Weapons of Maths Construction: how a combination of maths ... · Weapons of Math Destruction: How...

Weapons of Maths Construction: how a combination of maths theoryand algorithms conquers inequalities and constructs optimal solutions

Song Wang

Department of Mathematics and StatisticsCurtin University

CIC Seminar, June 2017

S. Wang (Maths & Stats, Curtin University) WMC CIC Seminar, June 2017 1 / 53

Outline

Outline I

Simulation and optima design of semiconductor devices.

Computational optimal feedback control.

Beyond classical engineering and physics – Numerical methods in financialengineering/actuarial mathematics.

Conclusions.


Introduction

Echoing a recent book I

Weapons of Math Destruction: How Big Data Increases Inequality and ThreatensDemocracy by Cathy O’Neil.

Maths and algorithms need to be used responsibly.

General-purposed algorithms can hardly solve complex real-world problems.

Understand the mathematical nature of a problem first.

Often a customised algorithm is needed.


Semiconductor device simulation

Semiconductor device simulation I

Booming time for computational maths & engineering and parallel computations -1980s-1990s.

Miniaturization of semiconductor devices during 1970s–1980s.

Costs increase in laboratory design of devices.

Semiconductor drift-diffusion model proposed in the 1950s, but hardly used indesign.

The drift-diffusion model is still used for solar panels - light inducedsemiconductors.



Semiconductor device simulation II

p

n

p-n junction

p-contact

n-contact

A semiconductor diode.

Drift-diffusion model.Need to find Ψ, n and p satisfying

∇2ψ − n + p = p − n + N,

∇ · (∇n − n∇ψ)− R(ψ, n, p) = 0,

−∇ · (∇p + p∇ψ) + R(ψ, n, p) = 0,

with appropriate boundary conditions, where

n- and p-regions are doped with different materials.

Doping profile/function — N = ND − NA, typically N = 1016∼20 in p-region and−1016∼20 in the n-regions.

Because of large jumps in D, interior layers appear along the p-n junctions.



Semiconductor device simulation III

Difficulties in Solving the Problem

Jump in the RHS function D⇓

Gradient becomes huge near the p-n junc-tions. (interior layers appear)

⇓Conventional numerical methods becomenumerically unstable

⇓Spurious oscillations in numerical results

⇓Errors in numerical results impracticallylarge Solution behaviour near a p-n junction.



Semiconductor device simulation IV

Classic Methods Do Not Work Properly!Results from a Finite Difference Method for

−(εu′ − u)′ = 2x , in (0, 1), u(0) = u(1) = 0.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

’fdm.dat’

’exact.dat’

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

’fdm.dat’

’exact.dat’

Figure: ε = 0.1 and 0.01. Spurious oscillations appear when ε = 0.01.



Semiconductor device simulation V

Our approach.

Approximate flux, rather than gradient, by a polynomial— eg, let εu′+ u = C locally, then u is an exponential function: exponential fitting!!!Based on this idea, we designed and analysed

Exponentially fitted finite volume method,Mixed finite element method – approximate flux and potential in different setting.A combination of conventional and exponentially fitted methods.

All these methods are proved to be unconditionally stable.

Upper error bounds are established theoretically.

Methods extended to Navier-Stokes and singular perturbation equations.

Optimal design of semiconductor devices.



Semiconductor device simulation VI

Numerical results. A 2D MOSFET (Metal-Oxide-Semiconductor Field EffectTransistor)

N =

1018, 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 1.2 ≤ z ≤ 1.5.

2× 1018, 3 ≤ x ≤ 4, , 0 ≤ y ≤ 2, 1.2 ≤ z ≤ 1.5,−1016, otherwise.

Gate height: 5× 10−3 µmBiases applied: Usb = Usc = 0, Udr = 0.2V , Ugt = 1, 2, ..., 25V .

n np

4unit: Micron

UU U

U

sc

gt dr

sb

21.2

1.5

0x

y z



Semiconductor device simulation VII

0

1

2

3

4

0

1

2

0

0.5

1

1.5

2

x 1018

electron concentration n

0

1

2

3

4

0

1

2

−5

0

5

10

x 1015

hole concentration p

Figure: Electron and hole concentrations.



Semiconductor device simulation VIII

0

1

2

3

4

0

1

2

−0.5

0

0.5

1

static potential ψ

-15

-10

-5

0

5

10

0 5 10 15 20 25L

og

arith

m o

f d

rain

cu

rre

nt

(A/c

m2

)Applied gate voltage (volt)

I-V characteristic

Figure: Static potential and I-V characteristic.



Semiconductor device simulation IX

Outcome: one PhD thesis.Our methods are featured in two major survey papers.

J.J.H. Miller, W.H.A. Schilders, S. Wang, Application of finite element methods tothe simulation of semiconductor devices, Reports on Progress in Physics, 63,No.3 (1999) 277-352.

F. Brezzi, L. D. Marini, S. Micheletti, P. Pietra, R. Sacco, S. Wang, Discretisation ofSemiconductor Device Problems (I), Chapter 4 in Handbook of NumericalAnalysis, Vol. XIII: Numerical Methods in Electromagnetics, eds. W.H.A.Schilders, E.J.W. ter Maten, Elsevier Science Publishers B.V.(North-Holland),(2005) 317-442.

Optimum choices of doping profiles and p-n junctions


Construction of optimal feed control laws

Construction of optimal feed control laws I

The Optimal Feedback Control Problem

minimise F (u) =

∫ 1

sL(t , x(t), u(t))dt + h(x(1)),

subject to

x(t) = f (t , x(t), u(t)), a.e.t ∈ [s, 1]x(s) = y ,

over a set of admissible controls U ⊂ Rm, where f = (f1, f2, . . . , fn)> and(s, y) ∈ [0, 1]× Rn.



Construction of optimal feed control laws II

Using the dynamic programming approach, this problem can be formulated as an HJBequation

−∂v∂t

+ supu∈U

[−∇x v · f (t , x(t), u(t))− L(t , x , u)

]= 0

(t , x) ∈ (0, 1)× Rn

with the initial/final conditionv(1, x) = h(x)

where v is the value function and ∇x v denotes the gradient of v with respect to x .There are two unknown functions in this equation – the value function v and theoptimal control u.



Construction of optimal feed control laws III

Difficulties in Solving the Equation

Solutions are usually not smooth.

Coupling of unknowns u and x .

Multiple dimensions – ‘Curse of Dimensionality’.

Pure initial value problem without any boundary conditions.

Stable and accurate methods we have developed1 Upwind schemes + explicit time-stepping – conditionally stable, for problems in 1

and 2 dimensions.2 Upwind and exponentially fitted finite volume schemes + implicit time-stepping –

unconditionally stable, but artificial boundary conditions are needed.3 B-spline function approximation + optimal mesh points.4 Radial basis function approximation + adaptive construction of sub-spaces.5 Domain decomposition + parallel computations.

Outcome: three PhD theses.



Construction of optimal feed control laws IV

The Radial Basis Function (RBF) approximationHuang, Wang, Li, Chen (2006), Alwardi, Wang, Jennings, Richardson (2012)Radial basis functions — Inverse Multi-quadratics

φ(x , y) =1√

||x − y ||22 + c2,

for x ∈ Rn, where ‖ · ‖2 denotes the Euclidean norm on Rn, y is a given point definingthe center of the RBF and c > 0 is the shape parameter.



Construction of optimal feed control laws V

HJB equation of the form

−∂v∂t

+ supu∈U

[−∇x v .f (t , x , u)− L(t , x , u)

]= 0, (t , x) ∈ [0,T ]× Ω,

v(T , x) = h(x), x ∈ Ω, (4)

where Ω ⊂ Rn denotes a bounded region.We now consider the problem on the finite region region Ω =

∏nh=1(ah, bh) containing

the region of interest, Ω, as a proper subset, where ah, bh ∈ R are two real constantssatisfying ah < bh for h = 1, 2, . . . , n. Write the problem as

−∂v∂t−[∇v · f (t , x , u∗)− L(t , x , u∗)

]= 0,

u∗ = arg supu∈U

[−∇v . f (t , x , u)− L(t , x , u)

].



Construction of optimal feed control laws VI

Time DiscretizationLet tkK

0 be a set of points in time satisfying T = t0 > t1 > · · · > tK = 0 and∆tk = tk − tk−1 < 0, where K ≥ 1 is a positive integer. We apply the Backward Eulerimplicit scheme to (5) and (6), yielding the following semi-discretized system:

−v k−1 − v k

−∆tk−[∇v k · f (tk , x , uk )− L(tk , x , uk )

]= 0,

uk = arg supu∈U

[−∇v k · f (tk , x , u)− L(tk , x , u)

]for k = 1, 2, ...,K , where v k (x) and uk (x) are respectively approximations to v(tk , x)and u(tk , x).



Construction of optimal feed control laws VII

Spatial DiscretizationFor any l = 0, 1, ...,K , let y l

j , j = 1, 2, . . . ,N l be a set of distinct points in Ω, where N l isa positive integer. Using φ(x , y), we approximate v k (x)

v lN l (x) =

N l∑j=1

αljφ(x , y l

j )

for l = 1, 2, ...,K , where αljN l

j=1 is a set of unknown coefficients. Replacing v k−1 andv k with the respective approximations defined above gives a system for the unknowncoefficients αk

j Nk

j=1.



Construction of optimal feed control laws VIII

To determine these unknowns, we choose a set of M distinct pointsxi ∈ Ω, i = 1, 2, . . . ,M and require the following system holds:

1∆tk

Nk−1∑j=1

αk−1j φ(xi , y k−1

j )− 1∆tk

Nk∑j=1

αkj φ(xi , y k

j )−

Nk∑j=1

αkj ∇φ(xi , y k

j ) · f (tk , xi , uki )− L(tk , xi , uk

i ) = 0, (10)

uki = arg sup

u∈U

− Nk∑j=1

αkj ∇φ(xi , y k

j ) · f (tk , xi , u)− L(tk , xi , u)

(11)

for i = 1, 2, . . . ,M and k = 1, 2, ...,K .A system of collocated equations for αk

j Nk

j=1.M > Nk , over-determined system and thus it is satisfied only in the least-squaressense.A coupled system for αk and uk . We decouple the two sub-systems by replacing theterm αk

j f (tk , xi , uki ) with αk

j f (tk , xi , uk−1i ).



Construction of optimal feed control laws IX

Decoupled system in matrix form:(Ak + Bk (uk−1)

)αk = Ak−1αk−1 + βk−1(uk−1), (12)

uki = arg sup

u∈U

[Bk (u)αk − βk (u)

]i

(13)

for k = 1, 2, ...,K and i = 1, 2, ...,M, whereAk = (Ak

ij ) – M × Nk matrix, αk – Nk × 1 column vector with Akij = φ(xi , y k

j )

αk = (αk1, α

k2 , . . . , α

kNk )>

Bk (u) = (Bkij (u)) – M × Nk matrix defined by Bk

ij (u) = ∆tk∇φ(xi , y kj ) · f (tk , xi , u),

βk−1i = −∆tk L(tk , xi , uk−1

i ).The system can be written as the following short form

Hkαk = dk−1,

where

Hk = Ak + Bk (uk−1) and dk−1 = Ak−1αk−1 + βk−1(uk−1). (17)

A least-squares system when M > Nk and we may solve the following system for αk :

(Hk )>Hkαk = (Hk )>dk−1.



Construction of optimal feed control laws X

Adaptivity and Matrix Inversion Technique (Alwardi, Wang, Jennings, Richardson,2012)

An algorithm to adaptively construct approximations to v to minimize the maximumerror of the difference between the right- and left-hand side vectors of the approximateequation.Suppose the maximum error occurs at Q points of x1, x2, ..., xM for a positive integerQ. We order these Q points as y k

Nk +1, ..., ykNk +Q and add a linear combination of the

basis functions centered at these points to the old to form a new approximation

vNk +Q(x , tk ) =Nk +Q∑

j=1

αkj φ(x , y k

j ),

where αk1, ..., α

kNk is a set of Nk + Q unknown parameters.

The collocation equation corresponding is[Hk Pk] αk = dk−1,



Construction of optimal feed control laws XI

where αk = (αk1, ..., α

kNk )>, Hk and dk−1 are the M × Nk and M × 1 matrices defined in

(17) and Pk = (Pkij ) is an M ×Q matrix defined by

Pkij = φ(xi , y k

Nk +j ) + ∆tk∇φ(xi , y kNk +j ) · f (tk , xi , uk−1

i )

for i = 1, 2, ...,M and j = 1, ...,Q.To solve the over-determined system for αk , we left-multiply it by the transpose of itscoefficient matrix, yielding[

(Hk )>Hk (Hk )>Pk

(Pk )>Hk (Pk )>Pk

]αk =

[Hk

Pk

]dk−1 (20).

This is an (Nk + Q)× (Nk + Q) linear system. The system matrix is normally denseand thus the inversion of this matrix requires O((Nk + Q)3) operations. Note that asimple matrix inversion technique which makes use of ((Hk )>Hk )−1 calculated fromthe step of the adaptive process will dramatically save the computational cost forinverting the system matrix of (20).



Construction of optimal feed control laws XII

Theorem: Let E be an (N + Q)× (N + Q) matrix of the following block form

E =

[E11 E12

E21 E22

](21)

where E11 is an N × N matrix, E12 and E>21 are N ×Q matrices and E22 is a Q ×Qmatrix. If both E11 and its Schur Complement S := E22 − E21E−1

11 E12 are non-singular,then we have

E−1 =

[E−1

11 + E−111 E12S−1E21E−1

11 −E−111 E12S−1

−S−1E21E−111 S−1

]. (22)

We comment that (22) with

E11 = (Hk )>Hk , E12 = E>21 = (Hk )>Pk , E22 = (Pk )>Pk

provides a formula for recursively calculating the inverse of the system matrix of (20).In this expression E−1

11 is available from the previous step and thus from (22) we seethat the inversion of E only involves the matrix multiplications and the inversion of theQ ×Q Schur complement S. Therefore, the computational cost for (22) is expected tobe small.



Construction of optimal feed control laws XIII

Non-overlapping Domain Decomposition RBFs Method (Alwardi, Wang, Jennings,2013)Main Idea

Divide Ω into S non-overlapped open subdomains Ωp, p = 1, 2, ...,S satisfying

Ω =S⋃

p=1

Ωp, Ωp ∩ Ωq = ∅, ∀p, q = 1, 2, . . . ,S, p 6= q.

It is necessary to extend each of these subdomains to a larger region Ωp, forp = 1, 2, . . . ,S.The original problem is then decomposed into a series of sub-problems as follows:

−∂vp

∂t+ sup

up∈U

[−∇x vp.f (t , x , up)− L(t , x , up)

]= 0, (23)

(t , x) ∈ [0,T ]× Ωp,

vp(T , x) = h(x), x ∈ Ωp, (24)

where vp and up are respectively the value function and the control of thesub-problem on Ωp, p = 1, 2, . . . ,S.



Construction of optimal feed control laws XIV

Let xi,pMpi=1 and y k

j,pNk

pj=1 be given sets of nodes in Ωp for any feasible p and k .

Applying the least-squares collocation RBF method to each of the problems, wehave the following linear systems

Hkpα

kp = dk−1

p , p = 1, 2, . . . ,S, (1)

where Hkp is an Mp × Nk

p matrix and bk−1p is an Mp × 1 column vector. This is a set

of least-squares problems when Mp > Nkp .

Merits: Parallelization; help to reduce the condition numbers.



Construction of optimal feed control laws XV

Algorithm ADDM (Adaptive Domain Decomposition Method)Let ΩpS

p=1 be the set of non-overlapped sub-domains of Ω defined above andΩpS

p=1 be the extended domains of ΩpSp=1. Choose a tolerance ε > 0. For

p = 1, 2, ...,S, perform the following steps (in parallel over p):

Choose a set of Mp distinct collocation points Xp = x1,p, x2,p, . . . , xMp,p ⊂ Ωp.Initialize v0

i,pMpi=1 using the terminal condition (24).

Choose a set of points Zp = z1,p, ..., zNp,p ⊆ Xp. Solve

Apα0p = (h(x1,p), . . . , h(xMp,p))>

for α0p and evaluate u0

p = (u01,p, ..., u

0Mp,p)> using

u0i,p = arg sup

u∈U

[−

Np∑j=1

α0j,p∇φ(xi,p; zj,p) · f (t0, xi,p, u)− L(t0, xi,p, u)

],

for i = 1, 2, . . . ,Mp, where Ap denotes the matrix defined in (14) with y l1, ..., y

lN l

replaced with Zp and h(·) is the terminal condition.

For k = 1, 2, . . . ,K , perform the following steps:



Construction of optimal feed control laws XVI

(a) Choose Y kp = yk

1,p, yk2,p, . . . , y

kNk

p ,p ⊂ Xp , where Nk

p represents the initial number of

RBFs in Ωp at tk . Find αkp by solving (18) with Nk = Nk

p .(b) Find an index Ip ∈ 1, 2, ...,Mp such that

Rp := |(dk−1p − Hk

pαkp)Ip | = max

1≤i≤Mp|(dk−1

p − Hkpα

kp)i |.

(c) If Rp < ε, goto (d). Otherwise,let yk

Nkp +1,p

= xIp,p ,

solve (20) for αkp using the matrix inversion technique (22) with Q = 1,

set Y kp := Y k

p ∪ ykNk

p +1,p, αk

p = αkp , Nk

p := Nkp + 1, and go(b).

(d) Evaluate ukp using (13).

For any k = 1, 2, ...,K , define v(x , tk ) approximating v(tk , x) on Ω

v(tk , x) =

∑Nkp

j=1 αkj,pφ(x , y k

j,p) x ∈ Ωp

12

(∑Nkp

j=1 αkj,pφ(x , y k

j,p) +∑Nk

qj=1 α

kj,qφ(x , yk

j,q)

),

x ∈ Ωp ∩ Ωq

if Ωp ∩ Ωq 6= ∅.

for p, q = 1, 2, ...,S.S. Wang (Maths & Stats, Curtin University) WMC CIC Seminar, June 2017 28 / 53


Construction of optimal feed control laws XVII

Test problem: Consider the following 2D HJB equation which arises from amultivariate optimal control problem:

−vt + sup0≤u≤1

[− xvx − yvy

]= 0,

where (t , x , y) ∈ [0, 1)× [−1, 1]2 and v(1, x , y) = −x − y . The exact value function vis given by

v(t , x) =

−(x + y)e1−t , if x + y > 0−(x + y) , if x + y ≤ 0.



Construction of optimal feed control laws XVIII

S 1 2× 2 4× 4 8× 8AN×N 961× 961 361× 361 169× 169 49× 49χ(A) 3.417e + 15 4.809e + 12 2.366e + 8 4.384e + 6L∞ 5.3e − 2 5.4e − 2 5.8e − 2 5.1e − 2Ttot 17537 2307 606 89Tinv 11 0.39 0.063 0.016Table: Computed errors using a classical matrix inversion technique.

S 1 2× 2 4× 4 8× 8AN×N 961× 961 361× 361 169× 169 49× 49χ(A) 3.417e + 15 4.809e + 12 2.366e + 8 4.384e + 6L∞ 5.3e − 2 5.4e − 2 5.8e − 2 5.1e − 2Ttot 16782 2140 466 87Tinv 0.625 0.047 0.016 0.005

Table: Computed errors using the new matrix inversion technique.



Construction of optimal feed control laws XIX

0 100 200 300 400 500 600 700 800 900 10000

2

4

6

8

10

12

Size of System Matrix

Max C

PU

tim

e (

sec)

Classical Matrix Inversion Technique

New Matrix Inversion Technique

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

3

3.5x 10

15

No of subdomains

Co

nd

itio

n N

um

be

rs

o

f th

e S

ys

te

m M

atrix

Condition Numbers of the System Matrix using our Scheme.

Figure: Comparison between two different matrix inversion techniques. Right – CPU time, Left –condition number.


Variational inequalities in financial engineering

Variational inequalities in financial engineering I

Black–Scholes equation for the price V of a European option on a stock:

LV := −∂V∂t− 1

2σ2x2 ∂

2V∂x2 − (r(t)x − D(x , t))

∂V∂x

+ rV = 0, (2)

for (x , t) ∈ I × [0,T ), with the boundary and final (or payoff) conditions

V (0, t) = g1(t) t ∈ [0,T ), (3)

V (X , t) = g2(t) t ∈ [0,T ), (4)

V (x ,T ) = g3(x) x ∈ I, (5)

whereI = (0,X ) ∈ R, σ > 0 — volatility of the asset , T > 0 — expiry date , r — interest rate,D — dividend. D = d(x , t)x



Variational inequalities in financial engineering II

Variational Inequality for pricing American options:

LV (x , t) ≥ 0 (6)

V (x , t)− V ∗(x) ≥ 0 (7)

LV (x , t) · (V (x , t)− V ∗(x)) = 0 (8)

in Ω := I × (0,T ) with proper terminal and boundary conditions, where V ∗ ≥ 0 is agiven function defining the theoretical lower bound for V .Can be written as the following HJB equation:

maxLV ,V − V ∗ = 0.

European option is a special case when V ∗ ≤ 0.



Variational inequalities in financial engineering III

Difficulties:

Non-smoothness of the exact solution.

Degeneracy at x = 0.

Combination of optimisation and linear/non-linear PDEs.

Requirement of non-negativity of solutions.

High-dimensionality.



Variational inequalities in financial engineering IV

What we have developed:

A fitted finite volume method for Black-Scholes operator (old wine in a new bottle!)— this is a landmark which has been used frequently by researchers and manylater methods are based on this work.

Penalty methods for solving the variational inequalities in both infinite and finitedimensions.

Numerical methods for pricing more complex options such as bond options,options with proportional transaction costs, options governed by fractionalBlack-Scholes equations, jump-diffusion, etc.

Interior penalty method for both infinite- and finite-dimensional variationalinequalities.

Outcomes: 4 PhD theses.


Penalty method for obstacle problems

Penalty method for obstacle/free boundary problems I

Find x ∈ Rn such that

x ≤ 0, (9)

F (x) ≤ 0, (10)

x>F (x) = 0, (11)

where F (x) is an n-dimensional vector-valued function defined on Rn.Let K = y ∈ Rn : y ≤ 0. We have, equivalently,Find x ∈ K such that for all y ∈ K,

(y − x)>F (x) ≥ 0. (12)

General-purposed algorithms are available.

F is usually a nonlinear sparse matrix.

F is strongly monotone, large-scale and an M-matrix.

General-purposed algorithms do not take advantage of these properties.



Penalty method for obstacle/free boundary problems II

Penalty formulation:Idea: constrained problem

minx∈Rn,x≤0

f (x)

can be approximated by

maxx∈Rn

(f (x) +

λ

1 + 1/k

n∑i=1

[x ]1+1/k+

).

Find xλ ∈ Rn such thatF (xλ) + λ[xλ]

1/k+ = 0,

where λ > 1 and k > 0 are parameters, [u]+ = maxu, 0 and yσ = (yσ1 , ..., yσn )> for

any y = (y1, ..., yn)> ∈ Rn and constant σ > 0.



Penalty method for obstacle/free boundary problems III

Mathematician’s job starts from here!Convergence.

A1. F is continuous on Rn.

A2. F is ξ-monotone, i.e., there exist constants α > 0 and ξ > 1 such that

(x − y)>(F (x)− F (y)) ≥ α||x − y ||ξ2, ∀x , y ∈ Rn.

Then, we haveTheoremLet x and xλ be the solutions to the original and penalized problems, respectively.There exists a constant C > 0, independent of xλ and λ, such that

||x − xλ||2 ≤Cλk/ξ . (13)



Penalty method for obstacle/free boundary problems IV

CorollariesWhen F is Holder continuous:

||F (x)− F (y)||2 ≤ β||x − y ||γ2 , ∀x , y ∈ Rn,

we have||x − xλ||2 ≤

Cλk/(ξ−γ)

for some positive constant C, independent of xλ and λ.If F is both Lipschitz continuous and strongly monotone, then

||x − xλ||2 ≤Cλk

NB. This estimate is not uniform in n.



Penalty method for obstacle/free boundary problems V

Bounded Nonlinear Complementarity Problems (Box Constrained).

minx∈Rm

F (x)

subject to −b ≤ x ≤ 0,

Optimality (KKT) conditions: find x , y ∈ Rn such that

f (x) + y ≤ 0, (14)

x ≤ 0, (15)

x>(f (x) + y) = 0, (16)

and

b − x ≤ 0, (17)

y ≤ 0, (18)

y>(b − x) = 0. (19)



Penalty method for obstacle/free boundary problems VI

Penalty formulation

w(zλ) + λ[zλ]1/k+ =

(f (xλ) + yλ

b − xλ

)+ λ

([xλ]

1/k+

[yλ]1/k+

)= 0, (20)

where

z =

(xy

)and w(z) =

(f (x) + y

b − x

). (21)



Penalty method for obstacle/free boundary problems VII

Rate of convergence

A1. f is Holder continuous on Rn, i.e., there exist constants β > 0 and γ ∈ (0, 1] suchthat

||f (x1)− f (x2)||2 ≤ β||x1 − x2||γ2 , ∀x1, x2 ∈ Rn. (22)

A2. f is ξ-monotone, i.e., there exist constants α > 0 and ξ ∈ (1, 2] such that

(x1 − x2)>(f (x1)− f (x2)) ≥ α||x1 − x2||ξ2, ∀x1, x2 ∈ Rn.

Let z := (x>, y>)> and zλ := (x>λ , y>λ )> be the solutions to the exact and penalized

problems, respectively. There exists a constant C > 0, independent of zλ, λ and k ,such that

||x − xλ||2 ≤ C max

1λk/(ξ−γ)

,1

λk/γ

, (23)

||y − yλ||2 ≤ C max

1λk ,

1λξk/[2(ξ−γ)]

,1

λγk/(ξ−γ)

, (24)

for sufficiently large λ.


Numerical results

Numerical Results I

An example of MNCP: Shape-preserving interpolation

−u(4)(s) + α(u′′(s))3 − u(s) ≤ f (s),

u′′(s) ≤ g(s),

(−u(4)(s) + α(u′′(s))3 − u(s)− f (s)) · (u′′(s)− g(s)) = 0

for s ∈ (0, 1) satisfying the Dirichlet boundary conditions u(0) = a, u(1) = b and thenatural boundary conditions u′′(0) = 0 = u′′(1),Introducing a new variable v = u′′, the above problem can be written as

−u′′ + v = 0,

−v ′′ + αv3 − u ≤ f ,

v ≤ g,

(−v ′′ + αv3 − u − f )(v − g) = 0

in (0, 1) satisfying the boundary conditions u(0) = a, u(1) = b and v(0) = 0 = v(1).


Numerical results

Numerical Results II

Application of the standard central finite difference scheme gives

Ax + y = c,

Ay − βx + αy3 ≤ d ,

y − e ≤ 0,

(y − e)>(Av − u + αu3 − d) = 0,

where

A =1h2

2 −1−1 2 −1

. . .. . .

. . .−1 2 −1

−1 2

.

The penalty equation:(A I−I A

)(xy

)+

(0αy3

)+ λ

(0

[y − e]1/k+

)−(

cd

)= 0, (25)

where I denotes the identity matrix in Rn−1. It is easy to show that the function isstrongly monotone since A is positive definite.


Numerical results

Numerical Results III

Rates of convergence

λ/k 5 10 20 40 80 1602 Err 1.81e-3 4.52e-4 1.13e-4 2.83e-5 7.07e-6 1.77e-6

R – 4.00 4.00 3.99 4.00 3.993 Err 3.27e-5 4.09e-6 5.12e-7 6.40e-85 8.00e-9 1.02e-9

R – 8.00 7.99 8.00 8.00 7.84

Table: Computed rates of convergence in λ.


Numerical results

Numerical Results IV

k 1 2 3 4λ = 5 Err 9.48e-2 1.81e-3 3.27e-5 6.07e-7

R – 52.38 55.35 53.87λ = 10 Err 4.78e-2 4.52e-4 4.09e-6 3.79e-8

R – 105.75 110.51 107.92

Table: Computed rates of convergence in k .


Numerical results

Numerical Results V

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

u

u.

0 0.2 0.4 0.6 0.8 1−30

−20

−10

0

10

20

30

xv

v = u′′.

Figure: Computed solutions.


Numerical results

Numerical Results VI

An example of Bounded NCP: Double obstacle problemFind u and v such that

−u′′(s) + αu3(s)− g(s) + v(s) ≤ 0,

u(s) ≤ 0,

u(s)(−u′′(s) + u3(s)− g(s) + v(s)) = 0,

p(s)− u(s) ≤ 0,

v(s) ≤ 0,

v(s)(p(s)− u(s)) = 0

in s ∈ (0, 1) satisfying the boundary conditions u(0) = u(1) = 0, where g and p aretwo given functions.The KKT conditions and the calculus of variations of the following ‘double obstacle’problem

minp≤u≤0

J(u) =

∫ 1

0

(12

(u′)2 +14

u4 − ug)

ds

satisfying that u is twice continuously differentiable and u(0) = u(1) = 0.


Numerical results

Numerical Results VII

We chooseg(s) = −4π2 sin(2πs) + sin3(2πs),

p(s) = sin(2πs)− 1.5.

Exact solution is u = − sin(2πs).


Numerical results

Numerical Results VIII

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

u and p0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−80

−70

−60

−50

−40

−30

−20

−10

0

10

v

Figure: Computed solutions u and v along with the lower bound p.


Numerical results

Numerical Results IX

Pricing American optionsTest Problem: American Put option. Parameters: X = 100, T = 1.5, r = 0.03, σ = 0.4and K = 50.

020

4060

80100 0

0.5

1

1.5

0

10

20

30

40

50

time t

share price x

op

tio

n v

alu

e V

V

0

20

40

60

80

100 0

0.5

1

1.5

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

time tshare price x

Delta o

f V

∆


Numerical results

Numerical Results X

020

4060

80100

0

0.5

1

1.5

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time t

share price x

Ga

mm

a o

f V

Γ

020

4060

80100

0

0.5

1

1.5−1

0

1

2

3

4

5

6

7

xtime t

V −

V*

V − V ∗

Figure: Computed value V , ∆ and Γ of the option, and the constraint V − V∗ for k = 4 andλ = 10.


Conclusions and future work

Conclusions and future work I

Real-world problems are full of singularities, non-smoothness and degeneracies.

To model a complex real-world system, it is important to first understand themathematical nature of the problem.

General-purposed algorithms often produce unsatisfactory numerical solutions.

High-resolution with a wrong method for a complex problem will increasecomputational errors.

Large-scale problems such as high-dimensional HJB equations are still nottractable by today’s technology and supercomputing computing power.

Synergy of mathematical theory and computational techniques is essential formany real-world problems.


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