Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation_2

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Computational performance of the M M OC in the

inverse design of the Doswell frontogenesis equation

Alexandre Franciscoa,∗, Umberto Biccarib, Enrique Zuazuab

Department of Mechanical Engineering, Universidade Federal Fluminense, 27255-125, Volta

Redonda, RJ, Brazil

Chair of Computational Mathematics, University of Deusto, 48007, Bilbao, Basque Country,

Spain

Abstract

Inverse design of transport equations can be addressed by using a gradient-

adjoint methodology. In this methodology numerical schemes used for the

adjoint resolution determine the direction of descent in its iterative algorithm,

and consequently the

CP U

time consumed by the inverse design. As the

CP U time constitutes a known bottleneck, it is important to employ light and

quick schemes to the adjoint problem. In this regard, we proposed to use the

Modiﬁed Method of Characteristics (

MMOC

). Despite not preserving identity

conservation, the

MMOC

is computationally competitive. In this work we

investigated the advantage of using the

MMOC

in comparison with the Lax-

Friedrichs and Lax-Wendroﬀ schemes for the inverse design problem. By testing

the Doswell frontogenesis equation, we observed that the

MMOC

can provide

more eﬃcient and accurate computation under some simulation conditions.

Keywords: Characteristics-based method, inverse design, Doswell frontogenesis.

2020 MSC: 00-01, 99-00

1. Introduction

The problem of inverse design of transport equations can be addressed by

using gradient-adjoint methodologies. Recently, Morales-Hernandez and Zuazua

∗Corresponding author

Email address: afrancisco@id.uff.br (Alexandre Francisco )

Preprint submitted to Computational and Applied Mathematics October 11, 2022

arXiv:2210.03798v1 [math.NA] 7 Oct 2022

[

] investigated the convenience of using low-order numerical schemes for the

adjoint resolution in the gradient-adjoint methodology. They focused on linear

scalar transport equations with heterogeneous time-independent ﬂux functions.

Morales-Hernandez and Zuazua analysed the numerical resolution of the

adjoint problem by means of ﬁrst-order and second-order numerical schemes.

They concluded that ﬁrst-order schemes are the best choices when dealing with

smooth functions since second-order schemes introduce high frequencies and

spurious oscillations in the solution. In addition, ﬁrst-order schemes require

shorter CP U times than second-order ones.

The numerical scheme used for the adjoint resolution determines the descent

direction in the gradient-adjoint method [1], and consequently it inﬂuences the

CP U

time consumed by the iterative algorithm. As the

CP U

time constitutes a

known bottleneck, we proposed a characteristic-based method, the called Modiﬁed

Method of Characteristics (

MMOC

) [

], for eﬃcient adjoint resolutions. The

MMOC

is based on the characteristic curves and so is very computationally

competitive for linear transport equations. Zhang et al. [

] has demonstrated

that the

MMOC

is feasible and reliable in forward and inverse simulations of

underwater explosion, for example.

The Doswell frontogenesis problem is a linear equation in which a non-uniform

and time-dependent ﬂow gives rise a challenging solution to be simulated. It

allows to assess the performance of inverse design simulations in the treatment

of moving vortex-type surfaces in two dimensions. The Doswell frontogenesis

equation can be used to describe the presence of horizontal temperature gradients

and fronts in the context of meteorological dynamics.

We performed numerical simulations in order to investigate the

MMOC

for

solving the Doswell frontogenesis problem. In this work the adjoint problem

is also a linear equation, worthing the use of the characteristic-based method.

For comparisons with the

MMOC

, we used the ﬁrst-order Lax-Friedrichs (

)

and the second-order Lax-Wendroﬀ (

) schemes. The

MMOC

provided

shorter

CP U

time and smaller error than the

and

schemes, under

some simulation conditions. Thus, the

MMOC

can be an eﬃcient and accurate

scheme for addressing the inverse design problem.

This work is presented as follows: this introduction is followed by a summa-

rized development of the gradient-adjoint methodology for the problem of inverse

design of transport equations. Next, the

MMOC

is introduced for solving linear

transport equations. Afterwards, the eﬃciency and accuracy of the

MMOC

is evaluated in comparison with the

and

for the Doswell frontogenesis

equation. Finally, the conclusions about the performance of the

MMOC

are

presented.

2. The gradient-adjoint methodology

Consider the following transport problem with a given time-independent

vector ﬁeld v=v(x):

∂u

∂t +∇ · (vu)=0, u(x,0) = u0.(1)

Obviously, the solution

(

x, t

) exists and it is unique and can be

determined by means of the method of characteristics provided

is smooth

enough (say,

). In this case, for all initial datum

u0∈L2

(

) there exists a

unique solution in the class C([0, T ]; L2(R2)).

The inverse design problem can be stated as follow: given Ω

⊂R2

, and a

target function

u∗

(

), determine the initial condition

such that the

corresponding solution of Eq. (1) satisﬁes u(x, T ) = uTfor all x∈Ω.

This problem can be easily addressed from the point of view of optimal

control. To this end, let us introduce the following functional measuring the

quadratic error with respect to the target function uT:

J(u0) = 1

2ZΩ

(u(·, T )−uT)2dx.(2)

The inverse design problem then translates in the following optimization one:

bu0= min

u0∈L2(RN)J(u0).(3)

This problem

(3)

is typically solved via a gradient descent (

) algorithm as

follows:

bu0= lim

k→+∞uk

uk+1

0=uk

0−η∇J(uk

0).(4)

We then need to compute the gradient

∇J

. To this end, let us ﬁrst rewrite

Eq. (1) in non-divergence form, that is











∂u

∂t +v· ∇u= 0,

u(x,0) = u0.

Let us now introduce the Lagrangian

L(u, σ) := 1

2ZΩ

(u(·, T )−uT)2dx+ZT

0ZΩ

σ−∂u

∂t −v· ∇udxdt

and compute the directional derivative δL(u, σ) as

δL(u, σ) = ZΩ

(u(·, T )−uT)δu(·, T )dx+ZT

0ZΩ

σ−∂δu

∂t −v· ∇δudxdt.

(5)

We now integrate by parts the last term in the above expression, obtaining

0ZΩ

σ−∂δu

∂t −v· ∇δudxdt

=−ZΩ

σ(·, T )δu(·, t)dx+ZΩ

σ(·,0)δu(·,0) dx

+ZT

0ZΩ∂σ

∂t +v· ∇σδu dxdt.

We then obtain from Eq. (5) that

δL(u, σ) = ZΩu(·, T )−u∗−σ(·, T )δu(·, T )dx+ZΩ

σ(·,0)δu(·,0) dx

+ZT

0ZΩ∂σ

∂t +v· ∇σδu dxdt

or, equivalently,

δL(u, σ) = ZΩ

σ(·,0)δu(·,0) dx

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摘要：

ComputationalperformanceoftheMMOCintheinversedesignoftheDoswellfrontogenesisequationAlexandreFranciscoa,,UmbertoBiccarib,EnriqueZuazuabaDepartmentofMechanicalEngineering,UniversidadeFederalFluminense,27255-125,VoltaRedonda,RJ,BrazilbChairofComputationalMathematics,UniversityofDeusto,48007,Bilbao,Ba...

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Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation_2

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