Elliptic boundary values problems with Gaussian white noise

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Computational Engineering and Technical Physics Technomathematics

Joël Right Dzokou Talla

ELLIPTIC BOUNDARY VALUES PROBLEMS WITH GAUSSIAN WHITE NOISE

Master’s Thesis

Examiners: Associate Professor Lassi Roininen Dr. Sari Lasanen

Supervisors: Associate Professor Lassi Roininen Dr. Sari Lasanen

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Lappeenranta-Lahti University of Technology School of Engineering Science

Computational Engineering and Technical Physics Technomathematics

Elliptic Boundary Values Problems with Gaussian White Noise

Master’s Thesis 2019

52 pages.

Examiners: Associate Professor Lassi Roininen Dr. Sari Lasanen

Keywords: Boundary value problem ·Gaussian random field · Gaussian white noise · inverse problem·Bayesian estimation.

This work focuses on the study of elliptic boundary value problems with Gaussian white noise on the formLX =W, whereL=−∆ +ν², withν > 0. First, the definitions and some properties of some important tools for this study such as, Gaussian random field and Gaussian white noise are given. Next, the boundary value problem is solved in dimensions d = 1,2, for the case of Dirichlet boundary condition. The spectral theorem is used to show that the operatorL admits eigenfunctions that form a complete orthonormal basis and the boundary value problem is solved in the eigenbasis ofL. Finally, the Bayesian estimation approach for inverse problem is derived, and as application, a one dimensional deconvolution problem is considered. The mathematical model is discretized on finite interval and the Bayesian estimation approach is used to estimate the truncated unknown.

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First of all, I would like to thankAlmighty God who has kept me this far and who has helped me to provide this work.

We could not have achieved such a work without the collaboration of many people.

I want to thank my supervisor, Dr.Sari Lasanen. She is incredibly kind. Interviews with Sari have always comforted me and given energy to continue. I admire her approach in research. I thank her especially for the knowledge she has impacted in me and the freedom she has given me during this work. I appreciate Associate ProfessorLassi Roininenfor his advices, the trust and freedom he has given me all through this work. I appreciate your kind personality.

I thank my family, especially my parents for their unconditional love and support they have provided me throughout my studies. I thank my friends and classmates for being there with me and for the good time we spent together.

Finally, I thank the department of Technomathematics, School of Computational Engi- neering and Technical Physics of LUT University for given me the opportunity to pursue this Master’s program. Thank you for the scholarship provided to aid my study and for allowing me to work in better conditions.

Lappeenranta, June 24, 2019

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1 INTRODUCTION

In general, many phenomena in nature are modelled using partial differential equations.

These phenomena can sometimes contain a certain level of uncertainty thus making modelling more difficult. Modelling using stochastic partial differential equations turn out to be the most appropriate way to capture the behaviour of such phenomena. The stochastic differential equation is used to define the prior is the statistical inverse problem [1, 2].

In this work, we study the elliptic Dirichlet boundary values problem modelled by the following stochastic partial differential equations







LX =W, inD, X = 0, on∂D.

(1.1)

whereDis a bounded Lipschitz domain, the operatorLis defined as−∆ +ν²withν >0 a constant, the unknownX is a random field andW is a zero-mean Gaussian white noise onD. In the analytical resolution of the above Equation (1.1), we encounter many difficulties due to the inherent complexity of partial differential equations and the irregularity of the noise thermW which appears in this equation.

In many problems such as the stochastic Navier-Stokes equations, the stochastic heat equation [1] and the inverse seismic problem [2], we encounter some irregular stochastic effects which make them more difficult to solve. Due to the irregularity of these stochastic effects, it becomes then very difficult to determine the unknownXfrom the mathematical model. Sometimes the problems may be ill-posed in the sense of Hadamard, that is, may not admit a unique solution or the solution may not depend continuously on the data. The ill-posed problems may be view as inverse problems [3]. To overcome these difficulties, some methods such as Bayesian statistical method in inverse problems turn out to be a promising alternative that offers more useful results [3, 4].

There exist a vast literature on solving Equation (1.1) in which many authors such as Walsh in [5], Buckdahn and Pardoux in [6] and Lasanen et al [7] have brought partial answers under various constraints more or less restrictive on the irregularity of the white noiseW. In [7] it is shown in the case of Dirichlet, Neumann and Robin boundary conditions that (1.1) has a unique Gaussian zero mean Banach space-valued random field solution.

Our main objectives in this work is first to base ourselves on the work done by Lasa- nen et al [7] to show the existence and the uniqueness of a random field solution of (1.1)

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in the eigenbasis of the operatorL, where the white noiseW has actually a representation in the eigenbasis ofL. Next, we will derive the Bayesian approach to inverse problem as in [4] and then we will use it to approximate the prior by truncation, that is, to approximate it by a finite sum.

In order to achieve this goal, we structure our work as follows: Chapter 2 is devoted to some preliminary results known on Sobolev spaces, Gaussian random variables and some properties of Gaussian white noise. In Chapter 3 we study the existence and uniqueness of a Gaussian random field X in the eigenbasis of operatorL in dimensionsd = 1,2.

In Chapter 4 we derive the Bayesian estimation approach for linear inverse problems. In Chapter 5 we apply the Bayesian approach to a discretized problem.

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2 PRELIMINARIES

In this section, we are going to present the concepts of Sobolev spaces, probability measure and Gaussian white noise that will be used later on to study the existence and uniqueness of the solution of Equation (1.1).

2.1 Banach and Hilbert spaces

We recall the definition and some properties of Banach and Hilbert spaces. The proofs of some results given in these subsections can be found in the books of Douglas Farenick [8]

and Sopka et al [11] .

Definition 2.1.1 ( [8, p. 13]). Let M be a non-empty set. We call distance on M the function

d:M ×M →[0,+∞[

satisfying the following properties, for allx, y ∈M 1. d(x, y) = 0⇔x=y,

2. d(x, y) =d(y, x),

3. d(x, z)≤d(x, y) +d(y, z), z ∈M.

The spaceM endowed with the distancedis calledmetric spaceand we denote it(M, d).

We are going to define a complete metric space which will allow us to define the concept of Banach space.

Definition 2.1.2( [11, p. 28]). Let(M, d)be a metric space and (x_n)_n∈

N a sequence in M. The sequence(x_n)_n∈

Nis said to beCauchy sequenceif(x_n)_n∈

Nsatisfies the following condition:

∀ε >0, there existsN ∈N such that d(x_m, x_n)< ε, ∀m, n≥N.

Definition 2.1.3( [11, p. 28]). A metric space(M, d)iscompleteif all Cauchy sequences inM are convergent.

Definition 2.1.4( [11, p. 59]). A normon a vector spaceV is a function k.k : V → R such that, for allu, v ∈V andα∈C, the following properties hold:

1. kuk ≥0

2. kuk= 0⇔u= 0 3. kαuk=|α| kuk

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4. ku+vk ≤ kuk+kvk.

If k·k is a norm on V, then the space V endowed with the norm k·k is called normed vector spaceand we denote it by(V,k·k).

Definition 2.1.5( [8, p. 170]). LetM be a vector space and letk·kbe a norm onM. A normed vector spaceM is said to be complete if all Cauchy sequences inM are convergent. A complete normed vector space is called Banach space. The spaceL^p(Ω), with Ω⊂C^dand1≤p < ∞defined as

L^p(Ω) =







f : Ω→C, Z

Ω

|f(x)|^pdx <∞







, (2.1)

is Banach space under the normkfk= R

Ω|f(x)|^pdx1/p

.

Proposition 2.1.1 ( [8, p. 170]). In finite dimensions, every normed vector space is a Banach space.

Definition 2.1.6( [8, p. 23]). LetM be a Banach space. A subspaceAofM is said to be denseinM if A = M, whereA denotes the closure ofA inM. That is, for an element a∈M, there exists a sequence inAthat converges toa.

Proposition 2.1.2( [8, p. 57]). A Banach spaceM isseparableif there exist a countable subsetA ⊂M which is dense inM.

Definition 2.1.7 ( [8, p. 195]). An inner product on a complex vector space H is a complex-valued function h·,·ion the Cartesian product H×H satisfying the following properties: for allξ, η, ξ₁, ξ₂ ∈H, and α₁, α₂ ∈C

1. hξ, ξi ≥0,

2. hξ, ξi= 0⇔ξ = 0,

3. hα₁ξ₁+α₂ξ₂, ηi=α₁hξ₁, ηi+α₂hξ₂, ηi, 4. hξ, ηi=hη, ξi.

The following lemma gives the Cauchy-Schwartz inequality in a complex vector space.

Its proof can be found in [8, p. 196].

Lemma 2.1.1. LetHbe a complex vector space endowed with an inner producth·,·i. For allξ, η ∈H, we have

|hξ, ηi| ≤ hξ, ξi¹²hη, ηi¹².

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Proposition 2.1.3( [8, p. 196]). Ifh., .iis an inner product on a vector spaceH, then

kξk=hξ, ξi¹² (2.2)

defines a norm onH.

Proof. The three first conditions of Definition 2.1.4 are immediate by using Definition 2.1.7. Now letξ, η ∈H, using the fact thathξ, ηi+hη, ξi= 2 Re(hξ, ηi), we have

kξ+ηk² =hξ+η, ξ+ηi

=kξk²+ 2 R_e(hξ, ηi) +kηk²

≤ kξk²+ 2|hξ, ηi|+kηk²

≤ kξk²+ 2kξk kηk+kηk² by using Lemma2.1.1

= (kξk+kηk)².

Hence (2.2) defines a norm onH.

Definition 2.1.8. A vector spaceHendowed with an inner product is calledinner product space.

Definition 2.1.9( [8, p. 197]). AHilbert spaceis an inner product space such thatH is a Banach space with respect to the normkξk=hξ, ξi¹².

Forp= 2, the Banach spaceL²(Ω)defined in Equation (2.1) is an Hilbert space with the norm induce by the inner product

hf, gi= Z

Ω

f(x)g(x)dx.

Definition 2.1.10( [8, p. 221]). LetBbe a Banach space. Thedual spaceofX denoted byB^∗, is the space of all continuous linear functionals onX. That is,

B^∗ ={T :B→C, T is continuous and linear}.

Proposition 2.1.4 ( [8, p. 221]). The dual space of a Banach space is also a Banach space.

Let us denote byB^∗∗ the dual space ofB^∗, we say that the Banach spaceBisreflexiveif Bis the dual of its dual, that is, the applicationj :B→B^∗∗define by

j(ξ)(Φ) = Φ(ξ), ∀ξ∈B, ∀Φ∈B^∗

is surjective.

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Definition 2.1.11( [8, p. 202]). LetH be a Hilbert space. A family(e_n)of elements ofH is anorthonormal basisofHif

1. ke_nk= 1, ∀n,

2. he_n, e_mi= 0, ∀n, m, n6=m;

3. The linear span of the family(e_n)is dense inH.

If(en)is an orthonormal basis ofH, then everyu∈Hcan be written as u=

∞

X

n=1

hu, e_nie_n, with kuk² =

∞

X

n=1

|hu, e_ni|².

Theorem 2.1.1( [8, p. 203]). Every separable Hilbert space admits an orthonormal basis.

2.2 Sobolev spaces

We introduce some function spaces calledSobolev spaceswhich have a great importance in the development of partial differential equations. To know more about these spaces, the reader can refer to Golse [12] and Brezis [10].

Definition 2.2.1( [10, p. 157]). LetΩbe an open set ofR^d. Let1≤p≤ ∞andm∈N. The linear spaceW^m,p(Ω)defined as

W^m,p(Ω) =

u∈L^p(Ω) ; D^αu∈L^p(Ω), ∀α∈N^d:|α| ≤m ,

where

D^α =

∂^|α|

∂x^α1¹ · · ·∂x^αN^d

is calledSobolev space. Here|α|=α₁+· · ·+α_d.

The spaceW^m,p(Ω)is a Banach space with respect to the norm

kuk_Wm,p(Ω) =









 P

|α|≤m

kD^αuk^p_Lp(Ω)

!¹_p

, if1≤p <∞

|α|≤mmaxkD^αuk_L∞(Ω), if p= +∞

(2.3)

We denote byW₀^m,p(Ω)the closure ofD(Ω)inW^m,p(Ω), for allm∈Nand1≤p <∞, where D(Ω) is the space of test functions, that is, the space of infinitely differentiable functions with compact support on Ω. If p = 2, we denote W^m,2(Ω) = H^m(Ω) and analogously,H₀^m(Ω) is the closure ofD(Ω)inH^m(Ω). For m ∈ Nand1 ≤ p < ∞the

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space

W^−m,p⁰(Ω) =







u∈ D⁰(Ω), u= X

|α|≤m

D^αu^α, u^α ∈L^p(Ω)







defines the dual space of W₀^m,p(Ω), with ¹_p + _p¹0 = 1, where D⁰(Ω) is the space of distribution on D(Ω), that is, the space of continuous linear forms on D(Ω). The space W^−m,p⁰(Ω)is a Banach space with the norm

kuk_W−m,p0

(Ω)(f) = sup

kvk_{W m,p(Ω)}=1

hu, vi

We can also define a Sobolev space of non-integer order using Fourier transform. Let us first recall the definition of Fourier transform.

Let us denote byC^∞(R^d)the space of infinitely differentiable functions.

Definition 2.2.2( [12, p. 150]). TheSchwartzspaceS(R^d)is the space ofC^∞(R^d)functions decreasing rapidly with all its derivatives, that is,

S R^d

=

u∈C^∞ :α, β ∈N^d, sup

x∈R^d

x^β|D^αu(x)|

<+∞

,

for all multi-indicesαandβ.

The functionu(x) = exp(−a|x|²), witha >0belongs to theSchwartzspaceS(R^d).

Definition 2.2.3 ( [12, p. 162]). A tempered distribution on R^d is a continuous linear form onS(R^d).

We denote the dual space of S(R^d)by S⁰(R^d) the space of continuous linear forms on S(R^d).

Definition 2.2.4( [12, p. 154]). The Fourier Transform ofu ∈S⁰(R^d)is the distribution Fudefined by

hFu, vi=hu,Fvi, for allv ∈S(R^d).

Definition 2.2.5( [12, p. 160]). Lets∈ R. We denote byH^s R^d

the space of functions u∈S⁰ R^d

such that

Z

R^d

1 +|ξ|²^s

|Fu(ξ)|²dξ < ∞.

For allu, v ∈H^s R^d

, the inner product is defined by (u, v)_Hs(R^d) =

Z

R^d

1 +|ξ|²^s

Fu(ξ)Fv(ξ)dξ.

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The associated norm is

kuk_Hs(R^d) =



 Z

R^d

1 +|ξ|²^s

|Fu(ξ)|²dξ





1 2

, u∈H^s R^d .

Proposition 2.2.1 ( [10, p. 125]). Lets ∈ RandΩ ⊂ C^d. The Sobolev spaceH^s(Ω) is an reflexive and separable Hilbert space.

2.3 Probability measures

We recall here some measure theoretic concepts that will help to define a Gaussian measure.

Definition 2.3.1 ( [8, p. 77]). LetΩbe an arbitrary non-empty set. Let Σ be a class of subsets ofΩ, that is,Σ⊆2^Ω. The classΣis calledσ-algebraif the following conditions are satisfied

1. Ω∈Σ

2. ifA ∈Σ, thenΩ\A∈Σ

3. ifA_i ∈Σ, fori= 1,2, . . . , then ^∞∪

i=1A_i ∈Σ.

Let a functionP : Σ→[0,1]and{A_i}^∞_i=1 ⊆Σbe a countable collection of disjoint sets.

ThenP is calledprobability measureif the following conditions are satisfied 1. P ∞

i=1∪ A_i

=

∞

P

i=1

P (A_i), and

2. P (Ω) = 1.

The triple(Ω,Σ, P)is calledprobability space, whereΣis aσ-algebra andP is a probability measure.

Remark 2.3.1. • The couple(Ω,Σ)is calledmeasurable space.

• The triple(Ω,Σ, P)is said to becompleteprobability space if for allB ∈Σ, with P (B) = 0and for allA⊂B one hasA∈Σ.

Definition 2.3.2 ( [8, p. 78]). Let C₀ be the collection of all open sets inΩ. The Borel σ-algebrais theσ-algebra generated byC₀and it is denoted byB(Ω).

An element ofB(Ω)is called aBorel measurable set, or simplyBorel set.

Definition 2.3.3 ( [13, p. 42]). Let (Ω,Σ, P) be a probability space and (E,Π) be a measurable space. Arandom variableis a functionX : Ω→E such thatX⁻¹(A)∈Σ, for allA∈Π.

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In general, if we takeE to be a Banach space andΠ = B(E), then in this caseX is an E-valued random variable.

A familyX = {X(t), t∈I}of random variables is calledrandom field, whereI is the index set.

Definition 2.3.4( [13, p. 1]). LetE(Ω)be the Borelσ-algebra. A measureµ:E(Ω)→R is calledBorel measure.

IfP is a Borel probability measure on(R,B(R)), thenP is Gaussian if and only if P(B) =

Z

B

√1

2πσexp

−(x−m)² 2σ²

dx, for all Borel setsB.

The parametersmandσ² aremeanandvarianceofP, respectively.

Remark 2.3.2. • Ifm = 0, themP is said to be centered or symmetric.

• Gaussian distribution are also called normal.

Definition 2.3.5( [13, p. 203]). Let(Ω,Σ, P)be a probability space. A random variable on(Ω,Σ, P)is Gaussian if its distribution is Gaussian, that is,

P(X ∈B) = Z

B

√1

2πσexp

−(x−m)² 2σ²

dx, for all Borel setsB.

A Gaussian random variable with a centered or symmetric distribution is called centered or symmetric. We can also define a Gaussian random variable on an infinite dimensional Banach spaceB.

Let us denote byB^∗ the topological dual ofBandh·,·i

B,B^∗ the duality betweenBandB^∗. Definition 2.3.6 ( [7, p. 5]). Let B be a separable Banach space and let P be a Borel probability measure on B. ThenP is Gaussian if and only if for each continuous linear functionalλ ∈B^∗, thepush-forwardmeasure

µ_λ =P ◦λ⁻¹ (2.4)

is Gaussian onR.

Definition 2.3.7( [7, p. 5]). LetXbe aB-valued random variable. Xis called Gaussian if and only ifhX, b^∗i

B,B^∗is Gaussian for allb^∗ ∈B^∗.

For sake of simplicity, let us take B to be a reflexive separable Banach space. We will adopt the following notations for the mean and the covariance operator of a B-valued

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random variableX. Let us denote bymthe mean ofX, that is, hm, b^∗i

B,B^∗ =EhX, b^∗i

B,B^∗ ∀b^∗ ∈B^∗

and we denote byCX the covariance operator ofX defined fromB^∗ toBsuch that hC_Xb^∗, b^∗i

B,B^∗ =EhX−m, b^∗i

B,B^∗hX−m, b^∗i

B,B^∗ ∀b^∗ ∈B^∗. Here

E[f] = Z

Ω

f(w)dP(w).

2.4 The Cameron-Martin space

Definition 2.4.1( [7, p. 5]). LetBbe a separable reflexive Banach space and letX be a GaussianB-valued random variable with zero mean and covariance operatorC_X which is non trivial. Set

kb^∗k_µ

X =q

hC_Xb^∗, b^∗i

B,B^∗, ∀b^∗ ∈B^∗ and we denote byB^∗µX the closure ofB^∗ in the normk·k_µ

X. Let us recall the definition ofµ_X-measurable linear functional.

Definition 2.4.2( [13, p. 80]). Letµ_X be a Gaussian measure on a Banach space B. A functionf onBis calledµ-measurable linear functional on(B, µX)if there exist a linear subspaceB¹of full measure and linear in the usual sense functionf0onB¹that coincides withf µ_X-almost surely.

If aµ_X-measurable linear functionalf : B → Ris defined in such a way thatf₀ will be linear on all ofB, then such a version off is called proper linear version.

Proposition 2.4.1 ( [13, p. 80]). Let µ_X be a centered Gaussian measure on B. If f is µ_X-measurable linear functional onX, thenf is a centered Gaussian random variable.

Through this work, we denote the elements of B^∗µX by fˆwhich can be identified with µ_X-measurable linear functionals onB.

Lemma 2.4.1. EveryfˆinB^∗µX can be identified with theµ_X-measurable linear functional b 7→fb(b)that are Gaussian random variables with zero mean on the space(B,B, µ_X).

Moreover, the covariance

f ,bbg

µ_X = Z

fb(b)bg(b)µ_X(db),

where f ,b bg ∈ B^∗µX, defines an inner product on B^∗µX and the corresponding norm is

f ,bfb

µX

= fb

2 µX

.

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Theorem 2.4.1( [1, p. 10]). (Fernique)

LetBbe a reflexive separable Banach space, and letX be aB-valued Gaussian random variable. There exists a constantα >0such thatE

exp αkXk²

<∞.

The covariance operator C_X : B^∗ 7→ B can be extended to a continuous mapping from B^∗µX to B. In order to show this, we use the fact thatEkXk^p < ∞ for all1 ≤ p < ∞ which is a consequence of Fernique theorem. Let us define a mapping onB^∗µX ×B^∗by

f ,b˜b^∗

7→Ef(X)b D X,˜b^∗E

B,B^∗

.

It is clear that this mapping is bilinear. Let us show that it is bounded. Using Cauchy- Schwartz inequality, we have:

|Efb(X)D X,˜b^∗E

B,B^∗

| ≤

Ef(X)b ²¹₂

EkXk²

B

˜b^∗

2 B^∗

¹₂

≤C fb

_µ

X

˜b^∗ B^∗

. (2.5) Since B is reflexive, then the covariance operator C_X can be extended to a continuous mapping also denoted byC_X fromB^∗µX toB. That is,

D

C_Xf , bb ^∗E

B,B^∗

=Efb(X)hX, b^∗i

B,B^∗, ∀fb∈B^∗µX, ∀b^∗ ∈B^∗.

Definition 2.4.3( [13, p. 60]). LetBbe a separable reflexive Banach space andX be a Gaussian B-valued zero mean random field with covariance operator C_X : B^∗µX → B. The Cameron-Martin space ofX is defined as the set

HµX =C_X(B^∗µ_X).

The inner product onHµX is defined as (f, g)_µ

X = Z

f(b)b bg(b)µ_X(db),

and the corresponding norm is

(f, f)_µ

X =kfk²_µ

X. For allf ∈HµX orfb∈B^∗µX, we denoteC_Xfb=f.

We have the following theorem that gives a fundamental property of the Cameron-Martin space.

Theorem 2.4.2 ( [13, p. 102]). Let X be a Gaussian random variable. The Cameron- Martin space and the spaceB^∗µX of all measurable linear functional onBare separable Hilbert spaces.

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2.5 White noise

Now, we are going to give the definition and some properties of Gaussian white noise.

Definition 2.5.1( [7, p. 102]). LetD⊂R^dandB(D)be Borel sets ofD. We definewhite noiseW as a collection{W(A) :A∈B}of random variables with zero mean and finite covarianceE[W(A)W(B)] =A∩B, for allA , B ∈B(D).

The white noise W is Gaussian if each random variable has a normal distribution. The way to construct functional is through stochastic integral

W(A) = Z

1_A(x)dW_x, (2.6)

with respect tod-dimensional field which is actually a Gaussian field with zero mean and covariance

E[W_xW_y] = min(x₁, y₁)· · ·min(x_d, y_d), ∀x y∈R^d.

We can define white noise functional by replacing characteristic functions1_Ain Equation (2.6) by functionsφ∈L²(D)such that

W(φ) = Z

φ(x)dWx. (2.7)

White noise defined by Equation (2.7) is a Gaussian random variable with zero mean and variance

EW(φ)W(φ) = E

|W(φ)|²

=E



 Z

D

φ(x)dW_x

2



=E



 Z

D

|φ(x)|²dx





=kφk²_L2(D).

In this work, we consider a version of white noise which actually has a representation W =

∞

X

k=1

W_ke_k, (2.8)

whereW_kare Gaussian random variable such thatW_k∼ N(0,1), and(e_k)is an orthonormal basis ofL²(D).

We are going to prove some properties of white noise which actually has representation

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(2.8). Let(e_k)be an orthonormal basis of L²(D)andW = P∞

k=1W_ke_k. By Proposition 2.2.1, L²(D) is a separable reflexive Banach space and the covariance operator of W denoted byC_W maps fromL²(D)toL²(D). Moreover, we have

hC_Wφ, φi_L² =EhW, φihW, φi=kφk²_L2

and by Definition 2.4.3, it follows that the Cameron-Martin space of the white noiseW is L²(D).

Proposition 2.5.1. LetW be as in Equation(2.8). The following conditions are satisfied:

1. W has zero mean and covariance

EhW, φihW, ψi=hφ, ψi_L², for allφ, ψ∈L². 2. Letφ ∈L², we have

Eexp(ihW, φi) = exp

−1 2kφk²

.

Proof. The random variableW has zero mean, sinceWk∼ N(0,1)and we have

EhW, φi=E

∞

X

k=1

W_kφ_k =

∞

X

k=1

EW_kφ_k = 0.

Forφ, ψ∈L², we have

EhW, φi hW, ψi=E

"* _∞ X

k=1

W_ke_k, φ

+ * _∞ X

k=1

W_ke_k, ψ +#

=E

" _∞ X

k=1

∞

X

l=1

he_k, φi he_l, ψiW_kW_l

!#

=

∞

X

k=1

he_k, φi he_k, ψiEW_k² =hφ, ψi.

SinceW_khas Gaussian distribution, we have:

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Eexp(ihW, φi) =

∞

Y

k=1

Eexp (ihW_ke_k, φi)

=

∞

Y

k=1

∞

Z

−∞

√1

2πexp

−x²_k 2

exp (ihx_ke_k, φi)dx_k

=

∞

Y

k=1

∞

Z

−∞

√1

2πexp

−x²

2 +ihx_ke_k, φi

dx_k

=

∞

Y

k=1

∞

Z

−∞

√1

2πexp

−x²_k

2 +ihek, φixk

dxk

=

∞

Y

k=1

∞

Z

−∞

√1

2πexp

−x²_k

2 +i(x_kφ_k) + 1

2(he_k, φi)²− 1

2(he_k, φi)²

dx_k

=

∞

Y

k=1

∞

Z

−∞

√1

2πexp (x_k− he_k, φi)² exp

−1 2φ²_k

dx_k

=

∞

Y

k=1

exp

−1 2φ²_k

Z^∞

−∞

√1

2π exp (x_k− he_k, φi)² dx_k

= exp

−1 2kφk²

,

we have thus the proof.

Proposition 2.5.2. The white noiseW is aH⁻^d²⁻(D)-valued Gaussian random variable.

Proof. We show the claim forD= [0,1]^d. Let us show thatW belongs toH⁻^d²⁻, for all >0according to Definition 2.2.5. LetW_ebe an extension ofW onR^dsuch that

W_e=

∞

X

k=1

W_ke_k1_[0,1]^d.

Let us denote by FWe the Fourier transform ofW. We have thatFWe ∈ S⁰. Now we just need to show that

Z

R^d

1 +|ξ|²s

|FW(ξ)|²dξ <∞.

By the definition of Fourier transform, we have

FW(ξ) = hW_e,exp(−iξ·x)i,

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and Z

R^d

1 +|ξ|²s

|FW(ξ)|²dξ = Z

R^d

1 +|ξ|²s

|hW_e,exp(−iξ·x)i|²dξ.

So, taking the expectation of this, we have

E



 Z

R^d

1 +|ξ|²s

|FW(ξ)|²dξ



=E



 Z

R^d

1 +|ξ|²s

|hW_e,exp(−iξ·x)i|²dξ





= Z

R^d

1 +|ξ|²^s E

|hW_e,exp(−iξ·x)i|² dξ

= Z

R^d

1 +|ξ|²^s Z

[0,1]^d

exp(−iξ·x) exp(iξ·x)dξdx

= Z

R^d

1 +|ξ|²s

dξ < C

∞

Z

0

1 +|r|²s

r^d−1dr

< C⁰

∞

Z

1

r^2s+d−1dr <+∞.

So whens <−^d₂, we can deduce Z

R^d

1 +|ξ|²^s

|FW(ξ)|²dξ < +∞

henceW_e belongs to Sobolev spaceH^s, and in particular toH⁻^d²⁻, for all > 0. Us- ing Pettis measurability theorem (see [13]), we deduce that W is a H⁻^d²⁻(D)-valued Gaussian random variable.

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3 EXISTENCE AND UNIQUENESS OF SOLUTION

Our main objective in this chapter is to show the existence and uniqueness of a Gaussian random fieldXsatisfying the following partial differential equation with Gaussian white noise

LX =W. (3.1)

3.1 Presentation of the equation

LetD ⊂R^dbe a open bounded domain. We define the differential operatorLonDby

L:=−∆ +ν², (3.2)

where ∆ is the Laplacian operator defined for any function φ : D → R as ∆φ = Pd

i=1

∂²φ

∂x²_i, andν > 0is a constant.

Definition 3.1.1( [8, p. 216]). LetHbe an Hilbert space. A linear operatorL:H →H isboundedif there existsM >0such that

kLφkH ≤MkφkH.

Definition 3.1.2( [8, p. 331]). LetH be an Hilbert space and h·,·ibe the inner product inH. A bounded linear operatorL:H →H is said to beself-adjointif

hLφ, ψi=hφ, Lψi, for allφ, ψ ∈H.

Definition 3.1.3( [8, p. 284]). LetHbe an Hilbert space. A linear operatorL:H →H iscompactif for every bounded sequence(u_k)k≥1inH, the image sequence(Lu_k)k≥1has a convergent subsequence.

Theorem 3.1.1( [8, p. 348]). (Spectral Theorem)

LetV be an infinite dimensional, real Hilbert space. LetA:V →V be a continuous linear, self-adjoint, compact operator. Then the eigenvalues ofAform a decreasing sequence (µ_k)k≥1of strictly positive reals which tends to zero and there exist an orthonormal basis (ψ_k)k≥1of eigenfunctions ofA.

Theorem 3.1.2( [14, p. 7]). (Lax-Milgram)

LetH be a Hilbert space and letabe a bilinear form onH×Hsuch that 1. ais bounded onH, that is,

|a(φ, ψ)| ≤Ckφk_Hkψk_H, for all φ, ψ∈H,

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2. aiscoercive, that is, there existc >0such that

|a(φ, φ)| ≥ckφk_H, for all φ∈H.

Then for anyf ∈H^∗, there exist a unique solutionφ ∈Hsuch that a(φ, ψ) = chf, φi_H, for all ψ ∈H.

Theorem 3.1.3( [10, p. 162]). (Poincaré Inequality)

LetD⊂R^dbe an open bounded domain. Then there exists a constantC_D, depending on Dandpsuch that, for allφ∈H₀¹(D)

kφk_L²_(D) ≤Ck∇φk_L²_(D).

Theorem 3.1.4( [10, p. 165]). (Rellich)

Let D ⊂ R^d be a bounded domain. The inclusion map form H₀¹(D) into L²(D) is a compact operator.

Now, let D ∈ R^d be an open bounded domain. We consider the following variational eigenvalue problem that consist of findingλ∈Randφ∈H₀¹(D)− {0}such that

a(φ, ψ) =λhφ, ψi_L², for all ψ ∈H₀¹(D), (3.3) whereais a bilinear symmetric form given by

a(φ, ψ) = Z

D

∇φ(x)∇ψ(x)dx+ν² Z

D

φ(x)ψ(x))dx.

Using Green’s formula, the above problem can be reformulated into the following

Lφ=λφ, inD, with φ= 0, on∂D (3.4)

that is λ and φ are eigenvalue and eigenfunction of L respectively. The solutions of Equation (3.3) can be given by the following theorem.

Theorem 3.1.5. The eigenvalues of Equation(3.3)form an increasing sequence(λk)k≥1

of positive reals which tends to infinity, and there exists an orthonormal basis(φ_k)_k≥1 of L²(D)of associated eigenvectors, that is,

φk∈H₀¹(D) and a(φk, ψ) =λkhφk, ψi_L², for all ψ ∈H₀¹(D). (3.5)

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Proof. Letf ∈L²(D). We define the following problem

Findφ ∈H₀¹(D)such that a(φ, ψ) = hf, ψi_L², for all ψ ∈H₀¹(D). (3.6) Using Theorem 3.1.2 we show that Equation (3.6) admits a unique solutionφ ∈ H₀¹(D).

If we takeψ =φin Equation (3.6), we have

φ∈H₀¹(D) such that a(φ, φ) = hf, φi_L², for all φ ∈H₀¹(D).

Set

T φ =hf, φi_L².

The bilinear formais continuous and bounded inH¹. Moreoverais coercive since

|a(φ, φ)|=|T φ|=|hf, φi_L²|= Z

D

f φdx

≤ kfk_L²kφk_L² ≤Ckfk_L²kφk_H¹

0,

since the embedding ofH₀¹(D)intoL²(D)is continuous by Rellich Theorem. It follows thatT andaare continuous. Furthermore, using Poincaré Inequality we have

|a(φ, φ)|= Z

D

|∇φ|²dx+ν² Z

D

|Af|²dx=k∇φk²_L2 +ν²kφk²_L2

≥ k∇φk²_L2 ≥ckφk²_H1 with c >0.

Henceais coercive. Let us denote byφ =Af the unique solution of (3.6), whereAis a linear operator defined fromL²(D)intoH₀¹(D)as follows:

Af ∈H₀¹(D), such that a(Af, ψ) = hf, ψi_L², for all ψ ∈H₀¹(D). (3.7) Since the operator A : L²(D) → H₀¹(D)and by Rellich Theorem H₀¹(D) is compactly embedded into L²(D), then it follows that the operator A : L²(D) → L²(D) and is compact. To show thatAis self-adjoint, let us takeψ =Af in Equation (3.7). Then we have

hf, Agi_L² =a(Af, Ag) = a(Ag, Af) =hg, Afi_L²

sinceais symmetric. Since all the hypothesis of Theorem 3.1.1 are satisfied, we can now apply it and we have that the eigenvalues of A form a decreasing sequence (µ_k)k≥1 of strictly positive reals which tends to zero and there exist an orthonormal basis(ψ_k)k≥1 of

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eigenfunctions ofA, with

Aψ_k=µ_kψ_k.

Coming back to Equation (3.3), we have

a(φ, ψ) =λhφ, ψi_L² =λa(Aφ, ψ), for all ψ ∈H₀¹(D), so,

a(φ−λAφ, ψ) = 0 ⇒φ=λAφ⇒Aφ= 1 λφ.

Hence the eigenvalues(λ_k)k≥1 from Equation (3.3) form an increasing sequence consist- ing of the inverses of eigenvalues(µ_k)k≥1 ofA. The eigenfunctions are exactly the same as(φ_k)k≥1 and form an orthonormal basis ofL²(D). Thus, the theorem is proved.

We have then shown that Ladmits an orthonormal basis of eigenfunctions, which form a complete orthonormal set ofL²(D). This complete set of orthonormal eigenfunctions form a basis for the infinite dimensional Hilbert space. Let us denote the eigenfunctions of Lbye_kunder Dirichlet conditions, and let us denote byλ_kthe corresponding eigenvalues.

ThenLmust satisfy the following problem

Le_k =λ_ke_k.

Next, we consider the Gaussian white noise W that we have presented in the previous chapter which has the following representation

W =

∞

X

k=1

W_ke_k, (3.8)

where W_k ∼ N(0,1) are independent, and (e_k) is a complete orthonormal basis in a Hilbert space.

Based on all what we have defined so far, we are now able to reformulate our main objective and thus we have the following problem:

Problem 3.1.1. Show the existence and uniqueness of a Gaussian random fieldXwhich is solution of the equation

LX =W,

in the eigenbasis ofL.

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3.2 Existence and uniqueness

In this subsection, we are interested solving Problem 3.1.1. We are going to solve the problem in dimensions one and two.

Lemma 3.2.1. LetD ⊂R^dbe a Lipschitz domain. LetHµX be the Cameron-Martin space of a zero meanH^−r-valued Gaussian random variableX, such thatHµX is continuously embedded into H¹(D). Then X has realizations in H¹⁻

d

2−(D) almost surely, for all >0.

The following result shows the existence of a unique random field solution of Problem 3.1.1.

Theorem 3.2.1. LetD ⊂ R be a bounded open interval. LetW be the Gaussian white noise define in Equation (3.8). Then there exists a zero mean H^−r(D)-valued Gaussian random fieldX, withr > ¹₂ −1 = ¹₂ such that

1. The Cameron-Martin space of X denoted by HµX(D) is continuously embedded intoH¹(D)and for allh∈HµX(D)we have∆h∈L²(D).

2. The operator L defined in Equation (3.2) has eigenspace representation and the solution of Problem(3.1.1)and can be represented as eigenspace expansion.

Moreover,H^−r(D)-valued Gaussian random fieldX is unique.

Proof. We prove the theorem for dimensionsd = 1,2. The prove can be done similarly for dimension for d ≥ 3. Let D = (0,1) ⊂ R. We have shown that L admits an orthonormal basis of eigenfunctions(e_k)andLmust satisfy the following problem:

Le_k =λ_ke_k.

Let us consider the case of Dirichlet boundary condition, the eigenvalues and eigenfunctions ofLcan be computed as follow. We know that

Le_k(x) = (−∆ +ν²)e_k(x) =λ_ke_k(x) x∈D= (0,1).

So

∆e_k(x) +αe_k(x) = 0, with α=λ−ν² >0, (3.9) since under Dirichlet boundary condition, the Equation (3.9) has solution only whenα >

0. Letα =b², the characteristic equation is

r²+b² = 0⇒r=±ib.

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Figure 1.Dirichlet eigenfunctions of operatorLin dimension one.

The general solution is given by

e_k(x) =c₁exp(ax) cos(bx) +c₂exp(ax) sin(bx),

whereais the real part ofr. Applying boundary conditione_k(0) =e_k(1) = 0we obtain e_k(x) = sin(kπx) and b_k=kπ, kinteger.

Hence we deduce that eigenfunctions and eigenvalues ofLare defined by:

e_k(x) =√

2 sin(kπx) and λ_k= (kπ)²+ν², kinteger andν > 0. (3.10) Eigenfunctions of operator L in dimension one for fixed value of parameter ν are presented in Figure 1.

We consider the expansion ofXin term of normed eigenfunction as X(x) =

+∞

X

k=1

X_ke_k(x), (3.11)

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and we writeLin its eigenspace as LX(x) =

+∞

X

k=1

X_kLe_k(x) =

+∞

X

k=1

X_kλ_ke_k(x) =

+∞

X

k=1

√2X_k[(kπ)² +κ²] sin(kπx).

Furthermore, the Gaussian white noise defined by Equation (3.8) can also be represent in term of eigenfunctions as

W =

+∞

X

k=1

√

2·W_ksin(kπx).

We have

LX(x) =W(x)⇒

+∞

X

k=1

√

2X_k[(kπ)²+ν²] sin(kπx) =

∞

X

k=1

√

2W_ksin(kπx).

Since, eigenfunctionse_k(x) = sin(kπx),k integer are orthonormal inL²(D), taking the inner product with a fixed eigenfunctione_k_o(x), we have

X_k[(kπ)²+ν²] =W_k.

It follows that

X_k= W_k

(kπ)²+ν², (3.12)

which is well defined since ν > 0andW_k ∼ N(0,1). Substituting Equation (3.18) into Equation (3.11), we obtain

X(x) =

+∞

X

k=1

√2·W_k

(kπ)²+ν² sin(kπx) (3.13) In order to show the uniqueness of a Gaussian random fieldX, we assume that there exists two Gaussian random fieldsX₁andX₂ which are solutions of Problem 3.1.1. That is,

LX₁ =W andLX₂ =W.

Then we have

L(X₁−X₂) = 0, almost surely. (3.14) It is known from deterministic problem

LX =f, withf ∈L²(D)

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Figure 2.Gaussian random fieldXin dimension one.

that there exists a unique solution, for any function. Since0is a function ofL²(D), we then have the that the solutionX =X1−X2 of Equation (3.14) is unique. Then

X₁−X₂ = 0, thus X₁ =X₂ almost surely.

The following Figure 2 shows the zero mean Gaussian random field for a one dimensional Problem 3.1.1 and Figure 3 shows its covariance for a fixed value of parameterν.

We are now going to solve Problem 3.1.1 in two dimensional case. We mainly prove Theorem 3.2.1 by consideringD = (0,1)² ⊂ R². We recall that in dimension two, that Laplace operator is defined as

∆e_k(x, y) = ∂ek(x, y)

∂x² + ∂ek(x, y)

∂y² .

From the same idea as in dimension one, we have

∆ek(x, y) +αek(x, y) = 0, with α =λ−ν² >0, (x, y)∈D= (0,1)². (3.15)

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Figure 3. Covariance of a Gaussian random fieldXin dimension one.

Letting

e_k(x, y) =u_k(x)v_k(y),

plugging into Equation (3.15) and separating gives u⁰⁰_k

uk

+ v_k⁰⁰ vk

=γ,

which can be true only if each term on the left hand side is a constant. So we have u⁰⁰_k+γ_xu_k = 0 and v_k⁰⁰+γ_yv_k = 0,

withγx andγy constant. Solving these two equations as in dimension one with Dirichlet boundary conditions

uk(0) = 0 =uk(1) and vk(0) = 0 = vk(1), and using the fact that

ek(x, y) =uk(x)vk(y),

we get eigenfunctions

e_k(x, y) = 2·e_m_k_,l_k(x, y) = sin(m_kx) sin(l_ky),

and eigenvalues

γ_k =γ_m_k_,l_k = (m_kπ)²+ (l_kπ)²+ν².

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Figure 4. Dirichlet eigenvalues of operatorLin dimension two.

Figure 4 and Figure 5 show respectively the Dirichlet eigenvalues and eigenfunctions of operatorLin dimension two for fixed value of parameterν.

We consider the expansion ofXin term of eigenfunctions as X(x, y) =

+∞

X

k=1

X_ke_k(x, y), (3.16)

and we writeLin its eigenspace as LX(x, y) =

+∞

X

k=1

X_kLe_k(x, y) =

+∞

X

k=1

X_kλ_ke_k(x, y)

=

+∞

X

k=1

2·Xk[(mkπ)²+ (lkπ)²+ν²] sin(mkπx) sin(lkπ).

Furthermore, the Gaussian white noise representation defined by (3.8) in dimension two is

W =

+∞

X

k=1

2·W_ksin(m_kπx) sin(l_kπy).

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Figure 5. Dirichlet eigenfunctions of operatorLin dimension two.

We have

LX(x, y) = W(x, y),

which implies

+∞

X

k=−∞

Xk[(mkπ)²+ (lkπ)²+ν²] sin(mkπx) sin(lkπy) =

∞

X

k=1

Wksin(mkπx) sin(lkπy).

(3.17) Since, eigenfunctionse_k(x, y) = sin(m_kπx) sin(l_kπx), m_k and l_k integer, are orthonormal inL²(D), taking the inner product with a fixed eigenfunctione_k_o(x, y), we have

X_k[(m_kπ)²+ (l_kπ)²+ν²] =W_k,

It follows that

X_k = W_k

(mkπ)²+ (lkπ)²+ν², (3.18) which is well defined since ν > 0andW_k ∼ N(0,1). Substituting Equation (3.18) into Equation (3.16), we obtain the solution

X(x, y) =

+∞

X

k=−∞

2·W_k

(m_kπ)²+ (l_kπ)²+ν² sin(m_kπx) sin(l_kπy). (3.19)

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Figure 6. Gaussian random fieldXin dimension two.

Figure 7.Covariance of a Gaussian random fieldXin dimension two.

By the same way as in dimension one, we prove that the Gaussian random field X is unique. The zero mean Gaussian random field solution of Problem 3.1.1 and its covariance are illustrated in the following Figure 7 for fixed value of parameterν.

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4 BAYESIAN ESTIMATION

In this section, we develop the Bayesian estimation approach that arises in Bayesian ap- proximation for inverse problems. We follow the same approach as in [4, 15, 16]. The Bayesian approach to inverse problem consist of determined the unknown from the mathematical model and finite-dimensional noisy measurement.

In the Bayesian approach, the solution of the inverse problem isposterior distribution and it is determined with the knowledge of the measurement, the mathematical model and the prior distribution. The posterior distribution gives us the degree of confidence that any candidate solution might be the true function. Theprior contains our beliefs about the unknown that we wish to update with the help of new information.

We start our discussion by considering the linear inverse problem that can be modelled in finite dimensions as

Y =GX+E, (4.1)

where, Y ∈ R^M is a finite dimensional vector of measurement, the functionG : R^N → R^M is an operator mapping the unknown Gaussian random variable X ∈ R^M into the measurement in the absence of measurement noise. We suppose that the measurement noiseE is independent of the unknownXand such thatE ∼ N(0, C), whereCis its the covariance matrix.

4.1 Bayes theorem

We recall that the Bayes formula from probability theory, that gives the conditional probability of an assumptionAknowing assumptionB. LetP be a probability measure given in Definition 2.3.1. LetAbe the assumption andB be that observed measurements under assumptionA. Then the Bayes formula is given by

P(A|B) = P(B|A)P(A)

P(B) , withP(B)6= 0. (4.2) We are going to adapt this version of Bayes formula to give an expression of the posterior distribution of the unknownXin Equation (4.1). We will now denote withπa probability density function (pdf) of a distribution of a random vector.

Theorem 4.1.1. LetX be a Gaussian random field which has a known prior distribution π(X), let Y be a vector of measurements such thatπ(Y) >0. Then the posterior pdf of

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X given the measurementY is given by

π(X|Y) = π(X)π(Y|X) π(Y) .

Note thatπ(Y)does not depend onX and therefore can just be considered as a normal- ization constant. The posterior probability distribution becomes

π(X|Y)∝π(Y|X)π(X), (4.3)

where the symbol ” ∝ ” means "is proportional to". The probability of observing the measurement Y, given the random field X defines the likelihood. Based on Equation (4.1), the likelihood can be written as

π(Y|X)∝exp

−1

2(Y −GX)^TC⁻¹(Y −GX)

.

SinceX is a Gaussian random variable, we choose the prior distribution to be Gaussian with a given zero mean and covariance matrixΣ. The prior density is then written as:

π(X)∝exp

−1

2X^TΣ⁻¹X

.

It follows from Equation (4.3) that the unnormalized posterior distribution is:

π(X|Y)∝exp

−1

2(Y −GX)^TC⁻¹(Y −GX)−1

2X^TΣ⁻¹X

. (4.4)

Let’s simplify Equation (4.4) and write it in a form of a Gaussian probability distribution function. Let

Q:=−1

2(Y −GX)^TC⁻¹(Y −GX)− 1

2X^TΣ⁻¹X.

We have Q=−1

2(Y −GX)^TC⁻¹(Y −GX)−1

2X^TΣ⁻¹X

=−1

2Y^TC⁻¹Y +1

2X^TG^TC⁻¹Y + 1

2X^TG^TC⁻¹Y − 1

2X^TG^TC⁻¹GX −1

2X^TΣ⁻¹X

=−1

2Y^TC⁻¹Y +1

2X^TG^TC⁻¹Y + 1

2Y^TC⁻¹GX − 1

2X^T(G^TC⁻¹G+ Σ⁻¹)X If we denote by

C_post = (G^TC⁻¹G+ Σ⁻¹)⁻¹, (4.5)

Elliptic boundary values problems with Gaussian white noise