homogenization and other multi-scale problems in...

Homogenization and other multi-scale problemsin Bellman-Isaacs equations

Martino Bardi

Dipartimento di Matemetica Pura ed ApplicataUniversità di Padova

New Trends in Analysis and Control of Nonlinear PDEsRoma, June 13–15, 2011

Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 1 / 29

1 Homogenization of non-coercive Hamilton–Jacobi equations(joint work with G. Terrone)

I Some previous results

I Examples of non-homogenization

I Homogenization on subspaces

I Convex-concave eikonal equations and differential games

2 Optimal control with stochastic volatility(joint work with A. Cesaroni and L. Manca)

I Motivation: financial models

I Singular perturbations of H-J-B equations

I Merton portfolio optimization with stochastic volatility

1 Homogenization of non-coercive Hamilton–Jacobi equations(joint work with G. Terrone)

I Some previous results

I Examples of non-homogenization

I Homogenization on subspaces

I Convex-concave eikonal equations and differential games

2 Optimal control with stochastic volatility(joint work with A. Cesaroni and L. Manca)

I Motivation: financial models

I Singular perturbations of H-J-B equations

I Merton portfolio optimization with stochastic volatility

1. Homogenization of Hamilton–Jacobi Equations

Problem: let ε → 0+ in

uεt + H

(x , x

ε , Duε)

= 0 in (0, T )× RN

uε(0, x) = h(x , x

ZN− periodic case:

H(x , ξ + k , p) = H(x , ξ, p), h(x , ξ + k) = h(x , ξ) ∀k ∈ ZN .

Goal: find continuous effective Hamiltonian and initial data H, h s.t.

uε → u locally uniformly, and u solves

ut + H (x , Du) = 0 in (0, T )× RN

u(0, x) = h (x) .

Classical result for coercive H

Coercivity in p is

lim|p|→∞

H(x , ξ, p) = +∞ uniformly in x , ξ.

P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92:

The effective Hamiltonian H = H(x , p) is the unique constant suchthat the cell problem, with x , p frozen parameters,

H(x , ξ, Dξχ + p) = H, in RN ,

has a ZN−periodic (viscosity) solution χ(ξ), called the corrector;

if h = h(x) is independent of xε , uε → u locally uniformly.

Coercivity in p is

lim|p|→∞

Coercivity in p is

lim|p|→∞

The effective initial data

If the initial data h = h(x , xε ) must find the initial condition h(x).

O. Alvarez - M. B. (ARMA ’03):Assume there exists the recession function of H

H ′(x , ξ, p) := limλ→+∞

H(x , ξ, λp)

Consider, for frozen x , wt + H ′(x , ξ, Dξw) = 0, w(0, ξ) = h(x , ξ).

If limt→+∞ w(t , ξ) = constant =: h(x),

then limt→0 limε→0 uε = h(x).

In the coercive case h (x) = minξh(x , ξ).

H ′(x , ξ, p) := limλ→+∞

H(x , ξ, λp)

H ′(x , ξ, p) := limλ→+∞

H(x , ξ, λp)

Bellman-Isaacs equations

In control theory the Hamiltonian is, for A compact,

H(x , ξ, p) = maxa∈A

{−f (x , ξ, a) · p − l (x , ξ, a)} , ξ =xε

and in 0-sum differential games it is, for B compact,

H(x , ξ, p) = minb∈B

maxa∈A

{−f (x , ξ, a, b) · p − l (x , ξ, a, b)} .

The associated control system in an oscillating periodic medium is

x = f(

x ,xε, a, b

)and cost - payoff

x(s),x(s)

ε, a(s), b(s)

)ds + h

(x(t),

x(t)ε

)that player a wants to minimize and player b to maximize.Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 6 / 29

Bellman-Isaacs equations

In control theory the Hamiltonian is, for A compact,

H(x , ξ, p) = maxa∈A

{−f (x , ξ, a) · p − l (x , ξ, a)} , ξ =xε

and in 0-sum differential games it is, for B compact,

H(x , ξ, p) = minb∈B

maxa∈A

{−f (x , ξ, a, b) · p − l (x , ξ, a, b)} .

The associated control system in an oscillating periodic medium is

x = f(

x ,xε, a, b

)and cost - payoff

x(s),x(s)

ε, a(s), b(s)

)ds + h

(x(t),

x(t)ε

)that player a wants to minimize and player b to maximize.Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 6 / 29

Alternative conditions to coercivity

coercivity of H is equivalent to Small Time Local Controllability ina time proportional to the distance (by one of the players for anycontrol of the other),

it is a STRONG assumption, can be replaced by

Controllability in a uniformly Bounded Time (by one of the playersfor any control of the other) (M.B. - Alvarez Mem. AMS 2010)Example: hypoelliptic eikonal equation

uεt +

k∑i=1

)· Duε| = l

)in (0, T )× RN

with f1, ..., fk ∈ C∞ satisfying Hörmander bracket-generatingcondition.Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10for control problems, Cardaliaguet ’10 for games).

uεt +

k∑i=1

)· Duε| = l

)in (0, T )× RN

uεt +

k∑i=1

)· Duε| = l

)in (0, T )× RN

Examples of non-homogenization

Example 1

uεt + |uε

x + uεy |β = cos

(2π x−y

)x , y ∈ R (β ≥ 1)

uε(0, x , y) = 0

is solved by uε(t , x , y) = t cos(2π x−y

Example 2

uεt + |uε

x + uεy |β = 0 x , y ∈ R

uε(0, x , y) = cos(2π x−y

)is solved by uε(t , x , y) = cos

(2π x−y

). In both cases

uε has no limit as ε → 0 if x 6= y .

Example 3

uεt + |uε

x | − |uεy | = cos

(2π x−y

)x , y ∈ R

uε(0, x , y) = 0

is solved by uε(t , x , y) = t cos(2π x−y

Example 4

uεt + |uε

x | − |uεy | = 0 x , y ∈ R

uε(0, x , y) =(2π x−y

)is solved by uε(t , x , y) = cos

(2π x−y

uε has no limit as ε → 0 if x 6= y .

H vanishing in a direction

The previous examples are special cases of

Proposition

Assume ∃ p0 ∈ ZN \ {0} such that H(λp0) = 0 ∀ λ ∈ R in

uεt + H(Dzuε) = l

), uε(0, z) = h

If h = 0 ∃ smooth ZN−periodic l such that limε→0 uε @

if l = 0 ∃ smooth ZN−periodic h such that limε→0 uε @

The 1st statement is SHARP if H is 1-homogeneous and convex:

∀ p0 ∈ ZN \ {0} H(p0) 6= 0 is nonresonance condition

⇒ homogenization!

H vanishing in a direction

The previous examples are special cases of

Proposition

Assume ∃ p0 ∈ ZN \ {0} such that H(λp0) = 0 ∀ λ ∈ R in

uεt + H(Dzuε) = l

), uε(0, z) = h

If h = 0 ∃ smooth ZN−periodic l such that limε→0 uε @

if l = 0 ∃ smooth ZN−periodic h such that limε→0 uε @

The 1st statement is SHARP if H is 1-homogeneous and convex:

∀ p0 ∈ ZN \ {0} H(p0) 6= 0 is nonresonance condition

⇒ homogenization!

Homogenization on subspaces

If H satisfies the assumption of the negative Proposition, must restrictthe data l , h to oscillate only in directions where H does not vanish.

Let V M–dimensional subspace of RN , zV := ΠV (z) projection

uεt + H

ε, Dzuε

), uε(0, z) = h

)Assume coercivity in V : ∀ p ∈ RN

lim|p|→+∞, p∈V

H(θ, p + p) = +∞, uniformly in θ ∈ V .

Theorem

∃ H(z, p) continuous such that, ∀h(z, θ) periodic in θ, uε convergesuniformly on compacta of (0,+∞)× RN to u, unique solution of

ut + H(z, Du) = 0, u(0, z) = minθ∈V

h(z, θ)

Homogenization on subspaces

If H satisfies the assumption of the negative Proposition, must restrictthe data l , h to oscillate only in directions where H does not vanish.

Let V M–dimensional subspace of RN , zV := ΠV (z) projection

uεt + H

ε, Dzuε

), uε(0, z) = h

)Assume coercivity in V : ∀ p ∈ RN

lim|p|→+∞, p∈V

H(θ, p + p) = +∞, uniformly in θ ∈ V .

Theorem

∃ H(z, p) continuous such that, ∀h(z, θ) periodic in θ, uε convergesuniformly on compacta of (0,+∞)× RN to u, unique solution of

ut + H(z, Du) = 0, u(0, z) = minθ∈V

h(z, θ)

Related results: Barles 2007, Viterbo 2008 (by symplectic methods).

Example 1

uεt + |uε

x + γuεy | = l

(x , y , x−y

)x , y ∈ RN/2

uε(0, x , y) = h(x , y , x−y

)there is homogenization ∀ l , h periodic in θ = x−y

ε ⇐⇒ γ 6= 1.

γ = 1 by the Examples 1 and 2 of non-homogenization;

γ 6= 1 fits in the Thm. with V = {(q,−q) : q ∈ RN/2}.

Example 1

uεt + |uε

x + γuεy | = l

(x , y , x−y

)x , y ∈ RN/2

uε(0, x , y) = h(x , y , x−y

ε ⇐⇒ γ 6= 1.

Example 1

uεt + |uε

x + γuεy | = l

(x , y , x−y

)x , y ∈ RN/2

uε(0, x , y) = h(x , y , x−y

ε ⇐⇒ γ 6= 1.

Convex-concave eikonal equation

uεt + g1

(x , y , x−y

)|Dxuε| − g2

(x , y , x−y

)|Dyuε| = l(x , y , x−y

uε(0, x , y) = h(x , y , x−yε )

there is homogenization if

(g1 − g2)(x , y , θ) ≥ γo> 0 ∀ x , y , θ,

or, symmetrically, g2 − g1 ≥ γo.Proof: by the Thm. with V = {(q,−q) : q ∈ RN/2}.

Motivation: Pursuit-Evasion type differential games, with pursuercontrolling the x variables, evader the y variables, and cost dependingon the distance between them.

The result is sharp in the next

Example 2

uεt + |uε

x | − γ|uεy | = l

(x , y , x−y

)x , y ∈ RN/2

there is homogenization ∀ l , h periodic in θ = x−yε ⇐⇒ γ 6= 1.

γ = 1 by the Examples 3 and 4 of non-homogenization;γ 6= 1 fits in the preceding slide.

Other resultsby control-game methods we can treat problems with differentassumptions: instead of strongly interacting fast variables,decoupling of x

ε and yε , e.g., h or l has a saddle point in this

entries;

many problems on homogenization of convex-concave H-Jequations remain open!

Example 2

uεt + |uε

x | − γ|uεy | = l

(x , y , x−y

)x , y ∈ RN/2

entries;

Example 2

uεt + |uε

x | − γ|uεy | = l

(x , y , x−y

)x , y ∈ RN/2

entries;

2. Optimal control with stochastic volatility

In Financial Mathematics the evolution of a stock S is described by

d log Ss = γ ds + σ dWs

and the classical Black-Scholes formula for the option pricing problemis derived assuming the parameters are constants.

However the volatility σ is not really a constant, it rather looks like anergodic mean-reverting stochastic process, so it is often modeled asσ(ys) with ys an Ornstein-Uhlenbeck diffusion process.It is argued in the book

Fouque, Papanicolaou, Sircar: Derivatives in financial markets withstochastic volatility, 2000,

that the process ys also evolves on a faster time scale than the stockprices. Next picture shows the typical bursty behavior.

The model with fast stochastic volatility σ proposed in [FPS] is

d log Ss = γ ds + σ(ys) dWs

dys = 1ε (m − ys) + ν√

For the option pricing problem the limit is given by the Black-Scholesformula of the model with (constant) mean historical volatility

d log Ss = γ ds + σ dWs, σ2 =

σ2(y)e−(y−m)2/2ν2

√2πν2

the mean being w.r.t. the (Gaussian) invariant measure of theOrnstein-Uhlenbeck process driving ys.

Merton portfolio optimization problem

Invest βs in the stock Ss, 1− βs in a bond with interest rate r .Then the wealth xs evolves as

d xs = (r + (γ − r)βs)xs ds + xsβs σ dWs

and want to maximize the expected utility at time t , E [h(xt)],for some h increasing and concave.

If h(x) = axδ/δ with a > 0, 0 < δ < 1, a HARA function, and theparameters are constants the problem has an explicit solution.

The version with fast stochastic volatility is

d xs = (r + (γ − r)βs)xs ds + xsβs σ(ys) dWs

dys = 1ε (m − ys) + ν√

QUESTIONS:1. Is the limit as ε → 0 a Merton problem with constant volatility σ?

2. If so, is σ an average of σ(·) ?Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 18 / 29

Merton portfolio optimization problem

Invest βs in the stock Ss, 1− βs in a bond with interest rate r .Then the wealth xs evolves as

d xs = (r + (γ − r)βs)xs ds + xsβs σ dWs

and want to maximize the expected utility at time t , E [h(xt)],for some h increasing and concave.

If h(x) = axδ/δ with a > 0, 0 < δ < 1, a HARA function, and theparameters are constants the problem has an explicit solution.

The version with fast stochastic volatility is

d xs = (r + (γ − r)βs)xs ds + xsβs σ(ys) dWs

dys = 1ε (m − ys) + ν√

QUESTIONS:1. Is the limit as ε → 0 a Merton problem with constant volatility σ?

2. If so, is σ an average of σ(·) ?Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 18 / 29

By formal asymptotic expansions [FPS] argue that the answers are

1. Yes, the limit is a Merton problem with constant volatility σ2. σ is a harmonic average of σ

σ2(y)dµ(y)

)−1/2

NOT the linear average as in Black-Scholes!Practical consequence:if I have N empirical data σ1, ..., σN of the volatility, in the Black-Scholesformula for option pricing I estimate the volatility by the arithmetic mean

σ2a =

N∑i=1

σ2i ,

whereas in the Merton problem I use the harmonic mean

σ2h =

N∑i=1

≤σ2a.

σ2(y)dµ(y)

)−1/2

σ2a =

N∑i=1

σ2i ,

σ2h =

N∑i=1

≤σ2a.

σ2(y)dµ(y)

)−1/2

σ2a =

N∑i=1

σ2i ,

σ2h =

N∑i=1

≤σ2a.

Conclusion: the correct average seems to depend on the problem!

QUESTIONS:

Can the formula for the Merton problem be rigorously justified andwhat is the convergence as ε → 0?

Is there a unified "formula" for the two problems and for othersimilar problems?

Note: the Black-Scholes model has no control, the associated PDE isa heat-type linear equation;

Merton is a stochastic control problem whose Hamilton-Jacobi-Bellmanequation is fully nonlinear and degenerate parabolic.

I’ll present a result on two-scale Hamilton-Jacobi-Bellman equationsthat will answer these (and other) questions.

The H-J approach to Singular Perturbations

Control system

dxs = f (xs, ys, αs) ds + σ(xs, ys, αs) dWs, xs ∈ Rn, αs ∈ A,

dys = 1ε g(xs, ys) ds + 1√

εν(xs, ys) dWs, ys ∈ Rm,

x0 = x , y0 = y

Cost functional Jε(t , x , y , α) :=∫ t

0 l(xs, ys, αs) ds + h(xt , yt)

Value function

uε(t , x , y) := infα∈A(t)

E [Jε(t , x , y , α)]

A(t) admisible controls on [0, t ].

GOAL: let ε → 0 and find a simplified model.

The H-J approach to Singular Perturbations

Control system

dxs = f (xs, ys, αs) ds + σ(xs, ys, αs) dWs, xs ∈ Rn, αs ∈ A,

dys = 1ε g(xs, ys) ds + 1√

εν(xs, ys) dWs, ys ∈ Rm,

x0 = x , y0 = y

Cost functional Jε(t , x , y , α) :=∫ t

0 l(xs, ys, αs) ds + h(xt , yt)

Value function

uε(t , x , y) := infα∈A(t)

E [Jε(t , x , y , α)]

A(t) admisible controls on [0, t ].

GOAL: let ε → 0 and find a simplified model.

Assumptions on data:

h continuous, supy h(x , y) ≤ K (1 + |x |2)

f , σ, g, τ, l Lipschitz in (x , y) (unif. in α) with linear growth

Then uε solves the HJB equation

∂uε

∂t+H

(x , y , Dxuε, D2

xxuε,1√ε

D2xyuε

)− 1

εLuε = 0 in R+×Rn ×Rm,

H (x , y , p, M, Z ) := maxa∈A

{−tr(σσT M)− f · p − l − tr(σνZ T )

}L := tr(ννT D2

yy ) + g · Dy .

Theorem (Da Lio - Ley). The value function uε is the unique viscositysolution with growth uε(t , x , y) ≤ C(1 + |x |2 + |y |2) of the Cauchyproblem with initial condition

uε(0, x , y) = h(x , y).

Next assume on the linear operator L

Ellipticity: ∃Λ(y) > 0 s.t. ∀ x ν(x , y)νT (x , y) ≥ Λ(y)I

Lyapunov: ∃w ∈ C(Rm), k > 0, R0 > 0 s.t.−Lw ≥ k for |y | > R0, ∀x ,

w(y) → +∞ as |y | → +∞.

the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen

(FS) dyτ = g(x , yτ ) dτ + ν(x , yτ ) dWτ ,

is uniquely ergodic, i.e., has a unique invariant measure µx .

Liouville property:any bounded subsolution of −Lv = 0 is constant.

w(y) → +∞ as |y | → +∞.

A typical sufficient condition: ellipticity + recurrence condition

lim sup|y |→+∞

[g(x , y) · y + tr(ννT (x , y))

Proof by choosing as Lyapunov function w(y) = |y |2.

Example: Ornstein-Uhlenbeck process

dyτ = (m(x)− yτ )dτ + ν(x)dWτ ,

where m, ν are bounded.

GOAL: let ε → 0 in the Cauchy problem for the HJB equation.

Similarity with homogenization problems: the "fast" variable y plays therole of x

Main difference with periodic homogenization: the state variables ys

are unbounded and uncontrolled.

A typical sufficient condition: ellipticity + recurrence condition

lim sup|y |→+∞

[g(x , y) · y + tr(ννT (x , y))

Proof by choosing as Lyapunov function w(y) = |y |2.

Example: Ornstein-Uhlenbeck process

dyτ = (m(x)− yτ )dτ + ν(x)dWτ ,

where m, ν are bounded.

GOAL: let ε → 0 in the Cauchy problem for the HJB equation.

Similarity with homogenization problems: the "fast" variable y plays therole of x

Main difference with periodic homogenization: the state variables ys

are unbounded and uncontrolled.

Convergence Theorem

Under the previous assumptions, the (weak) limit u(t , x) as ε → 0 ofthe value function uε(t , x , y) solves

∂u∂t +

∫H(x , y , Dxu, D2

xxu, 0)

dµx(y) = 0 in R+ × Rn

u(0, x) =∫

h(x , y) dµx(y)

If, moreover, either g, ν do not depend on x or

g(x , ·), ν(x , ·) ∈ C1 and g(·, y), ν(·, y) ∈ C1b

Dxg, Dyg, Dxν, Dyν are Hölder in y uniformly w.r.t. x ,

then u is the unique viscosity solution (with quadratic growth) of (CP)

and uε(t , x , y) → u(t , x) locally uniformly on (0,+∞)× Rn.

Examples

Denote 〈φ〉 :=∫

φ(y)dµx(y). Got simple formulas for effective H and h

H(x , p, M) = 〈H(x , ·, p, M, 0)〉, h(x) = 〈h(x , ·)〉.

Corollary [see also Kushner, book 1990]For split systems, i.e.,

σ = σ(x , y), f = f0(x , y) + f1(x , a), l = l0(x , y) + l1(x , a),

the linear averaging of the data is the correct limit, i.e.,

limε→0

uε(t , x , y) = u(t , x) := infα.

E[∫ t

0〈l〉(xs, αs) ds + 〈h〉(xt)

dxs = 〈f 〉(xs, αs) ds + 〈σσT 〉1/2(xs) dWs

Proof: H(x , p, M) = −trace(M〈σσT 〉)/2 + maxA {−〈f 〉 · p − 〈l〉} .

Examples

Denote 〈φ〉 :=∫

φ(y)dµx(y). Got simple formulas for effective H and h

H(x , p, M) = 〈H(x , ·, p, M, 0)〉, h(x) = 〈h(x , ·)〉.

Corollary [see also Kushner, book 1990]For split systems, i.e.,

σ = σ(x , y), f = f0(x , y) + f1(x , a), l = l0(x , y) + l1(x , a),

the linear averaging of the data is the correct limit, i.e.,

limε→0

uε(t , x , y) = u(t , x) := infα.

E[∫ t

0〈l〉(xs, αs) ds + 〈h〉(xt)

dxs = 〈f 〉(xs, αs) ds + 〈σσT 〉1/2(xs) dWs

Proof: H(x , p, M) = −trace(M〈σσT 〉)/2 + maxA {−〈f 〉 · p − 〈l〉} .

Example: we recover the Black-Scholes formula with stochasticvolatility.

In general, for system or cost NOT split,

H(x , p, M) = 〈maxA{...}〉 > max

A〈{...}〉

and the limit control problem is not obvious.We try to write H as a Bellman Hamiltonian in some other way in orderto find an explicit effective control problem approximating the singularlyperturbed one as ε → 0. We did it for

Merton problem with stochastic volatility (see next slides),

Ramsey model of optimal economic growth with (fast) randomparameters,

Vidale - Wolfe advertising model with random parameters,

advertising game in a duopoly with Lanchester dynamics andrandom parameters.

Merton portfolio optimization with stochastic volatility

Given a control βs the wealth xs evolves as

dxs = (r + (γ − r)βs)xs ds + xsβs σ(ys) dWs x0 = x

dys = 1ε (m − ys) ds + ν√

εdWs y0 = y

and the value functions is V ε(t , x , y) := supβ.E [h(xt)].

Let ρ = correlation of Ws and Ws. The HJB equation is

∂V ε

∂t− rxV ε

x −maxb

{(γ − r)bxV ε

x +b2x2σ2

xx +bxρσν√

εV ε

(m − y)V εy + ν2V ε

Assume the utility h has h′ > 0 and h′′ < 0. Then expect a valuefunction strictly increasing and concave in x , i.e., V ε

x > 0, V εxx < 0.

The HJB equation becomes

∂V ε

∂t− rxV ε

x +[(γ − r)V ε

x + xρσν√ε

V εxy ]2

σ2(y)2V εxx

=(m − y)V ε

y + ν2V εyy

By the Theorem, V ε(t , x , y) → V (t , x) as ε → 0 and V solves

∂V∂t

− rxVx +(γ − r)2V 2

σ2(y)dµ(y) = 0 in R+ × R+

This is the HJB equation of a Merton problem with the harmonicaverage of σ as constant volatility

σ2(y)dµ(y)

)−1/2

Therefore this is the limit control problem.

The limit of the optimal control βε,∗s as ε → 0 can also be studied.

The HJB equation becomes

∂V ε

∂t− rxV ε

x +[(γ − r)V ε

x + xρσν√ε

V εxy ]2

σ2(y)2V εxx

=(m − y)V ε

y + ν2V εyy

By the Theorem, V ε(t , x , y) → V (t , x) as ε → 0 and V solves

∂V∂t

− rxVx +(γ − r)2V 2

σ2(y)dµ(y) = 0 in R+ × R+

This is the HJB equation of a Merton problem with the harmonicaverage of σ as constant volatility

σ2(y)dµ(y)

)−1/2

Therefore this is the limit control problem.

The limit of the optimal control βε,∗s as ε → 0 can also be studied.

homogenization and other multi-scale problems in...

Documents

porous media and plasticity - homogenization for equations...

microsegregation and homogenization

homogenization of neuman boundary data with ... - university...

waveform relaxation for the computational homogenization of...

stability by homogenization of thermoviscoplastic...

homogenization silos

non periodic homogenization for elastic wave...

homogenization and parameter retrieval - home | em...

homogenization modeling for mechanical properties of...

biotic homogenization

alexandre girouard from steklov to neumann via...

some homogenization results for non-coercive … · some...

homogenization via p-fem for problems with...

homogenization for granular materials

upscaling , homogenization and hmm

homogenization gramberg

homogenization in porous media and associated spectral...

efficient methods for homogenization of random ... ·...

some inverse problems in periodic homogenization of ... ·...

papers de homogenization