Doubova Krasotchenko, Anna

idUS. Depósito de Investigación Universidad de Sevilla

Numerical solution of some geometric inverse problemsAnna DOUBOVADpto. E.D.A.N. - Univ. of Sevillajoint work withE. FERNA´NDEZ-CARA - Dpto. E.D.A.N. - Univ. of SevillaNumerical Resolution for Inverse ProblemsBCAM, Bilbao, 8-9 January 2015A. Doubova Numerical solution of some geometric inverse problemsmOutline1 Motivation: Elastography2 Wave equation3 Elasticity systems4 Reconstruction and numerical algorithms5 Numerical results2D wave equation2D Lame´ system3D wave equation3D Lame´ system6 Work in progressA. Doubova Numerical solution of some geometric inverse problemsmGeometric Inverse Problems governed by PDEsMotivation: ElastographyWe consider:Geometric inverse problemsWave equation and Lame´ systemsMotivation: ElastrographyA non-invasive method of tumor detection: when a mechanicalcompression or vibration is applied, the tumor deforms less than thesurrounding tissueA technique to detect elastic properties of tissue from acoustic wavegenerators (applications in Medicine)A. Doubova Numerical solution of some geometric inverse problemsmMotivationElastographyClassical detection methods in mammography:Figure: Palpation Figure: x-raysA. Doubova Numerical solution of some geometric inverse problemsmMotivationElastographyElastography (“imaging palpation”) is better suited than palpation and x-raystechniques:— Tumors can be far from the surface— or small— or may have properties indistinguishable through palpation or x-raysFigure: Stiffness is represented by a color spectrum, rangingfrom dark red (very stiff) through orange, yellow, and green, toblue (very soft).A. Doubova Numerical solution of some geometric inverse problemsmWave equationN-dimensional wave equation (N = 2 or 3)(a) Direct problem:Data: Ω, T > 0, ϕ, D and γ ⊂ ∂ΩResult: the solution u(1)utt −∆u = 0 in (Ω \ D)× (0,T )u = ϕ on (∂Ω)× (0,T )u = 0 on (∂D)× (0,T )u(x , 0) = u0, ut (x , 0) = u1 in ΩInformation:(2) α =∂u∂non γ × (0,T )(b) Inverse problem:(Partial) data: Ω, T , ϕ and γ ⊂ ∂Ω(Additional) information: αGoal: Find D such that the solution to (1) satisfies (2)A. Doubova Numerical solution of some geometric inverse problemsmN-dimensional wave equationUniquenessu itt −∆u i = 0 in Ω \ Di × (0,T ), i = 0, 1u i = ϕ in ∂Ω× (0,T )u i = 0 in ∂Di × (0,T )u i (x , 0) = 0, u it (x , 0) = 0 in Ω \ DiTheoremT > T∗(Ω, γ), D0,D1 are convex, ϕ 6= 0∂u0∂n=∂u1∂non γ × (0,T ) =⇒ D0 = D1Fundamental results: Ho¨rmander, LionsAttention: Weaker than the geometric condition (Only uniqueness, notobservability!)A. Doubova Numerical solution of some geometric inverse problemsmElasticity systemsIsotropic caseutt −∇ · (µ(x)(∇u +∇ut ) + λ(x)(∇ · u)Id.) = 0 in Ω \ D × (0,T )u = ϕ on ∂Ω× (0,T )u = 0 on ∂D × (0,T )u(0) = u0, ut (0) = u1 in Ω \ DObservation: σ(u) · n :=(µ(x)(∇u +∇ut ) + λ(x)(∇ · u)Id.)· n on γ × (0,T )Explanations:u = (u1, u2, u3) is the displacement vectorSmall displacements. Hence, linear elasticityIsotropy assumptions. The tissue is described by λ and µA. Doubova Numerical solution of some geometric inverse problemsmN-dimensional elasticity systemUniquenessu itt −∇ · (µ(x)(∇u i +∇)(u i )t ) + λ(x)(∇ · u i )Id.) = 0 in Ω \ Di × (0,T )u i = ϕ on ∂Ω× (0,T )u i = 0 on ∂Di × (0,T )u i (0) = 0, u it (0) = 0 in Ω \ DiTheorem (Constant coefficients)T > T∗(Ω, γ), D0,D1 are convex, ϕ 6= 0σ(u0) · n = σ(u1) · n on γ × (0,T )}=⇒ D0 = D1For uniqueness, the key point is Unique continuation property(Imanuvilov–Yamamoto, 2008, complex conditions on µ, λ)AD, E. Ferna´ndez-Cara, work in progress(∃ Other unique continuation results for stationary problems:Lin–Wang, 2005; Escauriaza, 2005; Alessandrini–Morasi, 2001;Nakamura–Wang, 2006; Imanuvilov–Yamamoto, 2012)A. Doubova Numerical solution of some geometric inverse problemsmReconstruction: 2-D wave equationCase of a ballResolution of an optimization problemOptimization problem: case of a ballGiven: α˜ = α˜(x , t).Find x0, y0 and r such that (x0, y0, r) ∈ Xb andJ(x0, y0, r) ≤ J(x ′0, y ′0, r ′) ∀ (x ′0, y ′0, r ′) ∈ Xbthe function J : Xb 7→ R is defined byJ(x0, y0, r) :=12∫∫γ×(0,T )|α[x0, y0, r ]− α˜|2 ds dtwithα[x0, y0; r ] :=∂u∂non γ × (0,T )andXb := { (x0, y0, r) ∈ R3 : B(x0, y0; r) ⊂ Ω }A. Doubova Numerical solution of some geometric inverse problemsmNumerical algorithms IAugmented Lagrangian method (ff-NLopt - AUGLAG)The problem formulation contains inequality constraints{Minimize f (x)Subject to x ∈ X0; ci (x) ≥ 0, 1 ≤ i ≤ IX0 = {x ∈ Rm : x j ≤ xj ≤ x j , 1 ≤ j ≤ m}Optimization problemMinimize LA(x , λk ;µk ) := f (x)−I∑i=1λki (ci (x)− si ) + 12µkI∑i=1(ci (x)− si )2Subject to x ∈ X0; si ≥ 0, 1 ≤ i ≤ Iλki : multipliers, µk : penalty parametersA. Doubova Numerical solution of some geometric inverse problemsmNumerical algorithms IIAugmented Lagrangian method (ff-NLopt - AUGLAG)Algorithm (Augmented Lagrangian: inequality constraints)(a) Fix µ1 and starting points x0 and λ1;(b) The, for given k ≥ 1, µk , xk−1, λk :(b.1) Unconstrained optimization: Find an approximate minimizer xk of LA( · , λk ;µk ),starting at xk−1: {Minimize LA(x, λk ;µk )Subject to x ∈ X0(b.2) Update the Lagrange multipliers:λk+1i = max(λki −ci (xk )µk, 0), 1 ≤ i ≤ I(b.3) Choose a new parameter and check whether a stopping convergence test is satisfied:µk+1 ∈ (0, µk )Subsidiary optimization algorithms for (b.1) (among others):CRS2 is a gradient-free algorithm a version of Controlled RandomSearch (CRS) for global optimizationDIRECTNoScal is variant of the DIviding RECTangles algorithm forglobal optimizationA. Doubova Numerical solution of some geometric inverse problemsmAUGLAG - CRS2 IControlled Random Search algorithm{Minimize f (x)Subject to x ∈ G(1)where G ⊂ Rm is either a box or some other region easy to sample andf : G ⊂ Rm 7→ R is continuous.CRS2: Main ideas1 Large initial sample of random points2 At each step:The current worst point xh is replaced by a new trial point x˜(generates from the current best point x` and other randompoints)A stopping condition fh − f` ≤ ε is checkedA. Doubova Numerical solution of some geometric inverse problemsmAUGLAG - DIRECT - DIRECTNoScal IDIviding RECTangles algorithmDIRECT: Main ideas(a) Normalize the domain to be the unit hyper-cube with center c1Find f (c1); set fmin = f (c1), i = 0, k = 1Evaluate f (c1 ± 13 ei ), 1 ≤ i ≤ m, and divide the hyper-cube: c1 ± 13 ei arethe centers of the new hyper-rectangles (see Figure)(b) Then, for given k ≥ 1:(b.1) Identify the set S of all potentially optimal rectangles(b.2) For each rectangle in S, identify the longest side(s), evaluate f at the center,divide in smaller rectangles and update fminPotentially optimal means:Best value at the center if the size is the sameOptimal value at the center if the size is minimalA. Doubova Numerical solution of some geometric inverse problemsmAUGLAG - DIRECT - DIRECTNoScal IIDIviding RECTangles algorithmFigure: Some interactions of DIRECT algorithmA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation ICase of a ballTest 1: T = 5, u0 = 10x , u1 = 0, ϕ = 10xx0des = -3, y0des = 0, rdes = 0.4x0ini = 0, y0ini = 0, rini = 0.6NLopt (AUGLAG + CRS2), NoIter = 1007, FreeFem++:x0cal = -2.998645439, y0cal = 0.000425214708rcal = 0.4001667063Figure: Initial mesh: triangles 992, vertices 526 Figure: The desired center and radius of the ballA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation IICase of a ballFigure: Computed center and radius: AUGALG + CRS2x0cal= -2.998645439, y0cal = 0.000425214708, rcal=0.4001667063A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation IIICase of a ball0 50 100 150 200 250 300 350 400 450 5000500100015002000250030003500IterationsCurrent Function ValuesFunction valueFigure: Evolution of J during the first 500 iterations of CRS2A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation IVCase of a ball0.2 0.3 0.4 0.5 0.6 0.7 0.8−0.200.20.40.60.811.21.41.61.8 r  J(x0,r)Figure: The functional J with respect to the variable rA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation VCase of a ball−5−4−3−2−1−2−1012020406080100120Figure: The functional J with respect to the variables x0 and y0A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation VICase of a ballTest 2: T = 5, u0 = 10x , u1 = 0, ϕ = 10xx0des = -3, y0des = 0, rdes = 0.4x0ini = 0, y0ini = 0, rini = 0.6NLopt (AUGLAG + DIRECT), NoIter = 1001, FreeFem++:x0cal = -2.962962963y0cal = -0.01219326322rcal = 0.4220164609A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation VIICase of a ball 20 ANNA DOUBOVA AND ENRIQUE FERNA´NDEZ-CARAFigure 14. Some experiences with AUGLAG and DIRECTNoScal asthe subsidiary optimization algorithm.Remark 3. As in the one-dimensional case, it is possible to introduce anothermethod for the partial identification on B(x0; r) relying on di↵erentiation withrespect to the domain. Thus, for each su ciently small m 2 R2 ⇥R, let us assumethat ↵ = @u@n | ⇥(0,T ) and ↵m = @um@n | ⇥(0,T ) are known, withm = (d, s), D +m = B(x0 + d; r + s).Here, u (reps. um) is the solution to (35) corresponding to B = B(x0; r) (resp.B = B(x0 + d; r + s)). Using domain variation techniques, we obtain the followingformula:↵m   ↵ = L(d, s) + 12Q((d, s), (d, s)) + o(|d|2 + |s)|2),whereL(d, s) =@z@n    ⇥(0,T ), Q((d, s), (d, s)) =@w@n    ⇥(0,T ),Figure: Some experiences with AUGLAG and DIRECTA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation VIIICase of a ballFigure: Desired and computed radius and centers of the ballA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation ICase of an ellipseOptimization problem: case of an ellipseGiven: α˜ = α˜(x , t).Find x0, y0 and θ and a, b such that (x0, y0, θ, a, b) ∈ Xe andJ(x0, y0, θ, a, b) ≤ J(x ′0, y ′0, θ′, a′, b′) ∀ (x ′0, y ′0, θ′, a′, b′) ∈ Xe, (2)the function J : Xe 7→ R is defined byJ(x0, y0, θ, a, b) :=12∫∫γ×(0,T )|α[x0, y0, θ, a, b]− α˜|2 ds dtwithα[x0, y0, θ, a, b] =∂u∂non γ × (0,T )Xe := { (x0, y0, θ, a, b) ∈ R5 : E(x0, y0, θ, a, b) ⊂ Ω }A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation IICase of an ellipseResolution of an optimization problem. Now, J = J(x0, y0, θ, a, b)Test 3: T = 5, u0 = 10x , u1 = 0, ϕ = 10xx0des=-3, y0des=-3, sin(thetades)=0, ades=0.8, bdes=0.4x0ini=-1, y0ini=-1, sin(thetaini)=0, aini=0.5, bini=0.5NLopt (AUGLAG + DIRECTNoScal), NoIter = 2001, FreeFem++x0cal = -2.963301665y0cal = -3.035106437sin(thetacal) = 0.112178021acal = 0.8446502058bcal = 0.4166666667A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D wave equation IIICase of an ellipseFigure: Computed center, radius, angle and semi-axisA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system ICase of a ballutt −∇ · σ(u) = 0 in Ω \ D × (0,T )u = ϕ on ∂Ω× (0,T )u = 0 on ∂D × (0,T )u(0) = u0, ut (0) = u1 in Ω \ Dσ(u) · n :=(µ(x)(∇u +∇ut ) + λ(x)(∇ · u)Id.)· n = σ˜ on γ × (0,T )Optimization problemGiven: σ˜ = σ˜(x , t)Find x0, y0 and r (x0, y0, r) ∈ Xb andJ(x0, y0, r) ≤ J(x ′0, y ′0, r ′) ∀ (x ′0, y ′0, r ′) ∈ Xb, (3)the function J : Xb 7→ R is defined byJ(x0, y0, r) :=12∫∫γ×(0,T )|σ[x0, y0, r ]− σ˜|2 ds dt .A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system IICase of a ballTest 4: T = 5, u01 = 10x , u02 = 10y , u11 = 0, u12 = 0,ϕ1 = 10x , ϕ2 = 10yx0des = -3, y0des = 0, rdes = 0.4x0ini = 0, y0ini = 0, rini = 0.6NLopt (AUGLAG + DIRECTNoScal), No Iter = 1000, FreeFem++x0cal = -3.000224338y0cal = -0.0005268693985rcal = 0.4000228624A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system IIICase of a ballFigure: Computed center and radiusA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system ICase of an ellipseTest 5: T = 5, u01 = 10x , u02 = 10y , u11 = 0, u12 = 0,ϕ1 = 10x , ϕ2 = 10yx0des=-3, y0des=0, sin(thetades)=0, ades=0.8, bdes=0.4x0ini=-1, y0ini=-1, sin(thetaini)=0, aini=0.5, bini=0.5NLopt (AUGLAG + DIRECTNoScal), No Iter = 2001, FreeFem++:x0cal = -3.002591068y0cal = -3.001574963sin(thetacal) = 0.00548696845acal = 0.8036351166bcal = 0.400617284A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system IICase of an ellipseFigure: Computed center, angle and semi-axisA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 2-D Lame´ system IIICase of an ellipseFigure: Computed solution at the final timeA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D wave equation ICase of a sphereTest 6: T = 5, u0 = 10x , u1 = 0, ϕ = 10xx0des = -2, y0des = -2, z0des = -2, rdes = 1x0ini = 0, y0ini = 0, z0des = 0, rini = 0.6NLopt (AUGLAG + DIRECTNoScal), No Iter = 438, FreeFem++x0cal = -1.975308642y0cal = -2.232383275z0cal = -2.305542854rcal = 1.05A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D wave equation IICase of a sphereFigure: Initial mesh. Points: 829, tetrahedra: 4023, faces: 8406, edges: 5210,boundary faces: 720, boundary edges: 1080A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D wave equation IIICase of a sphereFigure: Desired configurationA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D wave equation IVCase of a sphereFigure: Computed observation and meshA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D wave equation VCase of a sphereFigure: Solution of the wave equation corresponding to computed dataA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system ICase of a sphereTest 7: T = 5,u01 = 10x , u02 = 10y , u03 = 10zu11 = 0, u12 = 0, u13 = 0ϕ1 = 10x , ϕ2 = 10y , ϕ3 = 10zx0des = -2, y0des = -2, z0des = -2, rdes = 1x0ini = 0, y0ini = 0, z0des = 0, rini = 0.6NLopt (AUGLAG + DIRECTNoScal), No Iter = 444 , FreeFem++:x0cal = -1.981405274y0cal = -2.225232904z0cal = -2.148084171rcal = 0.9504115226A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system IICase of a sphereFigure: Initial mesh. Points: 829, tetrahedra: 4023, faces: 8406, edges: 5210,boundary faces: 720, boundary edges: 1080A. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system IIICase of a sphereFigure: Desired configurationA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system IVCase of a sphereA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system VCase of a sphereA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system VICase of a sphereA. Doubova Numerical solution of some geometric inverse problemsmNumerical results: 3-D Lame´ system VIICase of a sphereA. Doubova Numerical solution of some geometric inverse problemsmSome additional commentsWork in progressWork in progress:1 Evolution elasticity system:−utt −∇ · σ(u) = 0 in Ω \ D × (0,T )u = ϕ on ∂Ω× (0,T )u = 0 on ∂D × (0,T )u(0) = u0, ut (0) = u1 in Ω \ Dσkl (u) =3∑i,j=1aijklεij (u), εkl (u) =12(∂k ul + ∂luk )aijkl = λ(x)δijδkl + µ(x)(δikδjl + δilδjk), 1 ≤ i , j , k , l ≤ 3— Numerical results ?2 Ellipsoids, other more complicated geometries3 Internal observations ?: At present: new (emerging) techniques detectinternal waves via non-invasive techniques (a very precise description)A. Doubova Numerical solution of some geometric inverse problemsmReferencesAD, E. Ferna´ndez -Cara, Some geometric inverse problems for the linearwave equation, to appear in Inverse Problems and ImagingAD, E. Ferna´ndez -Cara, Geometric inverse problems concerning theLame´ system, work in progressJ. Nocedal, S. J. Wright, Numerical Optimization, Springer, 1999J. Ophir, I. Cespedes, H. Ponnekanti, Y. Yazdi X. Li, Elastography: Aquantitative method for imaging the elasticity of biological tissues,Ultrasonic Imaging, 13 (1991), 111 –134W. L. Price, A controlled random search procedure for globaloptimisation, The Computer Journal, 20 (1977), 367–370.D. E. Finkel, Direct optimization algorithm user guide, Center forResearch in Scientific Computation, North Carolina State University, 2.A. Doubova Numerical solution of some geometric inverse problemsm

Numerical solution of some geometric inverse problems

https://idus.us.es/xmlui/bitstream/11441/54649/1/Numerical%20solution%20of%20some%20geometric%20inverse%20problems.pdf

Numerical solution of some geometric inverse problems

Abstract

Similar works

Full text

Available Versions

idUS. Depósito de Investigación Universidad de Sevilla