text: 3 regularization terms that promote sparsity and steep variations is usually performed by assuming that the images are smooth, in the solution [11], [12], [13], [10]. which results in the matrix S being a high pass ﬁlter (e.g. a B. Traditional Super-Resolution Reconstruction Laplacian). The scalar αSRR is the regularization parameter. The objective of SRR techniques is to obtain one or more Note that the SRR process described considers imaging images of high resolution (HR) from a set of observed low sensors consisting of regular arrays of square pixels. EIT images resolution (LR) images. This is usually done by solving an from a ﬁnite element mesh, on the other hand, consist of a inverse problem, which is formulated based on a mathematical nonuniform array of irregular elements. This renders the direct model for the relationship between the different HR and application of the SRR methodology described above unfeasible. LR images. The super-resolution of images deriving from In the following, we shall represent the EIT image formation conventional imaging sensors has received a lot of interest, process as an optical imaging system with nonuniform pixels. and is already a consolidated methodology [14]. In this case, This will allow for the application of a SRR methodology − the acquisition process of an image at the t th discrete time previously designed for the reconstruction of images originating instant can be modeled as [15] from hypothetical sensors composed by Rhombus Penrose tiling [4], [5]. y(t) = DHx(t) + e(t) , (5) III. RESAMPLING-BASED SUPER-RESOLUTION FOR EIT × × The EIT image formation process, which is derived from where y(t) (N2 1) and x(t) (M2 1) are the lexicographic the EIT inverse problem in (3) or (4), does not resembles an (i.e. vectorized) representations of the degraded (LR) and × optical image acquisition system as described in (5). In this original (HR) images, respectively. Matrix D (N2 M2) case, each acquired image consists of a nonuniform mesh (i.e. a mode×ls the subsampling taking place at the sensor, and H ﬁnite element mesh - FEM) composed of elements of arbitrary (M2 M2) models the blurring in the acquisition process. sizes and shapes, each one presenting a constant conductivity These matrices are assumed to be known in advance. Vector × value. Moreover, the diffuse nature of the electrical currents e(t) (N2 1) models the observation noise in the sensor. injected in the boundary of the domain and the ill-posedness of In the case of video sequences, the relationship between the inverse problem inﬂict severe distortions on the resulting each pair of HR images can be modelled by [16] image. We will assume these distortions to be appropriately − represented by a combination of a space invariant blurring x(t) = G(t)x(t 1) + s(t) , (6) and a nonlinear, spatially variant distortion. Hence, the typical mathematical modeling of the EIT image formation is quite different from the model used in optical SRR algorithms. where G(t) is a warp matrix that represents the relative − displacement between the HR images x(t 1) and x(t) at In this work we propose to model the EIT image acquisition adjacent time instants. The motion between the images is process using a formulation similar to that used for tradi- estimated from the LR observations using image registration tional image systems. An overall illustration of the proposed techniques, and can be described either by simple parametric methodology can be seen in Figure 1. By considering each models (such as global rotations and translations) or by a element of the ﬁnite element mesh as a pixel of a nonuniform dense motion ﬁeld, where each pixel of the image can move sensor, we propose to employ the Penrose SRR strategy to independently of the others [17], [18]. Vector s(t) models the super resolve the low resolution tomography images. The innovations in x(t), which consists of unpredictable changes EIT images are upsampled from the ﬁnite element mesh − happening in the video sequence between time t 1 and t, into a uniform grid (IHR) befo−re being processed by a super i.e. the information in x(t) which canno−t be represented as a resolution algorithm↑. For t↑he t th time inst↑ant, this process linear combination of the pixels in x(t 1), usually due to is represented as y (t)↑=· y∆(t), where y (t) denotes the scene changes such as occlusions. upsampled image and ( ) the usampling operator. As will − The reconstruction of the t th image based on L previously be discussed later, the reconstruction method also requires the observed frames is performed by minimizing some metric of conjugate downsampling process to be performed. It consists the difference between the degraded HR (estimated) image and on the transition from the IHR to the LR grid, which is done by the LR observations. For the case of the squared L norm, the assigning to each element of the nonuniform mesh the average 2 minimization problem is formulated as [15]: value of all the pixels within the corresponding area of the (cid:110) (cid:88) IHR grid. This process is illustrated in Figure 2. − It should be noted that by assigning a position to the center L 1 (cid:107) − − (cid:111) (cid:107) xˆ(t) = arg min y(t l) DH M − x(t) 2 of each pixel in the regular IHR grid (which encloses the whole t l,t x(t) FEM domain), the value of each IHR pixel can be computed l=0 (cid:107) (cid:107) straightforwardly during the upsampling process by determining + αSRR Sx(t) 2 (7) which element of the FEM circumscribes its position. This is performed using the geometric information about the FEM, − where M − represents the total motion from the t th to which is available with the LR EIT images [2]. Since the − −t l,t ↑ the (t l) th image (which is obtained from the dynamical position of the center of each IHR pixel y (t) lies inside an i model (6)). The last term is a regularization that incorporates unique LR image element y (t) for some p, the upsampling ∆p additional a priori information about the desired solution. This process is identical to the uniform impulsive sampling of

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text: 5 Results for high SNR and ﬁner mesh Results for high SNR and ﬁner mesh × − × − Volume Ratio Hausdorff Distance MASD ( 10 2) Volume Ratio Hausdorff Distance MASD ( 10 2) LR SRR LR SRR LR SRR LR SRR LR SRR LR SRR NOSER 0.46 0.52/0.48 0.12 0.09/0.11 0.99 0.73/0.96 NOSER 0.78 0.91/0.83 0.083 0.033/0.036 0.23 0.05/0.11 TV 0.41 0.41/0.39 0.13 0.12/0.14 1.24 1.21/1.42 TV 0.70 0.87/0.80 0.174 0.086/0.081 0.81 0.30/0.31 TS 0.43 0.48/0.46 0.14 0.11/0.12 1.20 0.90/1.10 TS 0.83 0.92/0.84 0.064 0.030/0.035 0.13 0.04/0.10 Results for low SNR and coarser mesh Results for low SNR and coarser mesh × − × − Volume Ratio Hausdorff Distance MASD ( 10 2) Volume Ratio Hausdorff Distance MASD ( 10 2) LR SRR LR SRR LR SRR LR SRR LR SRR LR SRR NOSER 0.38 0.50/0.46 0.21 0.10/0.12 1.70 0.83/1.10 NOSER 0.51 0.81/0.75 0.154 0.076/0.081 1.14 0.25/0.33 TV 0.38 0.42/0.40 0.24 0.13/0.15 1.70 1.29/1.47 TV 0.46 0.79/0.73 0.216 0.085/0.087 1.85 0.31/0.39 TS 0.39 0.46/0.44 0.16 0.12/0.13 1.52 1.03/1.22 TS 0.69 0.84/0.78 0.100 0.064/0.067 0.46 0.18/0.24 TABLE I TABLE II EXAMPLE 1: AVERAGE ERROR METRICS FOR TRANSLATING T-SHAPED EXAMPLE 2: AVERAGE ERROR METRICS FOR SYNTHETIC LUNG OBJECT. OBJECT. SRR RESULTS ARE PRESENTED FOR THE CASE CONSIDERING SRR RESULTS ARE PRESENTED FOR THE MOTION ESTIMATED FROM THE KNOWN MOTION AND USING A REGISTRATION ALGORITHM (SIDE BY SIDE). KNOWN HR IMAGES AND FROM THE LR OBSERVATIONS (SIDE BY SIDE). conductivity. The matrix HB(t) models the EIT distortion with the linear known a priori and with motion estimated by a registration approximation previously discussed, which can be computed in algorithm. The second example, depicted in Figures 5 and 6, the form of a convolution operator. Matrix HD(t) is a spatially extends the ﬁrst one to EIT images formed by ellipsoids of variant kernel convolution matrix performing the transformation varying volume, which emulates the imaging of a breathing from the HR to the IHR grid. Differently from the decimation thorax. The motion estimation was performed twice, using performed in traditional SRR, HD(t) in (8) is a square matrix both the synthetic HR images and the observed IHR images as which is constructed based on the ﬁnite element mesh. When ↑ input, resulting in different levels of motion estimation errors. processed by this operator, each pixel yi (t) of the resulting In both Examples 1 and 2 the LR EIT images were generated IHR image lying within the area of the p-th nonuniform LR considering two different cases: one with small and another image element y(cid:52) (t) gets assigned the value corresponding with large measurement noise, using a ﬁner and a coarser FE p to the average of all pixels in the HR image xj(t) that falls mesh, respectively. This allows for a better assessment of the within the area of that speciﬁc element of the nonuniform LR performance of the SRR methodology. Finally, a third example, image. This process is illustrated in Figure 2, where matrix depicted in Figures 7 and 8, present the super resolution of HD(t) corresponds to downsampling the HR image to the LR EIT images of real in vivo data, considering two different grid and upsampling the resulting LR image back to the IHR experiments with voltage measurements acquired from healthy grid, which basically amounts to averaging the pixel values in adult subjects breathing. each LR region. The relationship between the HR images is described in the The EIT inverse problem was solved in EIDORS [19], using uniform HR grid, and does not depend on the ﬁnite element three different algorithms, namely, the Gauss-Newton solver mesh or reconstruction algorithm employed in the EIT inverse with the NOSER image prior [20], the Total Variation (TV) problem. Therefore, it can be described in the same way as in regularization [10] and the temporal regularization (TS) [21], the case of conventional SRR, by using the dynamical model (6). all considering difference imaging with the reference voltage In this case, the image registration (motion estimation) is set as the average of all measurements in the time window. performed performed based on the IHR images [17], [18]. These algorithms include both a simple solution (NOSER) Since the imaging model proposed for the EIT system and sophisticated methodologies (TV and TS), which help to given by (8) and (6) is now in the same form as the models illustrate the performance of the SRR methodology for input encountered for traditional optical systems, any traditional SRR images originating from a variety of algorithms, emphasizing its algorithm such as those described in Section can be applied to independence to the EIT inverse problem. The FEMs referred to super-resolve the EIT images, using the proposed observation in all examples are standard meshes available in EIDORS [19]. model and the upsampled EIT images as input. The mesh was also displayed over the reconstructed results in V. EXPERIMENTAL SETUP AND RESULTS the ﬁgures for ease of comparison. The hyperparameters of the algorithms were manually selected in order to achieve a good In this section we present experimental results illustrating performance. For the examples with synthetic images, these the super resolution of EIT images using the imaging model were generated with a background conductivity of 1 S/m and proposed in Section for both synthetic and real data. Three inclusions of 2 S/m. examples were devised to illustrate the application of the proposed method under different circumstances. The ﬁrst We have applied the Least Mean Squares SRR [16] algorithm example, depicted in Figures 3 and 4, presents the reconstruc- to super resolve the images in all the examples, with step tion of images of a T-shaped object subject to translational size µ = 0.01 and K = 100 iterations for each input image. displacements. The algorithm was tested both with motion The motion estimation was performed using the Horn & ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Results

label: table

text: for

text: high

text: SNR

text: and

text: ﬁner

text: mesh

text: Results

text: for

text: high

text: SNR

text: and

text: ﬁner

text: mesh

text: Volume

text: Ratio

text: Hausdorff

text: Distance

text: MASD

text: Volume

text: Ratio

text: Hausdorff

text: Distance

text: MASD

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: NOSER

text: NOSER

text: TV

text: TV

text: TS

text: TS

text: Results

text: for

text: low

text: SNR

text: and

text: coarser

text: mesh

text: Results

text: for

text: low

text: SNR

text: and

text: coarser

text: mesh

text: Volume

text: Ratio

text: Hausdorff

text: Distance

text: MASD

text: Volume

text: Ratio

text: Hausdorff

text: Distance

text: MASD

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: LR

text: SRR

text: NOSER

text: NOSER

text: TV

text: TV

text: TS

text: TS

text: TABLE

text: I

text: TABLE

text: II

text: EXAMPLE

text: AVERAGE

text: ERROR

text: METRICS

text: FOR

text: TRANSLATING

text: T-SHAPED

text: EXAMPLE

text: AVERAGE

text: ERROR

text: METRICS

text: FOR

text: SYNTHETIC

text: LUNG

text: OBJECT.

text: OBJECT.

text: SRR

text: RESULTS

text: ARE

text: PRESENTED

text: FOR

text: THE

text: CASE

text: CONSIDERING

text: SRR

text: RESULTS

text: ARE

text: PRESENTED

text: FOR

text: THE

text: MOTION

text: ESTIMATED

text: FROM

text: THE

text: KNOWN

text: MOTION

text: AND

text: USING

text: A

text: REGISTRATION

text: ALGORITHM

text: (SIDE

text: BY

text: SIDE).

text: KNOWN

text: HR

text: IMAGES

text: AND

text: FROM

text: THE

text: LR

text: OBSERVATIONS

text: (SIDE

text: BY

text: SIDE).

text: conductivity.

text: The

text: matrix

text: HB(t)

text: models

text: the

text: EIT

text: distortion

text: with

text: the

text: linear

text: known

text: a

text: priori

text: and

text: with

text: motion

text: estimated

text: by

text: a

text: registration

text: approximation

text: previously

text: discussed,

text: which

text: can

text: be

text: computed

text: in

text: algorithm.

text: The

text: second

text: example,

text: depicted

text: in

text: Figures

text: and

text: the

text: form

text: of

text: a

text: convolution

text: operator.

text: Matrix

text: HD(t)

text: is

text: a

text: spatially

text: extends

text: the

text: ﬁrst

text: one

text: to

text: EIT

text: images

text: formed

text: by

text: ellipsoids

text: of

text: variant

text: kernel

text: convolution

text: matrix

text: performing

text: the

text: transformation

text: varying

text: volume,

text: which

text: emulates

text: the

text: imaging

text: of

text: a

text: breathing

text: from

text: the

text: HR

text: to

text: the

text: IHR

text: grid.

text: Differently

text: from

text: the

text: decimation

text: thorax.

text: The

text: motion

text: estimation

text: was

text: performed

text: twice,

text: using

text: performed

text: in

text: traditional

text: SRR,

text: HD(t)

text: in

text: is

text: a

text: square

text: matrix

text: both

text: the

text: synthetic

text: HR

text: images

text: and

text: the

text: observed

text: IHR

text: images

text: as

text: which

text: is

text: constructed

text: based

text: on

text: the

text: ﬁnite

text: element

text: mesh.

text: When

text: input,

text: resulting

text: in

text: different

text: levels

text: of

text: motion

text: estimation

text: errors.

text: processed

text: by

text: this

text: operator,

text: each

text: pixel

text: yi

text: (t)

text: of

text: the

text: resulting

text: In

text: both

text: Examples

text: and

text: the

text: LR

text: EIT

text: images

text: were

text: generated

text: IHR

text: image

text: lying

text: within

text: the

text: area

text: of

text: the

text: p-th

text: nonuniform

text: LR

text: considering

text: two

text: different

text: cases:

text: one

text: with

text: small

text: and

text: another

text: image

text: element

text: (t)

text: gets

text: assigned

text: the

text: value

text: corresponding

text: with

text: large

text: measurement

text: noise,

text: using

text: a

text: ﬁner

text: and

text: a

text: coarser

text: FE

text: p

text: to

text: the

text: average

text: of

text: all

text: pixels

text: in

text: the

text: HR

text: image

text: xj(t)

text: that

text: falls

text: mesh,

text: respectively.

text: This

text: allows

text: for

text: a

text: better

text: assessment

text: of

text: the

text: within

text: the

text: area

text: of

text: that

text: speciﬁc

text: element

text: of

text: the

text: nonuniform

text: LR

text: performance

text: of

text: the

text: SRR

text: methodology.

text: Finally,

text: a

text: third

text: example,

text: image.

text: This

text: process

text: is

text: illustrated

text: in

text: Figure

text: where

text: matrix

text: depicted

text: in

text: Figures

text: and

text: present

text: the

text: super

text: resolution

text: of

text: HD(t)

text: corresponds

text: to

text: downsampling

text: the

text: HR

text: image

text: to

text: the

text: LR

text: EIT

text: images

text: of

text: real

text: in

text: vivo

text: data,

text: considering

text: two

text: different

text: grid

text: and

text: upsampling

text: the

text: resulting

text: LR

text: image

text: back

text: to

text: the

text: IHR

text: experiments

text: with

text: voltage

text: measurements

text: acquired

text: from

text: healthy

text: grid,

text: which

text: basically

text: amounts

text: to

text: averaging

text: the

text: pixel

text: values

text: in

text: adult

text: subjects

text: breathing.

text: each

text: LR

text: region.

text: The

text: relationship

text: between

text: the

text: HR

text: images

text: is

text: described

text: in

text: the

text: The

text: EIT

text: inverse

text: problem

text: was

text: solved

text: in

text: EIDORS

text: using

text: uniform

text: HR

text: grid,

text: and

text: does

text: not

text: depend

text: on

text: the

text: ﬁnite

text: element

text: three

text: different

text: algorithms,

text: namely,

text: the

text: Gauss-Newton

text: solver

text: mesh

text: or

text: reconstruction

text: algorithm

text: employed

text: in

text: the

text: EIT

text: inverse

text: with

text: the

text: NOSER

text: image

text: prior

text: the

text: Total

text: Variation

text: (TV)

text: problem.

text: Therefore,

text: it

text: can

text: be

text: described

text: in

text: the

text: same

text: way

text: as

text: in

text: regularization

text: and

text: the

text: temporal

text: regularization

text: (TS)

text: the

text: case

text: of

text: conventional

text: SRR,

text: by

text: using

text: the

text: dynamical

text: model

text: all

text: considering

text: difference

text: imaging

text: with

text: the

text: reference

text: voltage

text: In

text: this

text: case,

text: the

text: image

text: registration

text: (motion

text: estimation)

text: is

text: set

text: as

text: the

text: average

text: of

text: all

text: measurements

text: in

text: the

text: time

text: window.

text: performed

text: performed

text: based

text: on

text: the

text: IHR

text: images

text: These

text: algorithms

text: include

text: both

text: a

text: simple

text: solution

text: (NOSER)

text: Since

text: the

text: imaging

text: model

text: proposed

text: for

text: the

text: EIT

text: system

text: and

text: sophisticated

text: methodologies

text: (TV

text: and

text: TS),

text: which

text: help

text: to

text: given

text: by

text: and

text: is

text: now

text: in

text: the

text: same

text: form

text: as

text: the

text: models

text: illustrate

text: the

text: performance

text: of

text: the

text: SRR

text: methodology

text: for

text: input

text: encountered

text: for

text: traditional

text: optical

text: systems,

text: any

text: traditional

text: SRR

text: images

text: originating

text: from

text: a

text: variety

text: of

text: algorithms,

text: emphasizing

text: its

text: algorithm

text: such

text: as

text: those

text: described

text: in

text: Section

text: can

text: be

text: applied

text: to

text: independence

text: to

text: the

text: EIT

text: inverse

text: problem.

text: The

text: FEMs

text: referred

text: to

text: super-resolve

text: the

text: EIT

text: images,

text: using

text: the

text: proposed

text: observation

text: in

text: all

text: examples

text: are

text: standard

text: meshes

text: available

text: in

text: EIDORS

text: model

text: and

text: the

text: upsampled

text: EIT

text: images

text: as

text: input.

text: The

text: mesh

text: was

text: also

text: displayed

text: over

text: the

text: reconstructed

text: results

text: in

text: V.

text: EXPERIMENTAL

text: SETUP

text: AND

text: RESULTS

text: the

text: ﬁgures

text: for

text: ease

text: of

text: comparison.

text: The

text: hyperparameters

text: of

text: the

text: algorithms

text: were

text: manually

text: selected

text: in

text: order

text: to

text: achieve

text: a

text: good

text: In

text: this

text: section

text: we

text: present

text: experimental

text: results

text: illustrating

text: performance.

text: For

text: the

text: examples

text: with

text: synthetic

text: images,

text: these

text: the

text: super

text: resolution

text: of

text: EIT

text: images

text: using

text: the

text: imaging

text: model

text: were

text: generated

text: with

text: a

text: background

text: conductivity

text: of

text: S/m

text: and

text: proposed

text: in

text: Section

text: for

text: both

text: synthetic

text: and

text: real

text: data.

text: Three

text: inclusions

text: of

text: S/m.

text: examples

text: were

text: devised

text: to

text: illustrate

text: the

text: application

text: of

text: the

text: proposed

text: method

text: under

text: different

text: circumstances.

text: The

text: ﬁrst

text: We

text: have

text: applied

text: the

text: Least

text: Mean

text: Squares

text: SRR

text: algorithm

text: example,

text: depicted

text: in

text: Figures

text: and

text: presents

text: the

text: reconstruc-

text: to

text: super

text: resolve

text: the

text: images

text: in

text: all

text: the

text: examples,

text: with

text: step

text: tion

text: of

text: images

text: of

text: a

text: T-shaped

text: object

text: subject

text: to

text: translational

text: size

text: µ

text: and

text: K

text: iterations

text: for

text: each

text: input

text: image.

text: displacements.

text: The

text: algorithm

text: was

text: tested

text: both

text: with

text: motion

text: The

text: motion

text: estimation

text: was

text: performed

text: using

text: the

text: Horn

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text: 16 Y. CANZANI AND P. SARNAK Rule B is enforced to ensure that no holes are added between edges that join a struc- ture C with any of its children structures C . v (v,j) − Next, assume C ∈B . Note that for any edge e in E (C ) we have that the v join v function h takes the value 1 at their vertices, since those vertices belong to edges in E (C ) and the function h is deﬁned to be 1 on E (C ). We have the same picture ext v ext v if C ∈ B+, only that h takes the value −1 on the vertices of all the joining edges. We v therefore extend h to be deﬁned on e as follows. − C) Perturbation on E (C ). Let v ∈ T and assume C ∈ B . Given an edge in join v v E (C ) we already know that h takes the value 1 at the vertices of the edge. join v We extend h smoothly to the entire edge so that it takes the value −1 at the midpoint of the edge, and so that it only has two roots at which the absolute value of the derivative of h is ≥ 1. We further ask h to have zero derivative at the endpoints of the edge. For example, if the edge is {(a, b, z) : z ∈ [c, c + 1]} Z where a, b, c ∈ , we could take h(a, b, z) = cos(2πz). In the case in which C ∈ B+ we need h to take the value +1 at the midpoint of the edge. v Rule C is meant to glue the structures {E(C )}N through the middle point of the (v,j) j=1 edges that join them, without generating new holes. Remark 7. By construction the function h is smooth in the interior of each edge. Furthermore, since we ask the derivative of h to vanish at the vertices in K, the function R h can be extended to a function h ∈ C1(Ω) where Ω ⊂ 3 is an open neighborhood of K. Deﬁnition 1. Given a tree T , let h ∈ C1(Ω) be deﬁned following Rules A, B and C R and Remark 7, where Ω ⊂ 3 is an open neighborhood of K. Since K is compact and R R R 3\K is connected, Theorem 7 gives the existence of f : 3 → that satisﬁes −∆f = f and sup{|f − h| + (cid:107)∇f − ∇h(cid:107)} ≤ 1 . 100 K For ε > 0 small set u := u + εf. (cid:83) ε 0 We will show in Lemma 9 that the perturbation was built so that the nodal domain of u corresponding to v ∈ T is constituted by the deformed cubes in N E(C )\C ε j=1 (v,j) (v,j) after the perturbation is performed. We illustrate how Rules A, B, and C work in the following examples. In what follows we shall use repeatedly that the singularities of the zero set of u are on the edges and 0 vertices of the cubes. Therefore, the changes of topology in the zero set can only occur after perturbing the function u along the edges and vertices of the cubes. 0 ##LTLine##

text: Y.

text: CANZANI

text: AND

text: P.

text: SARNAK

text: Rule

text: B

text: is

text: enforced

text: to

text: ensure

text: that

text: no

text: holes

text: are

text: added

text: between

text: edges

text: that

text: join

text: a

text: struc-

text: ture

text: C

text: with

text: any

text: of

text: its

text: children

text: structures

text: C

text: v

text: (v,j)

text: Next,

text: assume

text: C

text: ∈B

text: Note

text: that

text: for

text: any

text: edge

text: e

text: in

text: E

text: (C

text: we

text: have

text: that

text: the

text: v

text: join

text: v

text: function

text: h

text: takes

text: the

text: value

text: at

text: their

text: vertices,

text: since

text: those

text: vertices

text: belong

text: to

text: edges

text: in

text: E

text: (C

text: and

text: the

text: function

text: h

text: is

text: deﬁned

text: to

text: be

text: on

text: E

text: (C

text: We

text: have

text: the

text: same

text: picture

text: ext

text: v

text: ext

text: v

text: if

text: C

text: B+,

text: only

text: that

text: h

text: takes

text: the

text: value

text: −1

text: on

text: the

text: vertices

text: of

text: all

text: the

text: joining

text: edges.

text: We

text: v

text: therefore

text: extend

text: h

text: to

text: be

text: deﬁned

text: on

text: e

text: as

text: follows.

text: C)

text: Perturbation

text: on

text: E

text: (C

text: Let

text: v

text: T

text: and

text: assume

text: C

text: B

text: Given

text: an

text: edge

text: in

text: join

text: v

text: v

text: E

text: (C

text: we

text: already

text: know

text: that

text: h

text: takes

text: the

text: value

text: at

text: the

text: vertices

text: of

text: the

text: edge.

text: join

text: v

text: We

text: extend

text: h

text: smoothly

text: to

text: the

text: entire

text: edge

text: so

text: that

text: it

text: takes

text: the

text: value

text: −1

text: at

text: the

text: midpoint

text: of

text: the

text: edge,

text: and

text: so

text: that

text: it

text: only

text: has

text: two

text: roots

text: at

text: which

text: the

text: absolute

text: value

text: of

text: the

text: derivative

text: of

text: h

text: is

text: We

text: further

text: ask

text: h

text: to

text: have

text: zero

text: derivative

text: at

text: the

text: endpoints

text: of

text: the

text: edge.

text: For

text: example,

text: if

text: the

text: edge

text: is

text: {(a,

text: b,

text: z)

text: z

text: [c,

text: c

text: Z

text: where

text: a,

text: b,

text: c

text: we

text: could

text: take

text: h(a,

text: b,

text: z)

text: In

text: the

text: case

text: in

text: which

text: C

text: B+

text: we

text: need

text: h

text: to

text: take

text: the

text: value

text: at

text: the

text: midpoint

text: of

text: the

text: edge.

text: v

text: Rule

text: C

text: is

text: meant

text: to

text: glue

text: the

text: structures

text: {E(C

text: )}N

text: through

text: the

text: middle

text: point

text: of

text: the

text: (v,j)

text: edges

text: that

text: join

text: them,

text: without

text: generating

text: new

text: holes.

text: Remark

text: By

text: construction

text: the

text: function

text: h

text: is

text: smooth

text: in

text: the

text: interior

text: of

text: each

text: edge.

text: Furthermore,

text: since

text: we

text: ask

text: the

text: derivative

text: of

text: h

text: to

text: vanish

text: at

text: the

text: vertices

text: in

text: K,

text: the

text: function

text: R

text: h

text: can

text: be

text: extended

text: to

text: a

text: function

text: h

text: where

text: Ω

text: is

text: an

text: open

text: neighborhood

text: of

text: K.

text: Deﬁnition

text: Given

text: a

text: tree

text: T

text: let

text: h

text: be

text: deﬁned

text: following

text: Rules

text: A,

text: B

text: and

text: C

text: R

text: and

text: Remark

text: where

text: Ω

text: is

text: an

text: open

text: neighborhood

text: of

text: K.

text: Since

text: K

text: is

text: compact

text: and

text: R

text: R

text: R

text: is

text: connected,

text: Theorem

text: gives

text: the

text: existence

text: of

text: f

text: that

text: satisﬁes

text: −∆f

text: f

text: and

text: sup{|f

text: h|

text: K

text: For

text: ε

text: small

text: set

text: u

text: u

text: εf.

text: ε

text: We

text: will

text: show

text: in

text: Lemma

text: that

text: the

text: perturbation

text: was

text: built

text: so

text: that

text: the

text: nodal

text: domain

text: of

text: u

text: corresponding

text: to

text: v

text: T

text: is

text: constituted

text: by

text: the

text: deformed

text: cubes

text: in

text: N

text: E(C

text: )\C

text: ε

text: (v,j)

text: (v,j)

text: after

text: the

text: perturbation

text: is

text: performed.

text: We

text: illustrate

text: how

text: Rules

text: A,

text: B,

text: and

text: C

text: work

text: in

text: the

text: following

text: examples.

text: In

text: what

text: follows

text: we

text: shall

text: use

text: repeatedly

text: that

text: the

text: singularities

text: of

text: the

text: zero

text: set

text: of

text: u

text: are

text: on

text: the

text: edges

text: and

text: vertices

text: of

text: the

text: cubes.

text: Therefore,

text: the

text: changes

text: of

text: topology

text: in

text: the

text: zero

text: set

text: can

text: only

text: occur

text: after

text: perturbing

text: the

text: function

text: u

text: along

text: the

text: edges

text: and

text: vertices

text: of

text: the

text: cubes.

text: ##LTLine##


text: ZERO SET OF MONOCHROMATIC RANDOM WAVES 23 Proof. First, we show that all the cubes in ∪N E(C )\C glue to form part of Ω j=1 (v,j) (v,j) v after the perturbation is performed. Assume, without loss of generality, that C ∈ B+. − v Then, C ∈ B for every child (v, j) of v. All the cubes in ∪N E(C )\C (v,j) j=1 (v,j) (v,j) have an edge in E (C ). Since such cubes are positive, and u takes the value ε on ext (v,j) ε E (C ), it follows that the cubes become part of the nodal domain that contains ext (v,j) E (C ). That is, all the cubes in ∪N E(C )\C become part of Ω after the ext (v,j) j=1 (v,j) (v,j) v perturbation is added to u . 0 Second, we show that no cubes, other than those in ∪N E(C )\C , will glue j=1 (v,j) (v,j) R to form part of Ω . Indeed, any other positive cube in 3\ ∪N E(C ) that touches v j=1 (v,j) ∂(∪N E(C )) does so through an edge in E (C ). Since the function u takes the j=1 (v,j) ext v ε value −ε on E (C ), those cubes will disconnect from ∪N E(C ) after we perturb. ext v j=−1 (v,j) On the other hand, any positive cube in ∪N C ∈ B is touching ∪N E(C ) j=1 (v,j) j=1 (v,j) through edges in ∪Nj E (C ) where N is the number of children of (v, j). Since i=1 ext (v,j,i) j f takes the value −ε on ∪Nj E (C ), the cubes in ∪N C will also disconnect i=1 ext (v,j,i) j=1 (v,j) (cid:3) from ∪N E(C )\C . j=1 (v,j) (v,j) It is convenient to deﬁne the partial collections of nested domains. Given a tree T , ∗ ∗ a perturbation u , and v ∈ T , we deﬁne the collection Ω = Ω (ε) of all nodal domains ε v ∗v that are descendants of Ωv as follows. If v i(cid:91)s a leaf then Ωv = Ωv. If v is not a leaf and has N children, we set N ∗ ∗ Ω := Ω ∪ Ω . v v (v,j) j=1 Remark 8. A direct consequence of Lemma 9 is the following. Let T be a tree and for each ε > 0 let u be the perturbation deﬁned in (1). For each v ∈ T , ε ∗ li→m Ωv(ε) = Cv. ε 0 We will use throughout this section that we know how the zero set behaves at a local scale (as described in Section ). Let T be a tree and for each ε > 0 let u be the ε perturbation deﬁned in (1). We shall prove that there is a subset of the nodal domains of u that are nested as prescribed by T . Since for every v ∈ T the set Ω is a nodal ε v domain of u , the theorem would follow if we had that for all v ∈ T ε ∗ ∗ (i) Ω ⊂ int(Ω ) for every (v, j) child of v. (v,j) v ∗ ∗ (ii) Ω ∩ Ω = ∅ for all j (cid:54)= k. (v,j) (v,k) R ∗ (iii) 3\Ω has no bounded component. v R Statements (i), (ii) and (iii) imply that 3\Ω has N + 1 components. One com- v ∗ ponent is unbounded, and each of the other N components is ﬁlled by Ω for some (v,j) j. We prove statements (i), (ii) and (iii) by induction. The statements are obvious for ##LTLine##

text: ZERO

text: SET

text: OF

text: MONOCHROMATIC

text: RANDOM

text: WAVES

text: Proof.

text: First,

text: we

text: show

text: that

text: all

text: the

text: cubes

text: in

text: ∪N

text: E(C

text: )\C

text: glue

text: to

text: form

text: part

text: of

text: Ω

text: (v,j)

text: (v,j)

text: v

text: after

text: the

text: perturbation

text: is

text: performed.

text: Assume,

text: without

text: loss

text: of

text: generality,

text: that

text: C

text: B+.

text: v

text: Then,

text: C

text: B

text: for

text: every

text: child

text: (v,

text: j)

text: of

text: v.

text: All

text: the

text: cubes

text: in

text: ∪N

text: E(C

text: )\C

text: (v,j)

text: (v,j)

text: (v,j)

text: have

text: an

text: edge

text: in

text: E

text: (C

text: Since

text: such

text: cubes

text: are

text: positive,

text: and

text: u

text: takes

text: the

text: value

text: ε

text: on

text: ext

text: (v,j)

text: ε

text: E

text: (C

text: it

text: follows

text: that

text: the

text: cubes

text: become

text: part

text: of

text: the

text: nodal

text: domain

text: that

text: contains

text: ext

text: (v,j)

text: E

text: (C

text: That

text: is,

text: all

text: the

text: cubes

text: in

text: ∪N

text: E(C

text: )\C

text: become

text: part

text: of

text: Ω

text: after

text: the

text: ext

text: (v,j)

text: (v,j)

text: (v,j)

text: v

text: perturbation

text: is

text: added

text: to

text: u

text: Second,

text: we

text: show

text: that

text: no

text: cubes,

text: other

text: than

text: those

text: in

text: ∪N

text: E(C

text: )\C

text: will

text: glue

text: (v,j)

text: (v,j)

text: R

text: to

text: form

text: part

text: of

text: Ω

text: Indeed,

text: any

text: other

text: positive

text: cube

text: in

text: ∪N

text: E(C

text: that

text: touches

text: v

text: (v,j)

text: ∂(∪N

text: E(C

text: does

text: so

text: through

text: an

text: edge

text: in

text: E

text: (C

text: Since

text: the

text: function

text: u

text: takes

text: the

text: (v,j)

text: ext

text: v

text: ε

text: value

text: −ε

text: on

text: E

text: (C

text: those

text: cubes

text: will

text: disconnect

text: from

text: ∪N

text: E(C

text: after

text: we

text: perturb.

text: ext

text: v

text: (v,j)

text: On

text: the

text: other

text: hand,

text: any

text: positive

text: cube

text: in

text: ∪N

text: C

text: B

text: is

text: touching

text: ∪N

text: E(C

text: (v,j)

text: (v,j)

text: through

text: edges

text: in

text: ∪Nj

text: E

text: (C

text: where

text: N

text: is

text: the

text: number

text: of

text: children

text: of

text: (v,

text: j).

text: Since

text: ext

text: (v,j,i)

text: j

text: f

text: takes

text: the

text: value

text: −ε

text: on

text: ∪Nj

text: E

text: (C

text: the

text: cubes

text: in

text: ∪N

text: C

text: will

text: also

text: disconnect

text: ext

text: (v,j,i)

text: (v,j)

text: from

text: ∪N

text: E(C

text: )\C

text: (v,j)

text: (v,j)

text: It

text: is

text: convenient

text: to

text: deﬁne

text: the

text: partial

text: collections

text: of

text: nested

text: domains.

text: Given

text: a

text: tree

text: T

text: a

text: perturbation

text: u

text: and

text: v

text: T

text: we

text: deﬁne

text: the

text: collection

text: Ω

text: Ω

text: (ε)

text: of

text: all

text: nodal

text: domains

text: ε

text: v

text: ∗v

text: that

text: are

text: descendants

text: of

text: Ωv

text: as

text: follows.

text: If

text: v

text: a

text: leaf

text: then

text: Ωv

text: Ωv.

text: If

text: v

text: is

text: not

text: a

text: leaf

text: and

text: has

text: N

text: children,

text: we

text: set

text: N

text: Ω

text: Ω

text: Ω

text: v

text: v

text: (v,j)

text: Remark

text: A

text: direct

text: consequence

text: of

text: Lemma

text: is

text: the

text: following.

text: Let

text: T

text: be

text: a

text: tree

text: and

text: for

text: each

text: ε

text: let

text: u

text: be

text: the

text: perturbation

text: deﬁned

text: in

text: For

text: each

text: v

text: T

text: ε

text: li→m

text: Ωv(ε)

text: Cv.

text: ε

text: We

text: will

text: use

text: throughout

text: this

text: section

text: that

text: we

text: know

text: how

text: the

text: zero

text: set

text: behaves

text: at

text: a

text: local

text: scale

text: (as

text: described

text: in

text: Section

text: Let

text: T

text: be

text: a

text: tree

text: and

text: for

text: each

text: ε

text: let

text: u

text: be

text: the

text: ε

text: perturbation

text: deﬁned

text: in

text: We

text: shall

text: prove

text: that

text: there

text: is

text: a

text: subset

text: of

text: the

text: nodal

text: domains

text: of

text: u

text: that

text: are

text: nested

text: as

text: prescribed

text: by

text: T

text: Since

text: for

text: every

text: v

text: T

text: the

text: set

text: Ω

text: is

text: a

text: nodal

text: ε

text: v

text: domain

text: of

text: u

text: the

text: theorem

text: would

text: follow

text: if

text: we

text: had

text: that

text: for

text: all

text: v

text: T

text: ε

text: (i)

text: Ω

text: int(Ω

text: for

text: every

text: (v,

text: j)

text: child

text: of

text: v.

text: (v,j)

text: v

text: (ii)

text: Ω

text: Ω

text: for

text: all

text: j

text: k.

text: (v,j)

text: (v,k)

text: R

text: (iii)

text: has

text: no

text: bounded

text: component.

text: v

text: R

text: Statements

text: (i),

text: (ii)

text: and

text: (iii)

text: imply

text: that

text: has

text: N

text: components.

text: One

text: com-

text: v

text: ponent

text: is

text: unbounded,

text: and

text: each

text: of

text: the

text: other

text: N

text: components

text: is

text: ﬁlled

text: by

text: Ω

text: for

text: some

text: (v,j)

text: j.

text: We

text: prove

text: statements

text: (i),

text: (ii)

text: and

text: (iii)

text: by

text: induction.

text: The

text: statements

text: are

text: obvious

text: for

text: ##LTLine##


text: 26 Y. CANZANI AND P. SARNAK − Join operation. Given C ∈ B+ ∪ B we distinguish two vertices using the lexico- − Z graphic order. For C ∈ B+ ∪ B , let Γ = C ∩ n be the set of its vertices. We let C v (C) be the largest vertex in Γ and v−(C) be the smallest vertex in Γ . Given the + C C vertex v (C) we deﬁne the edge e (C) to be the edge in ∂C that contains the vertex + + v (C) and is parallel to the hyperplane deﬁned by the x , . . . , x − coordinates. The + 1 n 2 edge e−(C) is deﬁned in the same way. Given C ∈ B+ and C ∈ B+ we deﬁne J(C , C ) ∈ B+ as follows. Let C˜ be the 1 2 1 2 2 translated copy of C for which e (C ) coincides with e−(C˜ ). We “join” C and C 2 + 1 2 1 2 as J(C , C ) = C ∪ C˜ . 1 2 1 2 In addition, for a single set C we deﬁne J(C) = C, and if there are multiple sets C , . . . , C we deﬁne J(C , . . . , C ) = J(C , J(C , J(C , . . . J(C − , C )))). 1 n 1 n 1 2 3 n 1 n Deﬁnition of the rough nested domains. Given a tree T we associate to each (cid:0) (cid:1)R node v ∈ T a structure C ⊂ n deﬁned as follows. If the node v ∈ T is a v leaf, then C is a cube of the adequate sign. For the rest of the nodes we set v C = J E(C ), . . . , E(C ) , where N is the number of children of the node v (v,1) (v,N) v. We continue to identify the original structures E(C ) with the translated ones (v,j) (cid:91) E˜(C ) that are used to build C . After this identiﬁcation, (v,j) v N C = E(C ). v (v,j) j=1 Building the perturbation. Let v ∈ T be a node with N children. We deﬁne the R sets of edges E (C ), E (C(cid:91)) and E (C ) in exactly the same way as we did in 3 ext v int v join v R (see Section ). We proceed to deﬁne a perturbation h : K → , where K = E (C ) ∪ E (C ) ∪ E (C ). ext v int v join v ∈ v T The function h is deﬁned by the rules A, B and C below. Let χ : [0, ∞] → [−1, 1] be a smooth increasing function satisfying χ(0) = −1, χ(1/2) = 0 and χ(t) = 1 for t ≥ 1. We also demand (cid:48) (cid:48) χ (0) = 0 and χ (1/2) ≥ 1. (30) − A) Perturbation on E (C ). Let v ∈ T and assume C ∈ B . We deﬁne h on ev- ext v v ery edge of E (C ) to be 1. If C ∈ B+, we deﬁne h on every edge of E (C ) ext v v ext v to be −1. B) Perturbation on E (C ). Let H (a , a ) be an edge that touches both E (C ) int v c i j ext −v and E (C ) for some of the child structures C of C . Assume C ∈ B . ext (v,(cid:96)) (v,(cid:96)) v v Then we know that we must have h|E = 1 and h|E = −1. Let ext(Cv) ext(C(v,(cid:96)))

text: Y.

text: CANZANI

text: AND

text: P.

text: SARNAK

text: Join

text: operation.

text: Given

text: C

text: B+

text: B

text: we

text: distinguish

text: two

text: vertices

text: using

text: the

text: lexico-

text: Z

text: graphic

text: order.

text: For

text: C

text: B+

text: B

text: let

text: Γ

text: C

text: n

text: be

text: the

text: set

text: of

text: its

text: vertices.

text: We

text: let

text: C

text: v

text: (C)

text: be

text: the

text: largest

text: vertex

text: in

text: Γ

text: and

text: v−(C)

text: be

text: the

text: smallest

text: vertex

text: in

text: Γ

text: Given

text: the

text: C

text: C

text: vertex

text: v

text: (C)

text: we

text: deﬁne

text: the

text: edge

text: e

text: (C)

text: to

text: be

text: the

text: edge

text: in

text: ∂C

text: that

text: contains

text: the

text: vertex

text: v

text: (C)

text: and

text: is

text: parallel

text: to

text: the

text: hyperplane

text: deﬁned

text: by

text: the

text: x

text: x

text: coordinates.

text: The

text: n

text: edge

text: e−(C)

text: is

text: deﬁned

text: in

text: the

text: same

text: way.

text: Given

text: C

text: B+

text: and

text: C

text: B+

text: we

text: deﬁne

text: J(C

text: C

text: B+

text: as

text: follows.

text: Let

text: C˜

text: be

text: the

text: translated

text: copy

text: of

text: C

text: for

text: which

text: e

text: (C

text: coincides

text: with

text: e−(C˜

text: We

text: “join”

text: C

text: and

text: C

text: as

text: J(C

text: C

text: C

text: C˜

text: In

text: addition,

text: for

text: a

text: single

text: set

text: C

text: we

text: deﬁne

text: J(C)

text: C,

text: and

text: if

text: there

text: are

text: multiple

text: sets

text: C

text: C

text: we

text: deﬁne

text: J(C

text: C

text: J(C

text: J(C

text: J(C

text: J(C

text: C

text: n

text: n

text: n

text: n

text: Deﬁnition

text: of

text: the

text: rough

text: nested

text: domains.

text: Given

text: a

text: tree

text: T

text: we

text: associate

text: to

text: each

text: node

text: v

text: T

text: a

text: structure

text: C

text: n

text: deﬁned

text: as

text: follows.

text: If

text: the

text: node

text: v

text: T

text: is

text: a

text: v

text: leaf,

text: then

text: C

text: is

text: a

text: cube

text: of

text: the

text: adequate

text: sign.

text: For

text: the

text: rest

text: of

text: the

text: nodes

text: we

text: set

text: v

text: C

text: J

text: E(C

text: E(C

text: where

text: N

text: is

text: the

text: number

text: of

text: children

text: of

text: the

text: node

text: v

text: (v,N)

text: v.

text: We

text: continue

text: to

text: identify

text: the

text: original

text: structures

text: E(C

text: with

text: the

text: translated

text: ones

text: (v,j)

text: E˜(C

text: that

text: are

text: used

text: to

text: build

text: C

text: After

text: this

text: identiﬁcation,

text: (v,j)

text: v

text: N

text: C

text: E(C

text: v

text: (v,j)

text: Building

text: the

text: perturbation.

text: Let

text: v

text: T

text: be

text: a

text: node

text: with

text: N

text: children.

text: We

text: deﬁne

text: the

text: R

text: sets

text: of

text: edges

text: E

text: (C

text: E

text: and

text: E

text: (C

text: in

text: exactly

text: the

text: same

text: way

text: as

text: we

text: did

text: in

text: ext

text: v

text: int

text: v

text: join

text: v

text: R

text: (see

text: Section

text: We

text: proceed

text: to

text: deﬁne

text: a

text: perturbation

text: h

text: K

text: where

text: K

text: E

text: (C

text: E

text: (C

text: E

text: (C

text: ext

text: v

text: int

text: v

text: join

text: v

text: v

text: T

text: The

text: function

text: h

text: is

text: deﬁned

text: by

text: the

text: rules

text: A,

text: B

text: and

text: C

text: below.

text: Let

text: χ

text: be

text: a

text: smooth

text: increasing

text: function

text: satisfying

text: and

text: χ(t)

text: for

text: t

text: We

text: also

text: demand

text: χ

text: and

text: χ

text: A)

text: Perturbation

text: on

text: E

text: (C

text: Let

text: v

text: T

text: and

text: assume

text: C

text: B

text: We

text: deﬁne

text: h

text: on

text: ev-

text: ext

text: v

text: v

text: ery

text: edge

text: of

text: E

text: (C

text: to

text: be

text: If

text: C

text: B+,

text: we

text: deﬁne

text: h

text: on

text: every

text: edge

text: of

text: E

text: (C

text: ext

text: v

text: v

text: ext

text: v

text: to

text: be

text: B)

text: Perturbation

text: on

text: E

text: (C

text: Let

text: H

text: (a

text: a

text: be

text: an

text: edge

text: that

text: touches

text: both

text: E

text: (C

text: int

text: v

text: c

text: i

text: j

text: ext

text: −v

text: and

text: E

text: (C

text: for

text: some

text: of

text: the

text: child

text: structures

text: C

text: of

text: C

text: Assume

text: C

text: B

text: ext

text: v

text: v

text: Then

text: we

text: know

text: that

text: we

text: must

text: have

text: h|E

text: and

text: h|E

text: Let

text: ext(Cv)


text: 30 Y. CANZANI AND P. SARNAK [Sod] M. Sodin. Lectures on random nodal portraits. Lecture Notes for a Mini-course Given at the St. Petersburg Summer School in Probability and Statistical Physics (2012). [SW] P. Sarnak and I. Wigman. Topologies of nodal sets of random band limited functions. Preprint arXiv:1312.7858 (2013). [Wh] H. Whitney. Analytic extension of diﬀerentiable functions deﬁned on closed sets. Transactions of the American Mathematical Society (1934) 36, 63-89. (Y. Canzani) University of North Carolina at Chapel Hill. Email address: canzani@email.unc.edu (P. Sarnak) Institute for Advanced Study and Princeton University. Email address: sarnak@math.ias.edu

text: Y.

text: CANZANI

text: AND

text: P.

text: SARNAK

text: [Sod]

label: reference

text: M.

text: Sodin.

text: Lectures

text: on

text: random

text: nodal

text: portraits.

text: Lecture

text: Notes

text: for

text: a

text: Mini-course

text: Given

text: at

text: the

text: St.

text: Petersburg

text: Summer

text: School

text: in

text: Probability

text: and

text: Statistical

text: Physics

text: [SW]

text: P.

text: Sarnak

text: and

text: I.

text: Wigman.

text: Topologies

text: of

text: nodal

text: sets

text: of

text: random

text: band

text: limited

text: functions.

text: Preprint

text: [Wh]

text: H.

text: Whitney.

text: Analytic

text: extension

text: of

text: diﬀerentiable

text: functions

text: deﬁned

text: on

text: closed

text: sets.

text: Transactions

text: of

text: the

text: American

text: Mathematical

text: Society

text: (Y.

text: Canzani)

text: University

label: author

text: of

text: North

text: Carolina

text: at

text: Chapel

text: Hill.

text: Email

text: address:

text: canzani@email.unc.edu

label: footer

text: (P.

text: Sarnak)

text: Institute

text: for

text: Advanced

text: Study

text: and

text: Princeton

text: University.

text: Email

text: address:

text: sarnak@math.ias.edu


text: ZERO SET OF MONOCHROMATIC RANDOM WAVES 7 That is, we can approximate f on compact subsets in the Ct-topology by elements of P and T respectively. 1 1 Proof. Let f ∈ E1. Since f is analy(cid:88)tic (cid:88)we can expand it in a rapidly convergent series in the Y (cid:96) ’s. That is, m ∞ d (cid:96) f(x) = a (|x|)Y (cid:96) (|x|). m,(cid:96) m x (cid:96)=0 m=1 (cid:90) Moreover, for r > 0, (cid:88) (cid:88) ∞ d (cid:96) |f(rx˙)|2 dσ(x˙) = |a (r)|2. (16) m,(cid:96) − Sn 1 (cid:96)=0 m=1 − R In polar coordinates, (r, θ) ∈ (0, +∞) × Sn 1, the Laplace operator in n is given by n − 1 1 ∆ = ∂2 + ∂ + ∆ − , r r r r2 Sn 1 and hence for each (cid:96), m we have that (cid:48)(cid:48) (cid:48) r2a (r) + (n − 1)ra (r) + (r2 − (cid:96)((cid:96) + n − 2))a (r) = 0. (17) m,(cid:96) m,(cid:96) m,(cid:96) where (cid:96) is some positive integer. There are two linearly independent solutions to − (17). One is r νJ (r) and the other blows up as r → 0. Since the left hand side of (cid:96)+ν (16) is ﬁnite as r → 0, it follows that the a ’s cannot pick up any component of the m,(cid:96) blowing up solution. That is, for r ≥ 0 J (r) (cid:96)+ν a (r) = c , m,(cid:96) (cid:96),m rν R (cid:18) (cid:19) for some c ∈ . Hence, (cid:88) (cid:88) m,(cid:96) ∞ d (cid:96) x J (|x|) f(x) = c Y (cid:96) (cid:96)+ν . (18) (cid:96),m m |x| |x|ν (cid:96)=0 m=1 Furthermore, this series converges absolutely and uniformly on compact subsets, as (cid:16) (cid:17) also do its derivatives. Thus, f can be approximated by members of P as claimed, by 1 simply truncating the series in (18). | | To deduce the same for T it suﬃces to approximate each ﬁxed Y (cid:96) |x| J(cid:96)+|ν|( x ). To 1 m x x ν − this end let ξ , −ξ , ξ , −ξ , . . . , ξ , −ξ be a sequence of points in Sn 1 which become 1 1 2 2 N N equidis(cid:88)trib(cid:16)uted with respect to dσ as N → ∞. (cid:17)Then, (cid:90)as N → ∞, N 1 − (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) e i x,ξj Y (cid:96) (ξ ) + (−1)(cid:96)ei x,ξ(cid:16)j Y (cid:96)(cid:17)(ξ ) −→ (cid:82) e i x,ξ Y (cid:96) (ξ) dσ(ξ). (19) 2N m j m j − m Sn 1 j=1 | | − (cid:104) (cid:105) The proof follows since (2π)n2 i(cid:96) Ym(cid:96) |xx| J(cid:96)+|xν|(νx ) = Sn−1 e i x,ξ Ym(cid:96) (ξ) dσ(ξ). In(cid:3)- deed, the convergence in (19) is uniform over compact subsets in x. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ZERO

text: SET

text: OF

text: MONOCHROMATIC

text: RANDOM

text: WAVES

text: That

text: is,

text: we

text: can

text: approximate

text: f

text: on

text: compact

text: subsets

text: in

text: the

text: Ct-topology

text: by

text: elements

text: of

text: P

text: and

text: T

text: respectively.

text: Proof.

text: Let

text: f

text: Since

text: f

text: is

text: can

text: expand

text: it

text: in

text: a

text: rapidly

text: convergent

text: series

text: in

text: the

text: Y

text: ’s.

text: That

text: is,

text: m

text: d

text: f(x)

text: a

text: (|x|)Y

text: (|x|).

text: m

text: x

label: equation

text: Moreover,

text: for

text: r

text: d

text: dσ(x˙)

text: |a

text: Sn

text: R

text: In

text: polar

text: coordinates,

text: (r,

text: θ)

text: Sn

text: the

text: Laplace

text: operator

text: in

text: n

text: is

text: given

text: by

text: n

text: ∂2

text: r

text: r

text: r

text: Sn

text: and

text: hence

text: for

text: each

text: m

text: we

text: have

text: that

text: (r)

text: (n

text: (r)

text: n

text: (r)

text: where

text: is

text: some

text: positive

text: integer.

text: There

text: are

text: two

text: linearly

text: independent

text: solutions

text: to

text: One

text: is

text: r

text: νJ

text: (r)

text: and

text: the

text: other

text: blows

text: up

text: as

text: r

text: Since

text: the

text: left

text: hand

text: side

text: of

text: is

text: ﬁnite

text: as

text: r

text: it

text: follows

text: that

text: the

text: a

text: ’s

text: cannot

text: pick

text: up

text: any

text: component

text: of

text: the

text: blowing

text: up

text: solution.

text: That

text: is,

text: for

text: r

text: J

text: (r)

text: a

text: (r)

text: c

text: rν

text: R

text: for

text: some

text: c

text: Hence,

text: d

text: x

text: J

text: (|x|)

text: f(x)

text: c

text: Y

text: m

text: |x|

text: |x|ν

text: Furthermore,

text: this

text: series

text: converges

text: absolutely

text: and

text: uniformly

text: on

text: compact

text: subsets,

text: as

text: also

text: do

text: its

text: derivatives.

text: Thus,

text: f

text: can

text: be

text: approximated

text: by

text: members

text: of

text: P

text: as

text: claimed,

text: by

text: simply

text: truncating

text: the

text: series

text: in

text: To

text: deduce

text: the

text: same

text: for

text: T

text: it

text: suﬃces

text: to

text: approximate

text: each

text: ﬁxed

text: Y

text: |x|

text: x

text: To

text: m

text: x

text: x

text: ν

text: this

text: end

text: let

text: ξ

text: −ξ

text: ξ

text: −ξ

text: ξ

text: −ξ

text: be

text: a

text: sequence

text: of

text: points

text: in

text: Sn

text: which

text: become

text: N

text: N

text: with

text: respect

text: to

text: dσ

text: as

text: N

text: N

text: N

text: e

text: i

text: x,ξj

text: Y

text: (ξ

text: Y

text: e

text: i

text: x,ξ

text: Y

text: (ξ)

text: dσ(ξ).

text: m

text: j

text: m

text: j

text: m

text: Sn

text: The

text: proof

text: follows

text: since

text: |xx|

text: e

text: i

text: x,ξ

text: (ξ)

text: dσ(ξ).

text: deed,

text: the

text: convergence

text: in

text: is

text: uniform

text: over

text: compact

text: subsets

text: in

text: x.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 B. KHOROMSKIJ AND S. REPIN (cid:107) · (cid:107) ∗ where V is a reﬂexive Banach space with the norm , V is the space conjugate to V ∗ V A → ∗ (the respective duality pairing is denoted by < v , v >), and : V V is a bounded linear A operator. It is assumed that the operator is positive deﬁnite and invertible, so that the P A problem (1.1) is well posed. However, is viewed as a very diﬃcult problem because is generated by a complicated physical structure, which may contain a huge amount details. Therefore, attempts to solve (1.1) numerically by standard methods may lead to enormous expenditures. Similar diﬃculties arise if we wish to verify the quality of a numerical solution. A Assume that the operator is approximated by a simpliﬁed positive deﬁnite operator A A A A ◦ and the inversion of ◦ is much simpler than inversion of . By means of ◦, we P A construct an iteration method based on solving a ”simple” problem : ◦u◦ = g. In other → A− 0 words, the method is based on the operation g ◦ 1g. It also includes the operation → A v v, which can be performed very eﬃciently by tensor type decomposition methods A provided that physical structures generated have low rank representations. We prove that iterations generate a sequence of functions converging to the exact solution of (1.1) with a geometrical rate. Furthermore, we deduce explicitly computable and guaranteed a posteriori error estimates adapted to this class of problems. They evaluate the accuracy of approximations computed on each step of the iteration algorithm. These estimates also use A → A only inversion of ◦ and operations of the type v v. In the iteration methods and error A estimates inversion of the operator is avoided. In the paper, we consider one class of problems associated with divergent type elliptic A ∗ A ∗ → equations where = Q ΛQ and = Q Λ◦Q. Here Λ : Y Y is a bounded operator → ∗ → ∗ induced by a complicated quasi–periodic structure while Q : V Y and Q : Y V are conjugate operators, i.e., ∗ ∀ ∈ ∈ (1.2) (y, Qw) =< Q y, w > y Y and w V, · · (cid:107) · (cid:107) where Y is a Hilbert space with the scalar product ( , ) and the norm . The operators ∗ Q and Q are induced by diﬀerential operators or certain ﬁnite dimensional approximations ∈ V V of them. Henceforth, it is assumed that f , where is a Hilbert space with the scalar · · ∗ ∈ V ∈ ∗ product ( , )V. This space is intermediate between V and V , i.e., V V . A ∗ The operator ◦ = Q Λ◦Q contains the operator Λ◦ generated by a simpliﬁed structure. We assume that the operators Λ and Λ◦ are Hermitiam (i.e., (Λy, z) = (y, Λz) and (Λ◦y, z) = (y, Λ◦z)) and satisfy the conditions ◦ (cid:107) (cid:107) ≤ ≤ ◦ (cid:107) (cid:107) ∀ ∈ (1.3) λ(cid:9) y 2 (Λ◦y, y) λ⊕ y 2 y Y, (cid:107) (cid:107) ≤ ≤ (cid:107) (cid:107) (1.4) λ(cid:9) y 2 (Λy, y) λ⊕ y 2, λ(cid:9) < λ⊕. Then, the structural operators Λ and Λ◦ are spectrally equivalent ≤ ≤ (1.5) c (Λ◦y, y) (Λy, y) c (Λ◦y, y), 1 2 where the constants are the minimal and maximal eigenvalues of the generalized spectral − ≥ ≤ problem Λy µΛ◦y = 0. Obviously, they satisfy the estimates c λ(cid:9)◦ and c λ⊕◦ (which 1 λ⊕ 2 λ(cid:9) may be rather coarse). Concerning the operator Q, we assume that there exists a positive constant c such that (cid:107) (cid:107) ≥ (cid:107) (cid:107) ∀ ∈ (1.6) Qw c w w V. V P P Generalized solutions of the problems and are deﬁned by the variational identities 0 ∀ ∈ (1.7) (ΛQu, Qw) =< f, w > w V, ##LTLine## ##LTLine##

text: B.

text: KHOROMSKIJ

text: AND

text: S.

text: REPIN

text: where

text: V

text: is

text: a

text: reﬂexive

text: Banach

text: space

text: with

text: the

text: norm

text: V

text: is

text: the

text: space

text: conjugate

text: to

text: V

text: V

text: A

text: (the

text: respective

text: duality

text: pairing

text: is

text: denoted

text: by

text: v

text: v

text: and

text: V

text: V

text: is

text: a

text: bounded

text: linear

text: A

text: operator.

text: It

text: is

text: assumed

text: that

text: the

text: operator

text: is

text: positive

text: deﬁnite

text: and

text: invertible,

text: so

text: that

text: the

text: P

text: A

text: problem

text: is

text: well

text: posed.

text: However,

text: is

text: viewed

text: as

text: a

text: very

text: diﬃcult

text: problem

text: because

text: is

text: generated

text: by

text: a

text: complicated

text: physical

text: structure,

text: which

text: may

text: contain

text: a

text: huge

text: amount

text: details.

text: Therefore,

text: attempts

text: to

text: solve

text: numerically

text: by

text: standard

text: methods

text: may

text: lead

text: to

text: enormous

text: expenditures.

text: Similar

text: diﬃculties

text: arise

text: if

text: we

text: wish

text: to

text: verify

text: the

text: quality

text: of

text: a

text: numerical

text: solution.

text: A

text: Assume

text: that

text: the

text: operator

text: is

text: approximated

text: by

text: a

text: simpliﬁed

text: positive

text: deﬁnite

text: operator

text: A

text: A

text: A

text: A

text: and

text: the

text: inversion

text: of

text: is

text: much

text: simpler

text: than

text: inversion

text: of

text: By

text: means

text: of

text: we

text: P

text: A

text: construct

text: an

text: iteration

text: method

text: based

text: on

text: solving

text: a

text: ”simple”

text: problem

text: ◦u◦

text: g.

text: In

text: other

text: A−

text: words,

text: the

text: method

text: is

text: based

text: on

text: the

text: operation

text: g

text: It

text: also

text: includes

text: the

text: operation

text: A

text: v

text: v,

text: which

text: can

text: be

text: performed

text: very

text: eﬃciently

text: by

text: tensor

text: type

text: decomposition

text: methods

text: A

text: provided

text: that

text: physical

text: structures

text: generated

text: have

text: low

text: rank

text: representations.

text: We

text: prove

text: that

text: iterations

text: generate

text: a

text: sequence

text: of

text: functions

text: converging

text: to

text: the

text: exact

text: solution

text: of

text: with

text: a

text: geometrical

text: rate.

text: Furthermore,

text: we

text: deduce

text: explicitly

text: computable

text: and

text: guaranteed

text: a

text: posteriori

text: error

text: estimates

text: adapted

text: to

text: this

text: class

text: of

text: problems.

text: They

text: evaluate

text: the

text: accuracy

text: of

text: approximations

text: computed

text: on

text: each

text: step

text: of

text: the

text: iteration

text: algorithm.

text: These

text: estimates

text: also

text: use

text: A

text: A

text: only

text: inversion

text: of

text: and

text: operations

text: of

text: the

text: type

text: v

text: v.

text: In

text: the

text: iteration

text: methods

text: and

text: error

text: A

text: estimates

text: inversion

text: of

text: the

text: operator

text: is

text: avoided.

text: In

text: the

text: paper,

text: we

text: consider

text: one

text: class

text: of

text: problems

text: associated

text: with

text: divergent

text: type

text: elliptic

text: A

text: A

text: equations

text: where

text: Q

text: ΛQ

text: and

text: Q

text: Λ◦Q.

text: Here

text: Λ

text: Y

text: Y

text: is

text: a

text: bounded

text: operator

text: induced

text: by

text: a

text: complicated

text: quasi–periodic

text: structure

text: while

text: Q

text: V

text: Y

text: and

text: Q

text: Y

text: V

text: are

text: conjugate

text: operators,

text: i.e.,

text: (y,

text: Qw)

text: Q

text: y,

text: w

text: y

text: Y

text: and

text: w

text: V,

text: where

text: Y

text: is

text: a

text: Hilbert

text: space

text: with

text: the

text: scalar

text: product

text: and

text: the

text: norm

text: The

text: operators

text: Q

text: and

text: Q

text: are

text: induced

text: by

text: diﬀerential

text: operators

text: or

text: certain

text: ﬁnite

text: dimensional

text: approximations

text: V

text: V

text: of

text: them.

text: Henceforth,

text: it

text: is

text: assumed

text: that

text: f

text: where

text: is

text: a

text: Hilbert

text: space

text: with

text: the

text: scalar

text: V

text: product

text: )V.

text: This

text: space

text: is

text: intermediate

text: between

text: V

text: and

text: V

text: i.e.,

text: V

text: V

text: A

text: The

text: operator

text: Q

text: Λ◦Q

text: contains

text: the

text: operator

text: Λ◦

text: generated

text: by

text: a

text: simpliﬁed

text: structure.

text: We

text: assume

text: that

text: the

text: operators

text: Λ

text: and

text: Λ◦

text: are

text: Hermitiam

text: (i.e.,

text: (Λy,

text: z)

text: (y,

text: Λz)

text: and

text: (Λ◦y,

text: z)

text: (y,

text: Λ◦z))

text: and

text: satisfy

text: the

text: conditions

text: y

text: (Λ◦y,

text: y)

text: λ⊕

text: y

text: y

text: Y,

text: y

text: (Λy,

text: y)

text: λ⊕

text: y

text: λ⊕.

text: Then,

text: the

text: structural

text: operators

text: Λ

text: and

text: Λ◦

text: are

text: spectrally

text: equivalent

text: c

text: (Λ◦y,

text: y)

text: (Λy,

text: y)

text: c

text: (Λ◦y,

text: y),

text: where

text: the

text: constants

text: are

text: the

text: minimal

text: and

text: maximal

text: eigenvalues

text: of

text: the

text: generalized

text: spectral

text: problem

text: Λy

text: µΛ◦y

text: Obviously,

text: they

text: satisfy

text: the

text: estimates

text: c

text: and

text: c

text: λ⊕◦

text: (which

text: λ⊕

text: may

text: be

text: rather

text: coarse).

text: Concerning

text: the

text: operator

text: Q,

text: we

text: assume

text: that

text: there

text: exists

text: a

text: positive

text: constant

text: c

text: such

text: that

text: Qw

text: c

text: w

text: w

text: V.

text: V

text: P

text: P

text: Generalized

text: solutions

text: of

text: the

text: problems

text: and

text: are

text: deﬁned

text: by

text: the

text: variational

text: identities

text: (ΛQu,

text: Qw)

text: f,

text: w

text: w

text: V,

text: ##LTLine##

text: ##LTLine##


text: (cid:101) RANK STRUCTURED METHOD 13 Here v and v are two consequent numerical approximations (e.g., ﬁnite element approxima- ρ I tions vk and vk+1 computed on a mesh . Then h h h − (cid:107) − (cid:107) η = ηk := vk vk+1 and δ = δk := vk vk+1 ◦ (cid:90) ρ(cid:90) h h h (cid:90) ρ h (cid:90) h (cid:90) (cid:0) (cid:1) are directly computable. Since a◦ is a ”simple” function, the integrals 1 − 1 − 1 (cid:48) 1 1 F = a◦ 1 dx, F = a◦ 1 g dx, F = a◦ ηk 2 dx, F = a◦ (µ¯ + g)2 dx, F = fηk dx 1 2 3 h 4 5 h (cid:90) (cid:90) (cid:90) (cid:90) 0 0 0 0 0 (cid:0) (cid:1) are easy to compute. Other integrals 1 − (cid:48) 1 (cid:48) (cid:48) 1 − (cid:48) 1 − (cid:48) G = a◦ 1a vk dx, G = a (vk) ηk dx, G = (µ¯ + g)a◦ 1a vk dx, G = a◦ 1 a2 vk ) 2 dx 1 h 2 h h 3 h 4 h 0 0 0 0 contain highly oscillating coeﬃcient a multiplied by piece wise polynomial mesh functions. If a has a low QTT rank tensor representation [20], then the integrals can be eﬃciently computed by ten(cid:101)sor type methods already discussed in [23]. We have − − G F I⊕2 (v, v ) = F + 2G + 2ρF + ρ2(F 2G + G ) =: εk, µ¯ = 1 2 . ρ 3 2 5 4 3 4 F 1 Here ρ = 2 is selected in accordance with Section 3. The respective contraction factor h−(cid:9)+h⊕ h⊕ h(cid:9) is q = . Now (4.8) yields easily computable lower and upper bounds of the error h(cid:9)+h⊕ encompassed in vk: h − δk εk ≤ (cid:107) − (cid:107) ≤ δk + εk vk u ◦ − 1 + q h 1 q Computation of M⊕ for 2d problems can be also reduced to the computation of one dimensional integrals. Certainly on the multidimensional case the amount of integrals is much larger. However the basic tensor decomposition methods remain the same. Below we brieﬂy discuss them with the paradigm of a simple case where f = f(1)(x )f(2)(x ) and a = a(1)(x )a(2)(x ). 1 2 1 2 Assume that approximations are represented in the form of series formed by one dimensional (cid:88) (cid:88) (cid:88) (cid:88) functions φ(1) and φ(2) (which may be supported locally or globally), so that i j (cid:101) (cid:101) n1 n2 n1 n2 v = γ φ(1)(x )φ(2)(x ), v = γ φ(1)(x )φ(2)(x ). ij i 1 j 2 ρ ij i 1 j 2 (cid:32) i=1 j=1 i=1 j(cid:33)=1 (cid:88) (cid:88) (cid:88) (cid:88) In this case, (cid:101) ∇ n1 n2 ∂φ(1) n1 n2 ∂φ(2) − η = ς i φ(2) , ς φ(1) j , where ς = γ γ . ρ ij ∂x j ij i ∂x ij ij ij 1 2 i=1 j=1 i=1 j=1 We deﬁne another set of one dimensional fun(cid:40)ctions W (1)(x ) and W (2)(x(cid:41)), which form the (cid:88) (cid:88) k 1 l 2 vector function m1 m2 ∂W (2) −∂W (1) (4.9) y = Υ + σ Υ , Υ = W (1) l ; k W (2) . 0 kl kl kl k ∂x ∂x l 2 1 k=1 l=1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RANK

text: STRUCTURED

text: METHOD

text: Here

text: v

text: and

text: v

text: are

text: two

text: consequent

text: numerical

text: approximations

text: (e.g.,

text: ﬁnite

text: element

text: approxima-

text: ρ

text: I

text: tions

text: vk

text: and

text: computed

text: on

text: a

text: mesh

text: Then

text: h

text: h

text: h

text: η

text: ηk

text: vk

text: and

text: δ

text: δk

text: vk

text: h

text: h

text: h

text: ρ

text: h

text: h

text: are

text: directly

text: computable.

text: Since

text: a◦

text: is

text: a

text: ”simple”

text: function,

text: the

text: integrals

text: F

text: a◦

text: dx,

text: F

text: a◦

text: g

text: dx,

text: F

text: a◦

text: ηk

text: dx,

text: F

text: a◦

text: (µ¯

text: dx,

text: F

text: fηk

text: dx

text: h

text: h

text: are

text: easy

text: to

text: compute.

text: Other

text: integrals

text: G

text: a◦

text: vk

text: dx,

text: G

text: a

text: (vk)

text: ηk

text: dx,

text: G

text: (µ¯

text: g)a◦

text: vk

text: dx,

text: G

text: a◦

text: vk

text: dx

text: h

text: h

text: h

text: h

text: h

text: contain

text: highly

text: oscillating

text: coeﬃcient

text: a

text: multiplied

text: by

text: piece

text: wise

text: polynomial

text: mesh

text: functions.

text: If

text: a

text: has

text: a

text: low

text: QTT

text: rank

text: tensor

text: representation

text: then

text: the

text: integrals

text: can

text: be

text: eﬃciently

text: computed

text: by

text: type

text: methods

text: already

text: discussed

text: in

text: We

text: have

text: G

text: F

text: (v,

text: v

text: F

text: G

text: εk,

text: µ¯

text: ρ

text: F

text: Here

text: ρ

text: is

text: selected

text: in

text: accordance

text: with

text: Section

text: The

text: respective

text: contraction

text: factor

text: h⊕

text: is

text: q

text: Now

text: yields

text: easily

text: computable

text: lower

text: and

text: upper

text: bounds

text: of

text: the

text: error

text: encompassed

text: in

text: vk:

text: h

text: δk

text: εk

text: δk

text: εk

text: vk

text: u

text: q

text: h

text: q

text: Computation

text: of

text: M⊕

text: for

text: problems

text: can

text: be

text: also

text: reduced

text: to

text: the

text: computation

text: of

text: one

text: dimensional

text: integrals.

text: Certainly

text: on

text: the

text: multidimensional

text: case

text: the

text: amount

text: of

text: integrals

text: is

text: much

text: larger.

text: However

text: the

text: basic

text: tensor

text: decomposition

text: methods

text: remain

text: the

text: same.

text: Below

text: we

text: brieﬂy

text: discuss

text: them

text: with

text: the

text: paradigm

text: of

text: a

text: simple

text: case

text: where

text: f

text: and

text: a

text: Assume

text: that

text: approximations

text: are

text: represented

text: in

text: the

text: form

text: of

text: series

text: formed

text: by

text: one

text: dimensional

text: functions

text: and

text: (which

text: may

text: be

text: supported

text: locally

text: or

text: globally),

text: so

text: that

text: i

text: j

text: v

text: γ

text: v

text: γ

text: ij

text: i

text: j

text: ρ

text: ij

text: i

text: j

text: In

text: this

text: case,

text: η

text: ς

text: i

text: ς

text: j

text: where

text: ς

text: γ

text: γ

text: ρ

text: ij

text: ∂x

text: j

text: ij

text: i

text: ∂x

text: ij

text: ij

text: ij

text: We

text: deﬁne

text: another

text: set

text: of

text: one

text: dimensional

text: W

text: and

text: W

text: which

text: form

text: the

text: k

text: l

text: vector

text: function

text: ∂W

text: −∂W

text: y

text: Υ

text: σ

text: Υ

text: Υ

text: W

text: l

text: k

text: W

text: kl

text: kl

text: kl

text: k

text: ∂x

text: ∂x

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RANK STRUCTURED METHOD 17 Figure 5.1. Example of the 2D periodic oscillating coeﬃcients (left) and the 1D factor a (x ). 1 1 We apply the FEM Galerkin discretization of equation (5.4) by means of tensor-product piecewise aﬃne basis functions (instead of ”linear ﬁnite elements”) { · · · } ∈ I { } ϕ (x) := ϕ (x ) ϕ (x ) , i = (i , . . . , i ), i = 1, . . . , n , (cid:96) = 1, . . . , d, i i1 1 id d 1 d (cid:96) (cid:96) (cid:96) where ϕ are 1D ﬁnite element basis functions (say, piecewise linear hat functions). ik { } We associate the univariate basis functions with the uniform grid ν , j = 1, . . . , n , on j (cid:96) [0, 1] with the mesh size h = 1/(n + 1). In this construction we have N = n n ...n basis (cid:96) 1 2 d functions ϕ . Notice that the univariate grid size n is of the order of n = O(1/(cid:15)) designating i (cid:96) (cid:96) the total problem size N = O(1/(cid:15)d). For ease of exposition we, ﬁrst, consider the case d = 2, and further assume that the scalar (cid:88) diﬀusion coeﬃcient a(x , x ) can be represented in the form 1 2 R a(x , x ) = a(1)(x )a(2)(x ) > 0 1 2 k 1 k 2 k=1 with a small rank parameter R. × The N N stiﬀness matrix is constructed by the standard mapping of the multi-index i into the N-long univariate index i representing all degrees of freedom. For instance, we use the so-called big-endian convention for d = 3 and d = 2 (cid:55)→ − − (cid:55)→ − i i := i + (i 1)n + (i 1)n n , i i := i + (i 1)n , 3 2 3 1 2 3 2 1 2 respectively. Hence all matrices and vectors are deﬁned on the long index i as usual, however, the special Kronecker structure allows the low-storage and low-complexity matrix vector multiplications when appropriate, i.e. when a vector also admits the low-rank Kronecker form representation. In particular, the basis function ϕ is designated via the long index, i.e. i ϕ = ϕ . i i First, we consider the simplest case R = 1 and let d = 2. We construct the Galerkin ∈ R × stiﬀness matrix A = [a ] N N in the form of a sum of Kronecker products of small ij × × ”univariate” matrices. Recall that given p q matrix A and p q matrix B, their × 1 1 2 2 Kronecker product is deﬁned as a p p q q matrix C via the block representation 1 2 1 2 ⊗ C = A B = [a B], i = 1, . . . , p , j = 1, . . . , q . ij 1 1 We say that the Kronecker rank of the matrix A in the representation above equals to 1. Now the elements of Galerkin stiﬀness matrix take a form ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: RANK

text: STRUCTURED

text: METHOD

text: Figure

text: Example

label: caption

text: of

text: the

text: periodic

text: oscillating

text: coeﬃcients

text: (left)

text: and

text: the

text: factor

text: a

text: (x

text: We

text: apply

text: the

text: FEM

text: Galerkin

text: discretization

text: of

text: equation

text: by

text: means

text: of

text: tensor-product

text: piecewise

text: aﬃne

text: basis

text: functions

text: (instead

text: of

text: ”linear

text: ﬁnite

text: elements”)

text: I

text: ϕ

text: (x)

text: ϕ

text: (x

text: ϕ

text: (x

text: i

text: (i

text: i

text: i

text: n

text: d,

text: i

text: id

text: d

text: d

text: where

text: ϕ

text: are

text: ﬁnite

text: element

text: basis

text: functions

text: (say,

text: piecewise

text: linear

text: hat

text: functions).

text: ik

text: We

text: associate

text: the

text: univariate

text: basis

text: functions

text: with

text: the

text: uniform

text: grid

text: ν

text: j

text: n

text: on

text: j

text: with

text: the

text: mesh

text: size

text: h

text: In

text: this

text: construction

text: we

text: have

text: N

text: n

text: n

text: ...n

text: basis

text: d

text: functions

text: ϕ

text: Notice

text: that

text: the

text: univariate

text: grid

text: size

text: n

text: is

text: of

text: the

text: order

text: of

text: n

text: designating

text: i

text: the

text: total

text: problem

text: size

text: N

text: For

text: ease

text: of

text: exposition

text: we,

text: ﬁrst,

text: consider

text: the

text: case

text: d

text: and

text: further

text: assume

text: that

text: the

text: scalar

text: diﬀusion

text: coeﬃcient

text: a(x

text: x

text: can

text: be

text: represented

text: in

text: the

text: form

text: R

text: a(x

text: x

text: k

text: k

text: with

text: a

text: small

text: rank

text: parameter

text: R.

text: The

text: N

text: N

text: stiﬀness

text: matrix

text: is

text: constructed

text: by

text: the

text: standard

text: mapping

text: of

text: the

text: multi-index

text: i

text: into

text: the

text: N-long

text: univariate

text: index

text: i

text: representing

text: all

text: degrees

text: of

text: freedom.

text: For

text: instance,

text: we

text: use

text: the

text: so-called

text: big-endian

text: convention

text: for

text: d

text: and

text: d

text: i

text: i

text: i

text: (i

text: (i

text: n

text: i

text: i

text: i

text: (i

text: respectively.

text: Hence

text: all

text: matrices

text: and

text: vectors

text: are

text: deﬁned

text: on

text: the

text: long

text: index

text: i

text: as

text: usual,

text: however,

text: the

text: special

text: Kronecker

text: structure

text: allows

text: the

text: low-storage

text: and

text: low-complexity

text: matrix

text: vector

text: multiplications

text: when

text: appropriate,

text: i.e.

text: when

text: a

text: vector

text: also

text: admits

text: the

text: low-rank

text: Kronecker

text: form

text: representation.

text: In

text: particular,

text: the

text: basis

text: function

text: ϕ

text: is

text: designated

text: via

text: the

text: long

text: index,

text: i.e.

text: i

text: ϕ

text: ϕ

text: i

text: i

text: First,

text: we

text: consider

text: the

text: simplest

text: case

text: R

text: and

text: let

text: d

text: We

text: construct

text: the

text: Galerkin

text: R

text: stiﬀness

text: matrix

text: A

text: [a

text: N

text: N

text: in

text: the

text: form

text: of

text: a

text: sum

text: of

text: Kronecker

text: products

text: of

text: small

text: ij

text: ”univariate”

text: matrices.

text: Recall

text: that

text: given

text: p

text: q

text: matrix

text: A

text: and

text: p

text: q

text: matrix

text: B,

text: their

text: Kronecker

text: product

text: is

text: deﬁned

text: as

text: a

text: p

text: p

text: q

text: q

text: matrix

text: C

text: via

text: the

text: block

text: representation

text: C

text: A

text: B

text: [a

text: B],

text: i

text: p

text: j

text: q

text: ij

text: We

text: say

text: that

text: the

text: Kronecker

text: rank

text: of

text: the

text: matrix

text: A

text: in

text: the

text: representation

text: above

text: equals

text: to

text: Now

text: the

text: elements

text: of

text: Galerkin

text: stiﬀness

text: matrix

text: take

text: a

text: form

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##