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Sometimes this method may be = useful.

Let us consider the following polynomial

f(x)=3D 240000 * x - 25000 * x^2 + 3500 * x^3/3 - 25*x^4 + = x^5/5

and the interval [x]=3D[-1000, 1]

First derivative of the function f has the following form

d(1)f(x)=3D 240000-50000*x+3500* x*x-100*x*x*x+x*x*x*x;

Interval extension of this function is equal to:

d(1)f([-1000, 1]) =3D [-1103310000,1103550240000]

We don’t know if this function is monotone.
Because of = that we can=20 calculate second derivative

d(2)f(x)=3D -50000 + 7000* x - 300* x*x + 4* x*x*x

Interval extension of this function is equal to:

d(2)f([-1000, 1]) =3D [-4307050000,4257000]

We don’t know if this function is monotone.
Because of = that we can=20 calculate third derivative

d(3)f(x)=3D 7000 - 600* x + 12*x*x

Interval extension of this function is equal to:

d(3)f([-1000,1])=3D[-5600,12607000]

We don’t know if this function is monotone.
Because of = that we can=20 calculate forth derivative

d(4)f(x)=3D -600+24*x

Interval extension of this function is equal to:

d(4)f([-1000,1]) =3D [-24600,-576]

We can see that the sign of the forth derivative is = constant.
Because of=20 that third derivative is monotone and
extreme value of this function = can be=20 calculated
using endpoints of given interval.

d(3)f(-1000) =3D 12607000
d(3)f(1) =3D  6412

We can see that the sign of the third derivative is = constant.
Because of=20 that second derivative is monotone and
extreme value of this = function can be=20 calculated
using endpoints of given interval.

d(2)f(-1000)=3D -4.3071e+009
d(2)f(1)=3D  -43296

We can see that the sign of the second derivative is = constant.
Because=20 of that first derivative is monotone and
extreme value of this = function can=20 be calculated
using endpoints of given interval.

d(1)f(-1000) =3D  1.1036e+012
d(1)f(-1) =3D  = 193401

We can see that the sign of the first derivative is = constant.
Because of=20 that function f is monotone and
extreme value of this function can = be=20 calculated
using endpoints of given interval.

f(-1000)=3D -2.2619e+014
f(1)=3D   2.1614e+005

Finally

f([-1000, 1])=3D[ -2.2619e+014, 2.1614e+005]

This is the exact range of the function f.

If this procedure fails,
we can divide the interval into = two parts=20 and repeat procedure again.
If the boxes are sufficiently small =
then we=20 can apply Taylor model.

This procedure can be also applied in multidimensional case.

Pownuk A., New inclusion functions in interval global optimization = of=20 engineering structures.
EUROPEAN CONFERENCE ON COMPUTATIONAL=20 MECHANICS,
Cracow, 26 - 29 June 2001, pp.460-461

Matlab code:

lowerX=3D-1000;
upperX=3D = 1;
x=3Dinterval(lowerX,upperX);

'First order derivative'
240000-50000*x+3500*=20 x*x-100*x*x*x+x*x*x*x
'Second order derivative'
-50000 + 7000* x - = 300*=20 x*x + 4* x*x*x
'Third order derivative'
7000 - 600* x + = 12*x*x
'Fourth=20 order derivative'
-600+24*x

x=3DlowerX;
y4_lower=3D-600+24*x
x=3DupperX;
y4_upper=3D-60= 0+24*x

x=3DlowerX;
y3_lower=3D7000 - 600* x + = 12*x*x
x=3DupperX;
y3_upper=3D7000=20 - 600* x + 12*x*x

x=3DlowerX;
y2_lower=3D-50000 + 7000* x - 300* x*x + 4*=20 x*x*x
x=3DupperX;
y2_upper=3D-50000 + 7000* x - 300* x*x + 4* = x*x*x

x=3DlowerX;
y1_lower=3D240000-50000 *x+3500* x*x-100*=20 x*x*x+x*x*x*x
x=3DupperX;
y1_upper=3D240000-50000 *x+3500* = x*x-100*=20 x*x*x+x*x*x*x

x=3DlowerX;
y0_lower=3D240000 * x - 25000 * x^2 + 3500 * x^3/3 - = 25*x^4 +=20 x^5/5
x=3DupperX;
y0_upper=3D240000 * x - 25000 * x^2 + 3500 * = x^3/3 - 25*x^4=20 + x^5/5

Regards,

Andrzej Pownuk

Chair of Theoretical = Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.p= l/~pownuk
I have worked on this problem since = 1995.
Now I=20 can solve system of 1000 equations with dependent interval = parameters.
(I=20 have solved system of 2300 equations
with two intervals = Young’s=20 modulus.)

http://157.158.187= .18/papers/Interval_FEM.pdf

This algorithm was presented on the following = conferences:

1) Pownuk A., Modelling of structures with uncertain = parameters.
XLI=20 Sympsion on Modeling in Mechanics,
18-22.02.2002, Wis=B3a,=20 Poland
(Theoretical background)

2) Pownuk A., Numerical solutions of fuzzy partial differential = equation
and its application in computational = mechanics.
Assessment and=20 New Directions for Research.
FUZZY PARTIAL DIFFERENTIAL = EQUATIONS,
FUZZY=20 RELATIONAL EQUATIONS,
FUZZY DIFFERENCE EQUATIONS.
March 15-17,=20 2002
University of California-Berkeley ,
California 94720 -=20 USA
(Applications)

Regards,

&n= bsp; =20 Andrzej Pownuk

...............................
Andrzej Pownuk
Chair of = Theoretical=20 Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.= pl/~pownuk/
...............................

>
> Dear colleagues,
>
> I am trying to = solve a 1D=20 heat conduction problem by  FInite Elements
> Method ( = Ax=3Db). My=20 stiffness matrix (A) is 51 x 51 ( row, column). This
> example = accessories=20 my Phd thesis.
> I am using the thermal conductivity (k) as = interval=20 number, with a little
> variation in k I get a large = overestimation in the=20 results.
> I have already tried the Preconditioning Gauss = Elimination and=20 Gauss-Seidel
> Methods,  the Combinatorial and the Powell=20 Optimization Method.
> Do anybody know how to treat and avoid this = so=20 large overestimation ?
> Papers, routines, books an examples are=20 welcome.
>
> Best Regards,
>
> Thank  you = very=20 much.
>
> Sebastiao Pereira
> Petrobras - Brazilian = Oil=20 Company
> http://www.petrobras.com.br
&= gt;=20 phone: 55 21 38656436
> fax: 55 21 3865 6441
>=20
>
Dear Andrzej,

I am very interested in your work and would very = much like=20 to read the publications you refer to below, however I am unable to open = the=20 download from the URL you give below.
Perhaps there is another version as a pdf file = that I=20 could open and read?

[Can anyone else in the reliable_computing group = help with=20 this?]

best wishes,

Ramon Moore
----- Original Message -----
From:=20 Andrzej Pownuk
To: reliable_comput= ing [at] interval [dot] louisiana.edu=20 ; sebastiaoc [at] cenpes [dot] petr= obras.com.br=20
Sent: Wednesday, March 20, 2002 = 2:55=20 PM
Subject: Re: Interval Linear = Systems of=20 Equations - Overestimation - Help !!

I have worked on this problem since = 1995.
Now=20 I can solve system of 1000 equations with dependent interval = parameters.
(I=20 have solved system of 2300 equations
with two intervals = Young’s=20 modulus.)

http://157.158.187= .18/papers/Interval_FEM.pdf

This algorithm was presented on the following = conferences:

1) Pownuk A., Modelling of structures with uncertain = parameters.
XLI=20 Sympsion on Modeling in Mechanics,
18-22.02.2002, Wis=B3a,=20 Poland
(Theoretical background)

2) Pownuk A., Numerical solutions of fuzzy partial = differential=20 equation
and its application in computational = mechanics.
Assessment and=20 New Directions for Research.
FUZZY PARTIAL DIFFERENTIAL = EQUATIONS,
FUZZY=20 RELATIONAL EQUATIONS,
FUZZY DIFFERENCE EQUATIONS.
March 15-17,=20 2002
University of California-Berkeley ,
California 94720 -=20 USA
(Applications)

Regards,

=
&n= bsp; =20 Andrzej Pownuk

...............................
Andrzej Pownuk
Chair of = Theoretical=20 Mechanics
Faculty of Civil Engineering
Silesian University of=20 Technology
E-mail: pownuk [at] zeus [dot] polsl.gliwice.pl=
URL:=20 http://zeus.polsl.gliwice.= pl/~pownuk/
...............................

>
> Dear colleagues,
>
> I am trying to = solve a 1D=20 heat conduction problem by  FInite Elements
> Method ( = Ax=3Db). My=20 stiffness matrix (A) is 51 x 51 ( row, column). This
> example=20 accessories my Phd thesis.
> I am using the thermal conductivity = (k) as=20 interval number, with a little
> variation in k I get a large=20 overestimation in the results.
> I have already tried the=20 Preconditioning Gauss Elimination and Gauss-Seidel
> = Methods,  the=20 Combinatorial and the Powell Optimization Method.
> Do anybody = know how=20 to treat and avoid this so large overestimation ?
> Papers, = routines,=20 books an examples are welcome.
>
> Best Regards,
> =
>=20 Thank  you very much.
>
> Sebastiao Pereira
>=20 Petrobras - Brazilian Oil Company
> http://www.petrobras.com.br
&= gt;=20 phone: 55 21 38656436
> fax: 55 21 3865 6441
>=20
>
------=_NextPart_000_0034_01C1D026.E9E024C0-- From owner-reliable_computing [at] interval [dot] louisiana.edu Wed Mar 20 15:48:58 2002 Received: (from daemon@localhost) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) id g2KLmv301504 for reliable_computing-outgoing; Wed, 20 Mar 2002 15:48:57 -0600 (CST) Received: from clmboh1-smtp3.columbus.rr.com (clmboh1-smtp3.columbus.rr.com [65.24.0.112]) by interval.louisiana.edu (8.11.3/8.11.3/ull-interval-math-majordomo-1.2) with ESMTP id g2KLmqr01500 for ; Wed, 20 Mar 2002 15:48:52 -0600 (CST) Received: from oemcomputer (dhcp065-024-174-102.columbus.rr.com [65.24.174.102]) by clmboh1-smtp3.columbus.rr.com (8.11.2/8.11.2) with SMTP id g2KLmpB14662 for ; Wed, 20 Mar 2002 16:48:51 -0500 (EST) Message-ID: <009901c1d058\$91a825a0\$66ae1841 [at] columbus [dot] rr.com> From: "Ramon Moore" To: "interval" Subject: sorry, my problem Date: Wed, 20 Mar 2002 16:45:35 -0500 MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0096_01C1D02E.A89FC300" X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.50.4133.2400 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4133.2400 Sender: owner-reliable_computing [at] interval [dot] louisiana.edu Precedence: bulk This is a multi-part message in MIME format. ------=_NextPart_000_0096_01C1D02E.A89FC300 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Dear People, Sorry for the trouble. The problem was at my end. I have the paper = Interval_FEM.pdf now. Thanks for your help Ramon Moore ------=_NextPart_000_0096_01C1D02E.A89FC300 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
Dear People,

Sorry for the trouble. The problem was at my = end. I have=20 the paper Interval_FEM.pdf
now.

Ramon Moore
Dear Colleagues,

This message is to inform you = that my address=20 and
official address of Reliable = Computing=20 journal changes tomorrow
from

slavanest [at] yahoo [dot] com

to

slnest [at] comset [dot] net

First address will work properly = during at=20 least one month and hopefully much more,
however please use the new = address for all=20 kind of communications with me.

Best regards,
Slava = Nesterov
Dear = Colleagues,=20

Robert=20 Mullen and I have been following your interesting and valuable = discussion for=20 the last week, and we would like to highlight some aspects of the short = history=20 of Interval Finite Element and the latest developments in this fast = growing=20 area.  To make this e-mail = brief,=20 you can download the short review from the following = link:

www.gtrep.= gatech.edu/~rmuhanna/IFEM_review.pdf

We do not=20 claim that our review is inclusive. =20 It is, however, complete to our knowledge and we would appreciate = list=20 members to send us additional information that we have missed.  We would also like to include = links to=20 the text of each paper where allowed by copyright = law.

We would=20 like to add the following comments to the discussion on interval finite = element=20 methods. While intervals are narrow in many practical applications, we = believe=20 that interval treatment in finite elements requires the use of wide=20 intervals.  For example, = the live=20 loads that a structure has to be designed for often vary between zero = and the=20 maximum design load. Material strengths are often uncertain by at least=20 10%.  In problems = associated with=20 computational mechanics we might face in the same problem narrow and = wide=20 intervals, i.e., modulus of elasticity (narrow) and load (wide).  We feel that restricting = solutions of=20 finite element problems to narrow intervals has only limited=20 usefulness.

More=20 important, successful application of interval methods to finite element = problems=20 requires that the issues of parameter dependency be included in the=20 analysis.  Failure to = consider=20 dependency results in interval solutions that are much too wide (see = paper by=20 Rao et al.)

While it is=20 well known that the general problem of solving interval system of = equations is=20 NP-hard, the system of equations associated with finite element problem = can have=20 different additional properties depending on the weak form used as a = basis of=20 the finite element formulation.  = For=20 example, for a class of structural problems, using a mixed finite = element=20 formulation, one can construct a multiplicative decomposition of the = system of=20 equations that diagonalizes the matrix of engineering (interval)=20 parameters.  Solution of = such a=20 problem is not NP-hard.

As we are striving to use = intervals=20 to represent uncertainty in our practical applications (analysis and = design),=20 consequently we would expect to enhance the reliability of final = products, and=20 we believe that cannot be done without rigorous verified methods.  Some times, approximate or = less rigorous=20 method (unless they are conservative) might lead to narrower solutions = in=20 comparison with the exact ones. =20 Such cases can have a catastrophic impact in design=20 problems.

In addition, we would like = to=20 comment on the remarks of Andrzej Pownuk. =20 In his last e-mail he wrote: "Mr. Muhanna in the paper Muhanna = R.L.,=20 Mullen R.L., Uncertainty in Mechanics Problems - Interval - Based = Approach.=20 Journal of Engineering Mechanics, Vol.127, No.6, 2001, 557-556, have = solved=20 problems with about 20 degree of freedom and one interval parameter. = What=20 happened when we will have thousand equations?"

In this paper we present = solutions=20 that can be compared to combinatorial solutions to demonstrate sharp = interval=20 bounds.  We cannot = calculate=20 combinatorial solutions to large problems and therefore did not included = larger=20 models in the paper.  In = other=20 conference papers, we have solved (using similar algorithms) problems up = to 1600=20 elements. It should also be noted that we allow for a different = stiffness for=20 each elements (not one parameter but one parameter per element).=20

www.gtr= ep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf

Finally, we are going to = host a=20 website dedicated to "Interval Finite Element Methods", and this website = will be=20 linked to the main interval website, i.e., http://www.cs.ute= p.edu/interval-comp/main.html.  It will include the latest = development=20 in the relevant development in the area of IFEM and related=20 publications.  We welcome = any=20 contribution from researchers in this field.  Moreover, we will try to = design the site=20 with Java-based interval computational = capabilities.

We found discussions on the interval list and = private=20 communications with Arnold Neumaier very useful during our investigation = of=20 Interval Finite Element and hope that this level of communications=20 continues.

Rafi Muhanna and Robert Mullen

________________________________________________

Rafi L. Muhanna

Department = of Civil=20 Engineering
Regional Engineering Program
Georgia Institute of=20 Technology

6001 Chatham Center Dr., Suite 350
Savannah, = GA=20 31405
USA

Email:       = ; =20 rafi.muhanna [at] gtrep [dot] gatech.e= du
Phone:      =20 (912)=20 651-7547
Fax:         &nb= sp;=20 (912) 651-7279
Dear Rafi and Robert,

Brilliant  work ! It opens the doors = for many=20 important applications of interval methods to problems in mechanics.=20

My sincere congratulations and best = wishes.

Ramon Moore
----- Original Message -----
From:=20 Rafi Muhanna
To: reliable_comput= ing [at] interval [dot] louisiana.edu=20
Cc: Rafi Muhanna
Sent: Tuesday, March 26, 2002 = 4:30=20 PM
Subject: Interval Linear = Systems

Dear = Colleagues,=20

Robert=20 Mullen and I have been following your interesting and valuable = discussion for=20 the last week, and we would like to highlight some aspects of the = short=20 history of Interval Finite Element and the latest developments in this = fast=20 growing area.  To make = this e-mail=20 brief, you can download the short review from the following=20 link:

www.gtrep.= gatech.edu/~rmuhanna/IFEM_review.pdf

We do not=20 claim that our review is inclusive. =20 It is, however, complete to our knowledge and we would = appreciate list=20 members to send us additional information that we have missed.  We would also like to = include links to=20 the text of each paper where allowed by copyright = law.

We would=20 like to add the following comments to the discussion on interval = finite=20 element methods. While intervals are narrow in many practical = applications, we=20 believe that interval treatment in finite elements requires the use of = wide=20 intervals.  For example, = the live=20 loads that a structure has to be designed for often vary between zero = and the=20 maximum design load. Material strengths are often uncertain by at = least=20 10%.  In problems = associated with=20 computational mechanics we might face in the same problem narrow and = wide=20 intervals, i.e., modulus of elasticity (narrow) and load (wide).  We feel that restricting = solutions of=20 finite element problems to narrow intervals has only limited=20 usefulness.

More=20 important, successful application of interval methods to finite = element=20 problems requires that the issues of parameter dependency be included = in the=20 analysis.  Failure to = consider=20 dependency results in interval solutions that are much too wide (see = paper by=20 Rao et al.)

While it=20 is well known that the general problem of solving interval system of = equations=20 is NP-hard, the system of equations associated with finite element = problem can=20 have different additional properties depending on the weak form used = as a=20 basis of the finite element formulation. =20 For example, for a class of structural problems, using a mixed = finite=20 element formulation, one can construct a multiplicative decomposition = of the=20 system of equations that diagonalizes the matrix of engineering = (interval)=20 parameters.  Solution of = such a=20 problem is not NP-hard.

As we are striving to = use=20 intervals to represent uncertainty in our practical applications = (analysis and=20 design), consequently we would expect to enhance the reliability of = final=20 products, and we believe that cannot be done without rigorous verified = methods.  Some times, = approximate=20 or less rigorous method (unless they are conservative) might lead to = narrower=20 solutions in comparison with the exact ones.  Such cases can have a = catastrophic=20 impact in design problems.

In addition, we would = like to=20 comment on the remarks of Andrzej Pownuk.  In his last e-mail he wrote: = "Mr.=20 Muhanna in the paper Muhanna R.L., Mullen R.L., Uncertainty in = Mechanics=20 Problems - Interval - Based Approach. Journal of Engineering = Mechanics,=20 Vol.127, No.6, 2001, 557-556, have solved problems with about 20 = degree of=20 freedom and one interval parameter. What happened when we will have = thousand=20 equations?"

In this paper we present = solutions=20 that can be compared to combinatorial solutions to demonstrate sharp = interval=20 bounds.  We cannot = calculate=20 combinatorial solutions to large problems and therefore did not = included=20 larger models in the paper.  = In=20 other conference papers, we have solved (using similar algorithms) = problems up=20 to 1600 elements. It should also be noted that we allow for a = different=20 stiffness for each elements (not one parameter but one parameter per = element).=20

www.gtr= ep.gatech.edu/~rmuhanna/IFEM_EBE_paper.pdf

Finally, we are going to = host a=20 website dedicated to "Interval Finite Element Methods", and this = website will=20 be linked to the main interval website, i.e., http://www.cs.ute= p.edu/interval-comp/main.html.  It will include the latest = development=20 in the relevant development in the area of IFEM and related=20 publications.  We = welcome any=20 contribution from researchers in this field.  Moreover, we will try to = design the=20 site with Java-based interval computational=20 capabilities.

We found discussions on the interval list and = private=20 communications with Arnold Neumaier very useful during our = investigation of=20 Interval Finite Element and hope that this level of communications=20 continues.

Rafi Muhanna and Robert Mullen

________________________________________________

Rafi L. = Muhanna

Department of Civil=20 Engineering
Regional Engineering Program
Georgia Institute of=20 Technology

6001 Chatham Center Dr., Suite = 350
Savannah, GA=20 = 31405
USA

Email:       = ; =20 rafi.muhanna [at] gtrep [dot] gatech.e= du
Phone:      =20 (912)=20 = 651-7547
Fax:         &nb= sp;=20 (912) 651-7279
&nbs= p;         =20 3-2002
&nb= sp;    =20 Reliable=20 Computing
=   =20 Volume 8, issue 3, 2002

&nbs= p; =20 Mathematical Research

Bounding Perturbations in Zeros = of Nonlinear=20 Systems
Michael A. Wolfe
177-188

Formal Solution to Systems of = Interval=20 Linear  or Non-Linear Equations
Miguel A. Sainz, Ernest = Gardenyes,=20 Lambert Jorba
189-211

In Case of Interval (or More = General)=20 Uncertainty,  No Algorithm Can
Choose the Simplest=20 Representative
Gerhard Heindl, Vladik Kreinovich, Maria=20 Rifqi
213-227

An Approach to Overcome Division = by Zero in=20 the Interval Gauss Algorithm
Jan Mayer
229-237

Intervals of Inverse = M-matrices
Charles R.=20 Johnson, Ronald L. Smith
239-243

&nbs= p; =20 Short communication

Rump's Example = Revisited
Eugene Loh, G.=20 William Walster
245-248