(y,s)

(

x

,

t

), (

x

,

t

) ∈ (∂)

T (2.3)

forafixedtimes∈

(

0

,

T

)

andthesamplingpoint y∈

. First,weanalyzethesolvabilityof(2.3).

Theorem2.1.Fory∈^{D ands}∈

(

0

,

T

)

,theequation(2.3)hasasolutionifandonlyiftheinteriortransmissionproblem

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(∂

t

− ∇ ·

^k

∇ )

w

=

^{0 inD}T

,

(∂

t

− )

v

=

^{0 inD}T

,

w

−

^v

=

^G_(y,s)

(

x

,

t

)

on

(∂

D

)

T

,

k

∂

_νw

− ∂

_νv

= ∂

_νG_(y,s)

(

x

,

t

)

on

(∂

D

)

T

,

w

=

^{0 att}

=

⁰

,

v

=

^{0 att}

=

⁰

(2.4)

issolvablewiththesolutionw andv satisfyingtheequations

(∂

t− ∇ ·

γ

_∇

)

w=^0and

(∂

t−

)

v=⁰ⁱⁿ

T,respectively.

Proof. Supposethat g∈^H−12,−14

((∂)

T

)

isasolutionto(2.3).Letw andv satisfy

⎧ ⎪

⎨

⎪ ⎩

(∂

_t

− ∇ · γ _∇ )

w

=

^{0 in}

_T

,

∂

_νw

=

^{g on}

(∂)

_T

,

w

=

^{0 att}

=

⁰

(2.5)

and

⎧ ⎪

⎨

⎪ ⎩

(∂

t

− )

v

=

^{0 in}

T

,

∂

νv

=

^{g on}

(∂)

T

,

v

=

^{0 att}

=

⁰

,

(2.6)

respectively.Thenwehave

w

−

^v

=

Dg

−

_∅g

=

^G_(y,s) ^on

(∂)

T

,

∂

_νw

− ∂

_νv

=

⁰

= ∂

_νG_(y,s) on

(∂)

T

,

andtherefore

w

−

v

=

G_(y,s) in

( \

D

)

T

.

Bythecontinuitiesofw andv across

∂

D,wefurtherhave

w

|

₋

−

v

|

₋

=

G_(y,s) on

(∂

D

)

T

,

k

∂

_νw

|

−

− ∂

_νv

|

−

= ∂

_νG_(y,s) on

(∂

D

)

_T

.

Totally,wehavetheinteriortransmissionproblem(2.4).

Conversely,assumethat

(

w

,

v

)

isasolutionto(2.4) andsatisfiestheequations

(∂

t− ∇ ·

γ

_∇

)

w=^{0 and}

(∂

t−

)

v=^{0 in}

T,respectively.Bythecontinuitiesofwandv across

∂

D,weobtainfromthetransmissionconditionsin(2.4) that

w

|

₊

−

^v

|

₊

=

^G_(y,s) ^on

(∂

D

)

T

,

∂

_νw

|

+

− ∂

_νv

|

+

= ∂

_νG_(y,s) on

(∂

D

)

T

.

Noticethat w−^{vsatisfiestheheatequation}

(∂

t−

)(

w−^v

)

=^{0 in}

(

\^D

)

T.Thenitfollowsfromtheuniquecontinuation principlethat

w

−

^v

=

^G_(y,s) ⁱⁿ

( \

^D

)

T

,

(2.7)

andhence

w

−

^v

=

^G_(y,s)

, ∂

ν

(

w

−

^v

) = ∂

νG_(y,s) on

(∂)

T

.

(2.8) Thisimpliesthat

∂

νw=

∂

νv on

(∂)

T andg:=

∂

νvisthesolutionto(2.3).Theproofisnowcomplete. 2

ThistheoremanalyzesthesolvabilityoftheNeumann-to-Dirichletmapgapequation(2.3). Furthermore,ifitdoesexist asolution,weexplainwhatisthesolutioninthefollowingtheorem.

Theorem2.2.Ifg isthesolutionto(2.3),weletv bethesolutionto(2.6)withtheboundarydata

∂

νv|(∂)T =^{g.Thenwecanconclude} that

v

=

^GD_(y,s)

(

x

,

t

) −

^G_(y,s)

(

x

,

t

)

inD_T

,

(2.9)

whereG^D_(y,s)

(

x

,

t

)

isdefinedby

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

(∂

_t

− ∇ ·

^k

∇ )

H₍^D_y,s)

=

^{0 inD}T

, (∂

t

− )

G₍^D_y,s)

= δ(

x

−

^y

) δ(

t

−

^s

)

inD_T

,

H₍^D_y,s)

−

^G₍^D_y,s)

=

^{0 on}

(∂

D

)

T

,

k

∂

νH₍^D_y,s)

− ∂

νG₍^D_y,s)

=

^{0 on}

(∂

D

)

T

,

H₍^D_y,s)

=

^{0 att}

=

⁰

,

G^D_(y,s)

=

^{0 att}

=

⁰

.

(2.10)

Proof. Set

G

˜

^D_(y,s)

(

x

,

t

) =

^G₍^D_y,s)

(

x

,

t

) −

^G1_(y,s)

(

x

,

t

),

(2.11) G

˜

_(y,s)

(

x

,

t

) =

^G_(y,s)

(

x

,

t

) −

^G1_(y,s)

(

x

,

t

).

(2.12) Notethat G˜_(y,s)

(

x

,

t

)

isthereflectedsolutionofthefundamentalsolutionG¹_(y,s)

(

x

,

t

)

in

T andsatisfies

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

(∂

t

− )

G

˜

_(y,s)

=

^{0 in}

T

,

∂

νG

˜

_(y,s)

= −∂

νG¹_(y,s)

(

x

,

t

)

on

(∂)

T

,

G

˜

_(y,s)

=

0 att

=

0

.

Weobservefrom(2.10) that

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(∂

t

− ∇ ·

^k

∇)

^H₍^D_y,s)

=

^{0 inD}T

, (∂

t

− )

G

˜

₍^D_y,s)

=

^{0 inD}T

,

H₍^D_y,s)

− ˜

G₍^D_y,s)

=

G¹_(y,s)

(

x

,

t

)

on

(∂

D

)

T

,

k

∂

_νH₍^D_y,s)

− ∂

_νG

˜

₍^D_y,s)

= ∂

_νG¹_(y,s)

(

x

,

t

)

on

(∂

D

)

_T

,

H₍^D_y,s)

=

^{0 att}

=

⁰

,

G

˜

^D_(y,s)

=

^{0 att}

=

⁰

.

Letwandv bethesolutionsto(2.5) and(2.6) withthesameboundarydata

∂

_νv|(∂)T=^g,respectively.Thenwehave

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(∂

t

− ∇ ·

^k

∇ )(

w

−

^H₍^D_y,s)

) =

^{0 inD}T

, (∂

t

− )(

v

− ˜

^GD_(y,s)

+ ˜

^G_(y,s)

) =

^{0 inD}T

, (

w

−

H₍^D_y,s)

) − (

v

− ˜

G₍^D_y,s)

+ ˜

G_(y,s)

) =

0 on

(∂

D

)

T

,

k

∂

_ν

(

w

−

^HD_(y,s)

) − ∂

_ν

(

v

− ˜

^G₍^D_y,s)

+ ˜

^G_(y,s)

) =

^{0 on}

(∂

D

)

_T

,

w

−

^H₍^D_y,s)

=

^{0 att}

=

⁰

,

v

− ˜

^G₍^D_y,s)

+ ˜

^G_(y,s)

=

^{0 att}

=

⁰

.

(2.13)

Theuniquenessofsolutionstotheinteriortransmissionproblem(2.4) saysthat v

= ˜

G^D_(y,s)

(

x

,

t

) − ˜

G_(y,s)

(

x

,

t

)

inDT

,

whichgives(2.9) andcompletestheproof. 2

Here we wouldliketo emphasizethat the reflectedsolution G˜_(y,s)

(

x

,

t

)

andthe Greenfunction G_(y,s)

(

x

,

t

)

are com- pletelydeterminedbythedomain

.Sotheyareactuallyknowninourinverseproblem.Thus,oncewehavethesolution

g to(2.3),wesolvetheinitial-boundaryvalueproblem(2.6) withtheboundarydata

∂

νv=^{g on}

(∂)

T andthenobtainthe Greenfunction G^D_(y,s)

(

x

,

t

)

=^v+^G_(y,s)

(

x

,

t

)

.Inthissense,ourreconstructionmethodinprincipleprovidesawayofcom- putingtheGreenfunctionG₍^D_y,s)

(

x

,

t

)

fortheinteriortransmissionproblem(2.4) fromtheNeumann-to-Dirichletmap

D.

For y∈

\^{D,wehaveproventhefollowingresultinourpreviouswork [17]:}

Theorem2.3.Fors∈

(

0

,

T

)

andy∈

\^{D,theequation(2.3)hasnosolution.}

AccordingtoTheorems2.2and2.3,wecannotguaranteethe solvability of(2.3).However,wecouldfindanapproxima- tion solutionin some sense thatcharacterizes the boundaryofthe inclusion.The theoretical resultsare obtainedin[17]

andcanbesummarizedasfollows.

Theorem2.4.Lets∈

(

0

,

T

)

befixed.Wehavethefollowingconclusions:

(1)ify∈^D,thenforany

ε >

0thereexistsafunctiong^y∈^H−12,−14

((∂)

T

)

satisfying

^{F gy}

−

^G₍y,s)

H^12,14((∂)T)

< ε

suchthat

ylim→∂^D

^gy

H^−12,−14((∂)T)

= ∞

^(2.14)

and

ylim→∂D

S g^y

_˜

H^1,12(D_T)

= ∞,

(2.15)

wheretheoperatorS isdefinedby

S

:

^H−12,−14

((∂)

T

) → ˜

^H1,12

(

D_T

),

g^y

→

^v

|

DT

,

withv beingthesolutionto(2.2);

(2)ify∈

\^D,thenforany

ε , η >

0thereexistsafunctiong^y∈^H−12,−14

((∂)

T

)

satisfying

^{F gy}

−

^G_(y,s)

H^12,14((∂)T)

< ε + η

suchthat ηlim→0

^gy

H⁻¹2,−¹₄((∂)T)

= ∞

^(2.16)

and

ηlim→0

S g^y

_˜

H^1,12(DT)

= ∞.

(2.17)

BasedonTheorem2.4,wedefinetheindicatorfunctionby I

(

y

) :=

g^y

H⁻¹2,−¹₄((∂)T)

andreconstructthegeometricinformationon D usingthefollowingalgorithm:

Algorithm2.5.

1. Fixs∈

(

0

,

T

)

andselectasetof“samplingpoints y”in

;

2. Computeanapproximatesolutiontotheequation(2.3) foreachsamplingpoint;

3. Assertthat y∈^{D ifandonlyifI}

(

y

)

≤^{C,wherethecut-offconstantC shouldbechosenproperly.}

In practice, the measurement data always contain some noise, so the operator F cannot be given exactly andsome regularizationtechniqueisnecessaryinnumericalimplementations.Denoteby F^δ theperturbedoperatorof F with

^Fδ

−

^F

≤ δ,

where

δ

is the noise level and · ^{is the operator norm. Using the Tikhonov} regularization method, we construct an approximatesolutiontotheperturbedNeumann-to-Dirichletmapgapequation

F^δg

=

^G_(y,s)

by

g_α^y_,δ

:=

α

I

+ (

F^δ

)

^∗F^δ

₋1

F^δ

_∗

G_(y,s)

.

(2.18)

Thenwehavethefollowingconvergenceresult:

Theorem2.6.Lety∈^{D andthe}Neumann-to-Dirichletmapgapequation(2.3)haveauniquesolutiong^y.Ifwechoosetheregular- izationparameter

α

properlysuchthat

α (δ)

→^0and_α3/2^δ(δ)→^0as

δ

→^{0,thenitholdsthat}

g_α^y_,δ

→

g^y as

δ →

0

.

(2.19)

Proof. Bydirectcalculations,wehave g_α^y_,δ

−

^gy

=

α

I

+

F^δ

∗

F^δ

₋₁

−

α

I

+

^F∗F

₋1

F^δ

∗

G_(y,s)

+

α

I

+

^F∗F

₋1

F^δ

∗

−

^{F∗ G}_(y,s)

+

α

I

+

F^∗F

₋1

F^∗

G_(y,s)

−

g^y

.

(2.20)

Fromtheestimate

α

I

+

F^δ

_∗ F^δ

₋1

−

α

I

+

F^∗F

₋1

≤

²

δ

α

^3/2

^,

wehave

α

I

+

F^δ

_∗

F^δ

₋1

−

α

I

+

^F∗F

₋1

F^δ

_∗

G_(y,s)

≤

F^δ

_∗

^G₍y,s)

²

δ

α

^3/2

^.

^(2.21)

Noticethat

( α

I

+

^B

)

⁻¹

≤

¹

α

holdsforanyself-adjointandpositiveoperatorB inHilbertspace.Thenweobtainthat

α

I

+

^F∗F

₋1

F^δ

_∗

−

^{F∗ G}_(y,s)

≤

^G(y,s)

δ

α ^.

^(2.22)

Inaddition,itfollowsfromstandardregularizationtheorythat

α

I

+

^F∗F

₋1

F^∗

G_(y,s)

→

^{gy as}

α →

⁰

.

Thus,theproofiscompletedbycombiningitwiththeestimates(2.21) and(2.22). 2

Remark2.7.Bythewell-posednessoftheinitial-boundaryvalueproblem(2.2) andtheequality(2.9),wecaneasilydeduce from(2.19) that

^S

[

^g_α^y_,δ

] −

G₍^D_y,s)

(

x

,

t

) −

^G_(y,s)

(

x

,

t

)

_˜

H^1,12(DT)

→

^{0 as}

δ →

⁰

,

(2.23)

whichleadstotheconvergenceforcomputingtheGreenfunctionG^D_(y,s)

(

x

,

t

)

fromtheNeumann-to-Dirichletmap

D.

3. Numericalschemeforsolvingthedirectproblem

Inthissection,wepresentanumericalschemetosolvetheforwardproblem(1.1) fortwo-dimensionalspatialdomains.

Based on the potential theory for the heat equation, we first reformulate the initial-boundary value problem (1.1) as a systemofboundaryintegral equations,andthenintroducea discretizationschemeforsolvingthissystem.Thisschemeis anextensionofthemethodproposedin[2,3] tothelayeredmediumcase.Ourproblem(1.1) involvestheoperator

∂

t−^a

witha=¹

,

k,sothediscretizationoftheintegraloperatorsforageneralconstanta willbeconsidered.

Definethefollowingheatlayerpotentials:

V^a_{i j}

[ ϕ ](

x

,

t

) :=

t 0

Si

G^a

(

x

,

t

;

y

,

s

) ϕ (

y

,

s

)

d

σ (

y

)

ds

, (

x

,

t

) ∈

Sj

× (

0

,

T

),

N^a_{i j}

[ ϕ _] (

x

,

t

) :=

t 0

S_i

∂

G^a

(

x

,

t

;

^y

,

s

)

∂ ν (

x

) ϕ (

y

,

s

)

d

σ (

y

)

ds

, (

x

,

t

) ∈

^Sj

× (

0

,

T

),

whereG^a

(

x

,

t;^y

,

s

)

isthefundamentalsolutiondefinedby(2.1).Laterwewilltakei

,

j=¹

,

2 with S1=

∂

Dand S2=

∂

. Assumethat D isahomogeneous inclusion,that is,kisaconstant.Weexpressthesolutionu to(1.1) bythefollowing single-layerpotentials:

u

(

x

,

t

) =

t 0

∂D

G^k

(

x

,

t

;

^y

,

s

) ϕ

1

(

y

,

s

)

d

σ (

y

)

ds

, (

x

,

t

) ∈

^DT

,

(3.1)

u

(

x

,

t

) =

t 0

∂D

G¹

(

x

,

t

;

y

,

s

) ϕ

2

(

y

,

s

)

d

σ (

y

)

ds

+

t 0

∂

G¹

(

x

,

t

;

^y

,

s

) ϕ

3

(

y

,

s

)

d

σ (

y

)

ds

, (

x

,

t

) ∈ ( \

^D

)

T

,

(3.2)

where

ϕ

1,

ϕ

2and

ϕ

3 aredensityfunctionstobedetermined.Usingjumprelationsofheatlayerpotentials[4],wecanverify that uexpressedby(3.1) and(3.2) isthesolutionto(1.1) providethat

ϕ

1,

ϕ

2 and

ϕ

3 satisfy

V₁₁^k

[ ϕ

1

] −

^V11¹

[ ϕ

2

] −

^V₂₁¹

[ ϕ

3

] =

⁰

,

(3.3)

ϕ

₁

₊

2kN^k₁₁

[ ϕ

₁

_{] +} ϕ

₂

₋

2N₁₁¹

[ ϕ

₂

_{] −}

2N¹₂₁

[ ϕ

₃

_{] =}

0

,

(3.4)

2N¹₁₂

[ ϕ

2

] + ϕ

3

+

^2N122

[ ϕ

3

] =

^2g

.

(3.5)

Notethat theintegral kernelsofV₂₁¹ ,N¹₂₁andN¹₁₂aresmooth,whilethoseof V₁₁^k ,N^k₁₁,V₁₁¹, N¹₁₁andN₂₂¹ aresingularat

(

x

,

t

)

=

(

y

,

s

)

.Tonumericallysolvetheequations(3.3)-(3.5),wepresentadiscretizationschemeasfollows[2,3].

Assumethattheboundaries

∂

D and

∂

havetheparametricrepresentations

∂

D

= {

^x1

( α ) :

^x1

( α ) = (

x₁₁

( α ),

x₁₂

( α )),

0

≤ α ≤

²

π } ,

∂ = {

^x2

( α ) :

^x2

( α ) = (

x₂₁

( α ),

x₂₂

( α )),

0

≤ α ≤

²

π } ,

where x_{i j}

(

x

)

(i

,

j=¹

,

2) are ofclass C² and2

π

-periodicfunctions. The unitoutward normalvectorson

∂

D and

∂

are givenby

ν

1

( α ) = (

x₁₂

( α ), −

x₁₁

( α ))

|

^x₁

( α ) | , ν

2

( α ) = (

x₂₂

( α ), −

x₂₁

( α ))

|

^x₂

( α ) | .

Set

ϕ

˜1

(β,

s

)

:=

ϕ

1

(

x1

(β),

s

)

,

ϕ

˜2

(β,

s

)

:=

ϕ

2

(

x1

(β),

s

)

,

ϕ

˜3

(β,

s

)

:=

ϕ

3

(

x2

(β),

s

)

,ri j

( α , β)

= |^xj

( α )

−^xi

(β)

|^.

Weapplyacollocationmethodusingpiecewiseconstantinterpolationwithrespecttothetimevariableontheequidis- tantgridtn:=^nT

/

N,n=⁰

,

1

,

· · ·

,

N.Thatis,weapproximatethedensity

ϕ

˜i

(β,

s

)

by

˜

ϕ

i

(β,

s

) ≈

N n=¹

˜

ϕ

i,n

(β)

n

(

s

),

where

ϕ

˜i,n

(β)

:= ˜

ϕ

i

(β,

tn

)

,i=¹

,

2

,

3 and

n

(

s

) :=

1

,

tn−¹

<

s

≤

^tn

,

0

,

otherwise

.

Take

K_{i j}^a,p

( α , β) :=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

1 4πaE₁

Nr_{i j}²(α, β) 4aT

,

p

=

⁰

,

1 4πaE₁

Nr²_{i j}(α, β) 4aT(p+¹)

−

_4π¹_a^E1

Nr²_{i j}(α, β) 4aT p

,

p

=

¹

, · · · ,

N

−

¹

,

(3.6)