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A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations

TL;DR: This work presents a new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids using separated representations and tensor product approximations basis for treating transient models.
Abstract: Kinetic theory models described within the Fokker-Planck formalism involve high-dimensional spaces (including physical and conformation spaces and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, E Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian Fluid Mech. 139 (2006) 153-176] for treating steady state kinetic theory models is extended here for treating transient models. (c) 2007 Elsevier B.V. All rights reserved.

Summary (3 min read)

1. Introduction

  • This paper constitutes the progression of a previous work [3] which focused on the steady solution of some classes of multidimensional partial differential equations (PDE) defined in spaces of dimension N (with N 1), and that is extended in the present work for treating multidimensional parabolic PDE.
  • In [3] a new solution technique based on a tensor product representation was introduced.
  • The authors would like to emphasize that they are not looking for a general numerical procedure for solving multidimensional PDE’s.
  • Moreover, some recent works [9] on sparse grids prove its ability for integrating functions defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives towards the solution of multidimensional partial differential equations.
  • Usual molecular descriptions make use of a configuration distribution function that is defined in a bounded domain and vanishes at its boundary.

2. Illustrating the solution strategy

  • Fkj are not given in advance but they are computed adaptively using the algorithm described later.
  • The construction of solution (Eq. (3)) consists of an iteration procedure involving, at each iteration n, three steps (see [3] for additional details).
  • The functions Fk(n+1) are finally obtained by normalizing the functions R1, R2, . . . , RN+1 after convergence of the non-linear problem.
  • Thus, in the steady case this ansatz leads to symmetric and positive definite linear systems (when the alternating direction strategy described in Section 3.2.1 applies) for the differential operators having these properties.

3.1. Solving the heat equation

  • As seen previously, the construction of such solution involves, at each iteration n, a projection, convergence checking and enrichment steps that the authors now describe.
  • Thus, Fj(x) (respectively,Gj(y) andHj(t)) and R(x) (respectively, S(y) and P(t)) are defined using a 1D finite element interpolation that, in their simulations, is assumed piecewise linear.
  • The authors denote byN (respectively,M and L) the vector containing the value of the p (respectively, q and r) shape functionsNp(x) (respectively, Mq(y) and Lr(t)).

3.2.1. Efficient non-linear solvers

  • In high-dimensional spaces, the discrete system (38) is strongly non-linear, and the convergence of non-linear solvers becomes a delicate task.
  • In their previous work [3] a Newton strategy was successfully applied.
  • That from their numerical experiments seems to be faster, simpler and more robust.
  • These linear systems are solved within an iteration procedure until convergence.
  • Other alternatives for solving this kind of strongly non-linear problems are being investigated, as is the case of the so-called asymptotic numerical technique [7].

3.2.2. Consistent time discretization

  • Another important numerical issue is the discretization of the time derivative term in the variational formulation (18).
  • If the authors consider the components of L as the usual piecewise linear shape functions, and perform the integral∫.
  • This is equivalent (dividing by the time step ht) to the central finite difference which is not consistent with the physics (justifying the use of backward time derivatives).
  • As the authors are discretizing the space-time variational formulation they cannot apply backward finite difference, and for this reason they propose to use an “upwind” Petrov–Galerkin formulation which leads to a backward time discretization.
  • Thus, the time discretization is now fully consistent.

3.3. Numerical example

  • The width of each representation of the 1D approximation functions illustrates the weight of that function, which is given by the corresponding α coefficient.
  • The computed solution is in perfect agreement with the one obtained by using a classical finite element method.

3.4. Convergence analysis

  • In order to conclude about the convergence rates of the proposed strategy the authors are considering a steady problem as well as a transient one whose large time solution coincides with the solution of the steady problem.
  • (51) This problem is solved in different multidimensional spaces considering the separated representation and an increasing number of nodes for the functional approximation in each dimension (for the sake of simplicity the authors are considering that all the one-dimensional functional approximations are defined from the same number of nodes nn).
  • The authors have reported in Fig. 3 the CPU time associated to each solution.
  • The same slope of two is noticed proving that the CPU time increases in a similar way with the space dimension and with the number of degrees of freedom considered for the one-dimensional functional approximations.
  • In these solutions only the number of nodes involved in the time discretization is increased.

4.1. Model equations

  • Finally, the link between the statistical distribution of dumbbell configurations and the polymer stress τp is provided, for the 1D molecular configuration considered in this section, by τp = 〈h(q)q2〉 − 1 = ∫ Ψ (q)h(q)q2 dq− 1. (59) Remark 3. (i) Eq. (56) defines the time evolution of the molecule distribution function.
  • The proposed method will be illustrated in homogeneous flows; the material derivative thus reduces to the partial derivative.

4.2. Problem discretization

  • The use of the variable separation strategy requires null boundary and initial conditions.
  • Due to both, the parabolic and convective character of Eq. (66), suitable stabilizations are required.

4.3. Extension to 1D multi-bead-spring models

  • The technique described above can be applied for simulating the Multi-Bead-Spring models of polymer chains.
  • The bead serves as an interaction point with the solvent and the spring contains the local stiffness information depending on local stretching (see [5] for more details).
  • Eq. (75) can be solved using a separated representation.

5. Numerical results

  • Fig. 6 shows the influence of the enrichment on the computed solution at tmax.
  • The polymer stress can then be calculated as a function of time using Eq. (59).
  • The authors must mention that the direct computation of the steady state is an issue for stochastic simulation techniques.
  • The last example that the authors are considering in the present work concerns the transient 2D FENE model in a starting shear flows.
  • (79) Fig. 14 depicts the three 2D space functions Fj(q) as well as the three associated time functions.

6. Conclusions and perspectives

  • This paper proves the ability of the proposed strategy to compute transient solutions of multidimensional parabolic PDE’s with homogeneous boundary conditions admitting a separated representation.
  • The novelty of the proposed technique justifies that many questions remain open: (i) the treatment of non-linear Fokker–Planck equations; (ii) optimal basis enrichment; (iii) analysis of complex flows involving non-homogeneous solution in the physical space; (iv) general initial conditions; (v) convergence analysis; (vi) stabilization of advection operators; . . ..
  • The use of a wavelet approximation, could offer an optimal refinement via the multiresolution properties of wavelets (additional degrees of freedom are only required in regions where the wavelets coefficients are not small enough).
  • The only remaining difficulty lies in the fact that nodes are assumed moving with the flow, which leads to highly distorted elements if one uses the finite element method (FEM) for solving momentum and mass balance equations.
  • At present the stabilization of the advection operators has been performed by considering a SUPG technique.

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A new family of solvers for some classes of
multidimensional partial dierential equations
encountered in kinetic theory modelling of complex
uids. Part II: Transient simulation using space-time
separated representations
Amine Ammar, Béchir Mokdad, Francisco Chinesta, Roland Keunings
To cite this version:
Amine Ammar, Béchir Mokdad, Francisco Chinesta, Roland Keunings. A new family of solvers for
some classes of multidimensional partial dierential equations encountered in kinetic theory modelling
of complex uids. Part II: Transient simulation using space-time separated representations. Journal of
Non-Newtonian Fluid Mechanics, Elsevier, 2007, 144 (2-3), pp.98-121. �10.1016/j.jnnfm.2007.03.009�.
�hal-01633241�

A new family of solvers for some classes of multidimensional
partial differential equations encountered in kinetic theory
modelling of complex fluids
Part II: Transient simulation using space-time separated representations
A. Ammar
a,
, B. Mokdad
a
, F. Chinesta
b
, R. Keunings
c
a
Laboratoire de Rh´eologie, INPG, UJF, CNRS (UMR 5520), 1301 rue de la Piscine, BP 53 Domaine Universitaire, F-38041 Grenoble Cedex 9, France
b
Laboratoire de M´ecanique des Syst`emes et des Proc´ed´es, UMR 8106 CNRS-ENSAM-ESEM, 151 Boulevard de l’Hˆopital, F-75013 Paris, France
c
CESAME, Universit´e Catholique de Louvain, Bat. Euler, Av. Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium
Kinetic theory models described within the Fokker–Planck formalism involve high-dimensional spaces (including physical and conformation
spaces
and time). One appealing strategy for treating this kind of problems lies in the use of separated representations and tensor product
approximations basis. This technique that was introduced in a former work [A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers
for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newtonian
Fluid Mech. 139 (2006) 153–176] for treating steady state kinetic theory models is extended here for treating transient models.
Keywords: Complex fluids; Kinetic theory; Model reduction; Multidimensional problems; Separation of variables; Numerical modeling; Fokker–Planck equation
1. Introduction
This paper constitutes the progression of a previous work [3] which focused on the steady solution of some classes of multidi-
mensional partial differential equations (PDE) defined in spaces of dimension N (with N 1), and that is extended in the present
work for treating multidimensional parabolic PDE.
In [3] a new solution technique based on a tensor product representation was introduced. In this technique the solution defined
in multidimensional domains is built by adding a certain number of products of N one-dimensional functions, each one defined in
a different space dimension. These functions are not known a priori, but they are constructed during the solution procedure. The
resulting technique was evaluated by solving some multidimensional elliptic problems as well as in some kinetic theory models.
The solution of multidimensional transient Fokker–Planck equations could be performed within an incremental time discretization.
However, the separated representation could be extended for representing the space-time solution. The main difficulties related to
such approach lies in the fact that the initial condition being non-zero the proposed procedure cannot be applied in a direct manner.
Another difficulty lies in the fact that space approximations are built using standard piece-wise functions, and when this kind of
approximation is retained to construct the time interpolation, the well known instability related to centered differences of time
derivatives is encountered. This paper proposes alternatives to circumvent both difficulties.
Despite the fact that tensor product spaces allow a significant reduction of the number of degrees of freedom, i.e. N × n
n
instead
of (n
n
)
N
(where n
n
is the number of degrees of freedom on each coordinate axis and N is the dimension of the problem), the resulting
Corresponding author. Tel.: +33 4 76 82 52 94; fax: +33 4 76 82 51 64.
E-mail addresses: Amine.Ammar@ujf-grenoble.fr (A. Ammar), Bechir.Mokdad@ujf-grenoble.fr (B. Mokdad), francisco.chinesta@paris.ensam.fr (F. Chinesta),
rk@inma.ucl.ac.be (R. Keunings).
1

discrete system is strongly non-linear. In this work, we propose an efficient alternating directions strategy for solving those non-linear
problems.
The novelty of the proposed technique justifies that a lot of questions are still open: (i) the treatment of non-linear Fokker–Planck
equations; (ii) optimal basis enrichment; (iii) analysis of complex flows involving non-homogeneous solution in the physical space;
(iv) general initial conditions; (v) analysis of convergence; (vi) stabilization of advection operators; ....
We would like to emphasize that we are not looking for a general numerical procedure for solving multidimensional PDE’s. In this
context, the sparse grids or sparse tensor product basis [6] are excellent candidates, even if it was argued in [1] its inability for treating
highly multidimensional problems (N 20). The sparse grid technique has been applied for solving parabolic equations in moderate
high-dimensional spaces [13](where the numerical examples concerned N 20) as well as kinetic theory models also involving
moderate high-dimensional spaces [8]. Moreover, some recent works [9] on sparse grids prove its ability for integrating functions
defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives
towards the solution of multidimensional partial differential equations. The technique proposed in our former work [3] and extended
in the present one to transient simulations, is, in our opinion, a suitable choice when dealing with highly multidimensional parabolic
PDE’s with homogeneous boundary conditions. Despite the apparent loss of generality induced by these assumptions, usual molecular
descriptions make use of a configuration distribution function that is defined in a bounded domain and vanishes at its boundary.
2. Illustrating the solution strategy
Let us define the following heat problem in a space of dimension N:
∂T
∂t
−T = f (x
1
,x
2
,...,x
N
), (1)
where T is a scalar function of space and time coordinates, i.e. T (x
,t) = T (x
1
,x
2
,...,x
N
,t). Problem (1) is assumed defined in
the domain Ω = Ω
x
× Ω
t
=] L, +L[
N
×]0,t
max
] and T vanishes at the space boundary, i.e. T (x ∂Ω
x
,t) = 0, as well as at the
initial time, i.e. T (x
,t = 0) = 0.
The problem solution is assumed in the form:
T (x
1
,x
2
,...,x
N
,t) =
j=1
α
j

N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
, (2)
where F
kj
is the jth basis function which only depends on the k th space coordinate. Thus, within the Galerkin framework the solution
can be approximated in the form:
T (x
1
,x
2
,...,x
N
,t)
J
j=1
α
j

N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
, (3)
where functions F
kj
are not given in advance but they are computed adaptively using the algorithm described later.
The construction of solution (Eq. (3)) consists of an iteration procedure involving, at each iteration n, three steps (see [3] for
additional details).
Step 1. Projection of the solution onto a discrete basis
If we assume the functions F
kj
(j [1,...,n]; k [1,...,N + 1]) known (verifying the boundary and the initial conditions,
that is vanishing on ∂Ω
x
), the coefficients α
j
can be computed by introducing the approximation of T into the Galerkin variational
formulation associated with Eq. (1):
Ω
T
∂T
∂t
dΩ +
Ω
Grad T
· Grad T dΩ =
Ω
T
f dΩ. (4)
Introducing the approximation of T and T
:
T (x
1
,x
2
,...,x
N
,t) =
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t), (5)
and
T
(x
1
,x
2
,...,x
N
,t) =
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t), (6)
2

we obtain
Ω
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
×
∂t
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
dΩ
+
Ω
Grad
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
× Grad
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
dΩ
=
Ω
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t)
f dΩ. (7)
Now, we assume that f (x
1
,...,x
N
) can be written in the form
f (x
1
,...,x
N
) =
m
h=1
N
k=1
f
kh
(x
k
). (8)
Eq. (7) involves integrals of a product of N + 1 functions, each one defined in a different dimension. Let (
N
k=1
g
k
(x
k
))g
N+1
(t)be
one of these functions to be integrated. The integral over Ω can be performed by integrating each function over its definition interval
and then multiplying the N + 1 computed integrals according to:
Ω
N
k=1
g
k
(x
k
)
g
N+1
(t)dΩ =
N
k=1
L
L
g
k
(x
k
)dx
k
×
t
max
0
g
N+1
(t)dt. (9)
This makes possible the numerical integration in highly dimensional spaces.
Now, due to the arbitrariness of coefficients α
j
, Eq. (7) allows us to compute the n approximation coefficients α
j
, by solving the
resulting linear system of size n × n. This problem rarely exceeds the order of tens of degrees of freedom.
Step 2. Checking convergence
From the solution of T at iteration n just calculated and given by Eq. (5), we compute the residual Re related to Eq. (1):
Re =
Ω
((∂T/∂t) −T f (x
1
,...,x
N
))
2
T
. (10)
where the norm of the solution is computed according to:
T =
Ω
T
2
dΩ, (11)
with all the integrals are computed in the space-time domain.
If Re<(epsilon is a small enough parameter), the iteration process stops, the solution T (x
1
,...,x
N
,t) being given by Eq. (5).
Otherwise, the iteration procedure continues.
Step 3. Enrichment of the approximation basis
From the coefficients α
j
just computed, the approximation basis can be enriched by adding the new function
(
N
k=1
F
k(n+1)
(x
k
))F
(N+1)(n+1)
(t). For this purpose, we solve the non-linear Galerkin variational formulation related to Eq. (1):
Ω
T
∂T
∂t
dΩ +
Ω
Grad T
· Grad T dΩ =
Ω
T
f dΩ, (12)
using the approximation of T given by:
T (x
1
,x
2
,...,x
N
,t) =
n
j=1
α
j
N
k=1
F
kj
(x
k
)
F
(N+1)j
(t) +
N
k=1
R
k
(x
k
)
R
N+1
(t), (13)
and the test function
T
(x
1
,x
2
,...,x
N
,t) = R
1
(x
1
) ×···×R
N
(x
N
) × R
N+1
(t) +···+R
1
(x
1
) ×···×R
N
(x
N
) × R
N+1
(t)
+R
1
(x
1
) ×···×R
N
(x
N
) × R
N+1
(t). (14)
3

This leads to a non-linear variational problem, whose solution allows us to compute the N functions R
k
(x
k
) as well as R
N+1
(t). The
functions F
k(n+1)
are finally obtained by normalizing the functions R
1
,R
2
,...,R
N+1
after convergence of the non-linear problem.
Remark 1. Each one of these one-dimensional functions R
k
(x
k
) are approximated in this work using a piecewise finite element
interpolation built from n
n
k
nodes. Other possibilities have being investigated, as for example the use of spectral approximations that
seem an appealing choice for increasing the convergence rates of the method, or the use of wavelet basis in order to take advantage
of its multiresolution character which could lead to efficient adaptive strategies.
Remark 2. The particular choice of the ansatz (14) was justified in the first part of this work [3] in the vartiational framework.
Thus, in the steady case this ansatz leads to symmetric and positive definite linear systems (when the alternating direction strategy
described in Section 3.2.1 applies) for the differential operators having these properties. In the general case (general differential
operators or parabolic problems) this ansatz vanishes on the domain boundary and when one proceeds using the alternating direction
strategy for solving the resulting non-linear discrete problem (as described in Section 3.2.1) this choice does not introduce additional
numerical difficulties and it worked perfectly in all the problems until now considered. However, we are aware that a further analysis
on this questions should be addressed in future works.
3. Matrix form and some numerical issues
3.1. Solving the heat equation
In the 2D case, Eq. (1) reduces to:
∂T
∂t
−T = f (x, y), (15)
where T = T (x, y, t). Eq. (15) is solved in the domain Ω = Ω
x
× Ω
t
=] L, +L[
2
×]0,t
max
] with zero boundary and initial
conditions.
The solution is now sought in the form:
T (x, y) =
j=1
α
j
F
j
(x)G
j
(y)H
j
(t). (16)
As seen previously, the construction of such solution involves, at each iteration n, a projection, convergence checking and enrichment
steps that we now describe.
In the enrichment step, the approximation basis is enriched by adding the new functions F
n+1
(x), G
n+1
(y) and H
n+1
(t) according
to
T (x, y, t) =
n
j=1
α
j
F
j
(x)G
j
(y)H
j
(t) + R(x)S(y)P(t), (17)
where F
n+1
(x), G
n+1
(y), H
n+1
(t) are obtained by normalizing the functions R(x), S(y) and P (t) using the L
2
norm.
From a practical point of view, these functions must be defined in a discrete form. Thus, F
j
(x) (respectively, G
j
(y) and H
j
(t)) and
R(x) (respectively, S(y) and P(t)) are defined using a 1D finite element interpolation that, in our simulations, is assumed piecewise
linear. We denote by N
(respectively, M and L) the vector containing the value of the p (respectively, q and r) shape functions N
p
(x)
(respectively, M
q
(y) and L
r
(t)). Finally F
j
, G
j
, H
j
, R, S and P are the nodal description of those functions.
3.1.1. Computing of the coefficients α
j
: projection stage
Now, we consider the variational formulation related to Eq. (15):
Ω
T
∂T
∂t
dΩ +
Ω
Grad T
Grad T dΩ =
Ω
T
f (x, y)dΩ, (18)
where the fact that T
vanishes at the boundary of Ω
x
has been introduced. Moreover, we assume that f (x, y) can be also written
in the form:
f (x, y) =
m
h=1
a
h
(x)b
h
(y). (19)
4

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Cites methods from "A new family of solvers for some cl..."

  • ...This technique has already been used in the context of stochastic problems [19] and multidimensional problems [1, 2, 6]....

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  • ...The PGD has been successfully applied to more general problems with different denominations: “Generalized Spectral Decomposition” for stochastic problems in [19], by replacing the time variable by a stochastic one ; “separated representation technique” for multivariable problems in [1, 2, 6], by adding new coordinates in the separated representation....

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References
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Journal ArticleDOI
TL;DR: The dimension–adaptive quadrature method is developed and presented, based on the sparse grid method, which tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators, and leads to an approach which is based on generalized sparse grid index sets.
Abstract: We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high-dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower-dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself.The dimension-adaptive quadrature method which is developed and presented in this paper aims to find such an expansion automatically. It is based on the sparse grid method which has been shown to give good results for low- and moderate-dimensional problems. The dimension-adaptive quadrature method tries to find important dimensions and adaptively refines in this respect guided by suitable error estimators. This leads to an approach which is based on generalized sparse grid index sets. We propose efficient data structures for the storage and traversal of the index sets and discuss an efficient implementation of the algorithm.The performance of the method is illustrated by several numerical examples from computational physics and finance where dimension reduction is obtained from the Brownian bridge discretization of the underlying stochastic process.

578 citations


"A new family of solvers for some cl..." refers background in this paper

  • ...Moreover, some recent works [9] on sparse grids prove its ability for integrating functions defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives towards the solution of multidimensional partial differential equations....

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Journal ArticleDOI
TL;DR: This work states thatKinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support using a reduced approximation basis within an adaptive procedure making use of an efficient separation of variables.
Abstract: Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables. (c) 2006 Elsevier B.V. All rights reserved.

546 citations


"A new family of solvers for some cl..." refers background or methods or result in this paper

  • ...13 and compared to the steady state simulation done in [3]....

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  • ...Introduction This paper constitutes the progression of a previous work [3] which focused on the steady solution of some classes of multidimensional partial differential equations (PDE) defined in spaces of dimension N (with N 1), and that is extended in the present work for treating multidimensional parabolic PDE....

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  • ...In our previous work [3] a Newton strategy was successfully applied....

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  • ...The technique proposed in our former work [3] and extended in the present one to transient simulations, is, in our opinion, a suitable choice when dealing with highly multidimensional parabolic PDE’s with homogeneous boundary conditions....

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  • ...These functions can be accurately represented using spherical coordinates or rectangular coordinates combined with a penalty technique as considered in [3]....

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Journal ArticleDOI
TL;DR: This paper further develops the separated representation by discussing the variety of mechanisms that allow it to be surprisingly efficient; addressing the issue of conditioning; and presenting algorithms for solving linear systems within this framework.
Abstract: Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to be surprisingly efficient; (ii) addressing the issue of conditioning; (iii) presenting algorithms for solving linear systems within this framework; and (iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schrodinger equation in quantum mechanics. Numerical examples are given.

393 citations

Book
01 Jul 2005
TL;DR: In this article, the finite element method and adaptive mesh refinement were used to calibrate local volatility with European options and with American options, respectively, in the context of option pricing.
Abstract: Preface 1. Option pricing 2. Black-Scholes equation mathematical analysis 3. Finite differences 4. The finite element method 5. Adaptive mesh refinement 6. American options 7. Sensitivities and calibration 8. Calibration of local volatility with European options 9. Calibration of local volatility with American options Bibliography Index.

340 citations


"A new family of solvers for some cl..." refers background in this paper

  • ...In this context, the sparse grids or sparse tensor product basis [6] are excellent candidates, even if it was argued in [1] its inability for treating highly multidimensional problems (N 20)....

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Journal ArticleDOI
TL;DR: It is proved that this algorithm gives an L 2 (Ω)-error of O(N-p ) for u (x,T) where N is the total number of operations, provided that the initial data satisfies with e > 0 and that u(x,t) is smooth in x for t>0 .
Abstract: We consider the numerical solution of diffusion problems in (0,T ) x Ω for and for T > 0 in dimension d d ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order p d ≥ 1, and hp discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2 (Ω)-error of O(N-p ) for u(x,T) where N is the total number of operations, provided that the initial data satisfies with e > 0 and that u(x,t) is smooth in x for t>0 . Numerical experiments in dimension d up to 25 confirm the theory.

152 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. part ii: transient simulation using space-time separated representations" ?

Ammar et al. this paper proposed a new family of solvers for solving non-linear Fokker-planck PDEs in space-time separated representations. 

For this purpose the authors consider that each node used in the discretization of the physical domain moves along its flow trajectory, allowing the integration of the advection operator of the Fokker–Planck equation using the method of characteristics. One possibility to circumvent this difficulty consists in using a meshless natural element method ( NEM ) [ 11 ] instead usual finite elements, because the accuracy of the NEM does not require any geometrical quality of the nodal distribution. Other fixed mesh techniques can be used, as the ones based on the characteristics-Galerkin scheme. In the case of general initial conditions, the authors can compute a separated representation of that initial condition ( as the sum of a product of terms involving the different coordinates ) using for example an alternating least squares technique [ 4 ].