An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry
AAn extremal problem arising in the dynamicsof two-phase materials that directly reveals information about the internal geometry
Ornella Mattei , Graeme W. Milton , and Mihai Putinar Department of Mathematics, San Francisco State University, CA 94132, USA, Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA, Department of Mathematics, University of California at Santa Barbara, CA 93106, USA, andSchool of Mathematics, Statistics and Physics, Newcastle University, NE1 7RU Newcastle uponTyne, UK.Emails: [email protected], [email protected], [email protected],[email protected]
Abstract
In two phase materials, each phase having a non-local response in time, it has beenfound that for appropriate driving fields the response somehow untangles at specifictimes, and allows one to directly infer useful information about the geometry of thematerial, such as the volume fractions of the phases. Motivated by this, and to findhow the appropriate driving fields may be designed, we obtain approximate, measureindependent, linear relations between the values that Markov functions take at a givenset of possibly complex points, not belonging to the interval [-1,1] where the measureis supported. The problem is reduced to simply one of polynomial approximationof a given function on the interval [-1,1] and to simplify the analysis Chebyshev ap-proximation is used. This allows one to obtain explicit estimates of the error of theapproximation, in terms of the number of points and the minimum distance of thepoints to the interval [-1,1]. Assuming this minimum distance is bounded below by anumber greater than 1/2, the error converges exponentially to zero as the number ofpoints is increased. Approximate linear relations are also obtained that incorporate aset of moments of the measure. In the context of the motivating problem, the analysisalso yields bounds on the response at any particular time for any driving field, and al-lows one to estimate the response at a given frequency using an appropriately designeddriving field that effectively is turned on only for a fixed interval of time. The approxi-mation extends directly to Markov-type functions with a positive semidefinite operatorvalued measure, and this has applications to determining the shape of an inclusion ina body from boundary flux measurements at a specific time, when the time-dependentboundary potentials are suitably tailored.
Keywords : Composites, best rational approximation, volume fraction estimation,bounds on transient response, Calderon problem, Markov functions a r X i v : . [ m a t h - ph ] J u l Introduction
Following the tradition in approximation theory, Cauchy transforms of positive measureswith compact support on the real line are called in the present note
Markov functions . Someauthors may suitably call them Herglotz functions, or Nevanlinna functions, or Stieltjestransforms.Suppose F µ ( z ) is a Markov function having the integral representation F µ ( z ) = (cid:90) − dµ ( λ ) λ − z , (1.1)where the Borel measure µ is positive with unit mass: (cid:90) − dµ ( λ ) = 1 . (1.2)Given m (possibly complex) points z , z , . . . , z m not belonging to the interval [ − , α , α , . . . , α m such that m (cid:88) k =1 α k F µ ( z k ) ≈ µ . Optimal bounds correlating the possible values of the m -tuple ( F µ ( z ) , F µ ( z ) , . . . , F µ ( z m )) as µ varies over all probability measures are well known,as derived from the well charted analysis of the Nevanlinna-Pick interpolation problem [29].Indeed, the nonlinear constraints among the values F µ ( z ) , F µ ( z ) , . . . , F µ ( z m ), and standardconvexity theory provide optimal bounds on the range of the left hand side of (1.3) for givenconstants α , α , . . . , α m , see [29] for details. But this is not our main concern.We would rather like to choose m points z , z , . . . , z m , and find associated constants α , α , . . . , α m , for every prescribed integer m , having the propertysup µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k F µ ( z k ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m (1.4)for some bound (cid:15) m subject to (cid:15) m → m → ∞ . The geometry of the locus of these pointsis obviously essential and it will be detailed in the sequel. The faster the convergence, thebetter.Since we deal with probability measures, condition (1.4) is equivalent tosup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m . (1.5)And this is good news. We seek the minimal deviation from zero on the interval [ − , R ( z ) satisfying R ( ∞ ) = 1 and possessing simple poles at the points z , . . . , z m . Or equivalently, denoting q ( z ) = ( z − z )( z − z ) . . . ( z − z m ) and w ( λ ) = | q ( λ ) | − we aim at finding the minimal deviation from zero of a monic polynomial p of degree m , withrespect to the weighted norm (cid:107) p w (cid:107) ∞ = sup λ ∈ [ − , | p ( λ ) w ( λ ) | . w ( λ ) , λw ( λ ) , λ w ( λ ) , . . . (1.6)form a Chebyshev system on the interval [ − , w ( λ ) , λw ( λ ) , λ w ( λ ) , . . . , λ m w ( λ ) has at most m zeros in [ − , p of degree m minimizing the norm (cid:107) p w (cid:107) ∞ : this polynomial is characterized by the fact that | p ( λ ) | attains its maximal value at m + 1 points, and the sign of p ( λ ) alternates there, seealso [34]. In case w ( λ ) = 1, the optimal polynomial is of course the normalized Chebyshevpolynomial of the first kind: p ( λ ) = T m ( λ )2 m , T m (cos x ) = cos( mx ) , m ≥ . The constructiveaspects of weighted Chebyshev approximation are rather involved, see for instance the earlyworks of Werner [54, 55, 56]. In the same vein, the asymptotics of the optimal bound of ourminimization problem inherently involves potential theory or operator theory concepts. Wecite for a comparison basis a few remarkable results of the same flavor [16, 45, 6].Without seeking sharp bounds and guided by the specific applications we aim at, wepropose a compromise and relaxation of our extremal problem:inf p (cid:107) p w (cid:107) ∞ ≤ (cid:107) w (cid:107) ∞ inf p (cid:107) p (cid:107) ∞ . (1.7)At this point we can invoke Chebyshev original theorem and his polynomial T m , obtainingin this way the benefit, very useful for applications, of computing in closed form the residues α k . Details and some ramifications will be given in the Section 3. The motivation for studying this problem comes from our work [31, 33] where we derivedmicrostructure-independent bounds on the viscoelastic response at a given time t of two-phase periodic composites (in antiplane shear) with prescribed volume fractions f and f =1 − f of the phases and with an applied average stress or strain prescribed as a function oftime. We found that the bounds were sometimes extremely tight at particular times t = t :see Figure 1. This was quite a surprise because the response of each phase is nonlocal intime, yet somehow this response is untangled at these particular times. Thus, the boundscould be used in an inverse fashion to determine the volume fractions from measurementsat time t . While they were very tight at specific times in some examples, they were farfrom tight at all times in other examples: see Figure 2. The question arose as to whetherdifferent input functions could produce the desired tightness at specific times, and if so, how3 Figure 1: Comparison between the lower and upper bounds on the output average stress withan input applied average strain of H ( t ), where H ( t ) is the Heaviside function, 0 for t < t ≥
0. This is called a stress relaxation test. One phase is purely elastic ( G = 6000),while the other phase is viscoelastic and modeled by the Maxwell model ( G = 12000 and η = 20000) (the results are normalized by the response of the elastic phase). The followingthree cases are graphed: no information about the composite is given; the volume fractionof the components is known ( f = 0 . v ( t )] of the material at a specificmoment of time t is totally measure independent, while Re[ v ( t )] has a smooth dependenceon t , with Re[ v ( t )] → t → −∞ . The example of Figure 1 suggests that it may bepossible.Determining volume fractions of phases is important in the oil industry, where one wantsto know the proportions occupied by oil and water in the rock, to detecting breast cancer, toassessing the porosity of sea-ice and other materials, and even to determining the volume ofholes in swiss cheese. So an extension of the results to these problems, not yet within reach,would be very valuable. Perhaps also the analysis may ultimately shed light on show electric4 Figure 2: Comparison between the lower and upper bounds on the output stress relaxationin the “badly-ordered case”, when the responses on the pure phases as a function of time donot cross with an input applied average strain of H ( t ), where H ( t ) is the Heaviside function.Here the purely elastic phase has shear modulus G = 12000, while the Maxwell parametersfor the viscoelastic phase are G = 6000 and η = 20000 (again, the results are normalizedby the response of the elastic phase). The three subcases are the same as for the previousfigure. However the bounds remain quite wide except near t = 0. Reproduced from Figure6.5 in [33]. The approach developed in this paper can yield tight bounds with a suitablydesigned input function as shown in Figure 3.eels, electric fish [50], sharks, rays, swordfish, [21] appropriately tailor electrical signals tolocate prey (see the Wikipedia entry https://en.wikipedia.org/wiki/Electroreception for many more examples). Although the electrical response of the water is fast, the responsetime for currents to be generated in the prey is much slower.Without going into the specific details, as these will be provided later in Section 7, 8 and9, in many linear systems with an input function u ( t ) varying with time t , of the form u ( t ) = m (cid:88) k =1 β k e − iω k ( t − t ) , (2.1)where the ω k are a set of (possibly complex) frequencies, and t is a given time, the outputfunction v ( t ) takes the form v ( t ) = m (cid:88) k =1 α k a F µ ( z ( ω k )) e − iω k ( t − t ) , (2.2)in which the function F µ ( z ) is given by (1.1), α k = β k c ( ω k ) , (2.3)5 Figure 3: Comparison between the lower and upper bounds on the output stress relaxation inthe “badly-ordered case” ( G = 12000 for the elastic phase, and G = 6000 and η = 20 ,
000 forthe viscoelastic Maxwell phase), when the input function is chosen accordingly to equation(3.3), which represents the main result of this paper. Specifically, equation (3.3) providesthe amplitude of the applied field that gives extremely tight bounds at a chosen moment oftime (here t = 0) when the volume fraction is known. Indeed, the bounds incorporating thevolume fraction (the innermost bounds, in red) take the value 0 . t = 0, which coincidesexactly with the volume fraction of the viscoelastic phase. Here, the applied loading is thesum of three time-harmonic fields with frequencies ω = 0 . , . , . z ( ω ) and c ( ω ) depend on ω in some known way: z = z ( ω ) and c = c ( ω ).The real constant a > dµ depend on the system. In ourviscoelasticity study [33] the connection with Markov functions comes from the fact thatthe effective shear modulus G ∗ ( ω ), that relates the average stress to the average strain atfrequency ω , as a function of the shear moduli G ( ω ) and G ( ω ) of the two phases, has theproperty that [( G ∗ /G ) − / (2 f ), in which f is the volume fraction of phase 1, is a Markovfunction of z = ( G + G ) / ( G − G ) taking the form (1.2) [7, 35, 15].Henceforth, we adopt the notational simplification f ( z ) = F µ ( z ) . Thus, at time t = t , the output function is v ( t ) = a m (cid:88) k =1 α k f ( z k ) with z k = z ( ω k ) , (2.4)and we seek an input signal so that the output v ( t ) is almost system independent with v ( t ) ≈ a . So, by measuring v ( t ) we can determine the system parameter a . In the6iscoelastic problem that we studied [31, 33], a is the volume fraction f (see also [7]) and itis useful to be able to determine this from indirect measurements. Typically, one may assumethe frequencies ω k have a positive imaginary part so that the input signal u ( t ) is essentiallyzero in the distant past. In (2.1) one could just take a signal with m − ω k , k = 1 , , . . . , m −
1. Then, we have v ( t ) + α m a f ( z ( ω m )) = a m (cid:88) k =1 α k f ( z k ) ≈ a with z k = z ( ω k ) . (2.5)Then, if a is known, a measurement of v ( t ) will allow us to estimate the output a f ( z ( ω m )) e − iω m ( t − t ) at a desired (possibly real) frequency ω m given the time harmonic input e − iω m ( t − t ) .In many systems, such as the viscoelastic problem, it is only the real part of v ( t ) thathas a direct physical significance and, hence, one might want to find constants α k such that,say, 2 Re[ v ( t )] = a (cid:32) m (cid:88) k =1 α k f ( z k ) + m (cid:88) k =1 α k f ( z k ) (cid:33) = a (cid:32) m (cid:88) k =1 α k f ( z k ) + m (cid:88) k =1 α k f ( z k ) (cid:33) ≈ a , (2.6)where the overline denotes complex conjugation. This, again, reduces to a problem of theform (1.3) where, after renumbering, the complex values of z k come in pairs, z k and z k +1 = z k and we may take α k +1 = α k so that the left hand side of (1.3) is real.It may be the case that the first n moments of the probability measure dµ are known, M (cid:96) = (cid:90) − λ (cid:96) dµ ( λ ) , (cid:96) = 1 , , . . . , n, (2.7)in addition to the zeroth moment M = 1 and that m (possibly complex) points z , z , . . . , z m not on the interval [ − ,
1] are given. We then may seek complex constants α , α , . . . , α m and γ , γ , γ , . . . γ n , with say γ n = 1, such that m (cid:88) k =1 α k f ( z k ) ≈ n (cid:88) (cid:96) =0 γ (cid:96) M (cid:96) (2.8)for all probability measures µ with the prescribed moments. We will treat this problem inSection 3. We can use these results to determine an approximate linear relation among the n moments if v ( t ) is measured. This may be used to estimate one moment if the rest areknown. This can be useful when the moments have an important physical significance: in theviscoelastic problem, for instance, M depends only on the volume fraction f if one assumesthat the composite has sufficient symmetry to ensure that its response remains invariant asthe material is rotated [7]. So, incorporating the moment M and measuring the responseat time t , then, allows us to obtain tighter bounds on f as shown in [31, 33].Mutatis mutandis, we may seek complex constants α , α , . . . , α m and γ , γ , . . . γ n (eachconstant depending both on m and n ) such thatinf A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k =1 α k [ A − z k I ] − − n (cid:88) (cid:96) =0 γ (cid:96) A (cid:96) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:15) ( n ) m , (2.9)7here the infimum is over all self adjoint operators A with spectrum in [ − ,
1] and for fixed n , (cid:15) ( n ) m → m → ∞ . Our analysis extends easily to treat this problem too. The relevanceis that in many linear systems with an input field u ( t ) varying with time t , of the form u ( t ) = m (cid:88) k =1 β k e − iω k ( t − t ) u , (2.10)the output field v ( t ) takes the form v ( t ) = m (cid:88) k =1 α k e − iω k ( t − t ) a [ A − z ( ω k ) I ] − u with α k = β k c ( ω k ) , (2.11)where the real constant a and the self-adjoint operator A characterize the response of thesystem, and the system parameters z ( ω ) and c ( ω ) depend on the frequency ω in some knownway. Then, the bound (2.9) implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v ( t ) − a n (cid:88) (cid:96) =0 γ (cid:96) A (cid:96) u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ a (cid:15) ( n ) m | u | . (2.12)We emphasize that our results are applicable not just to determining the volume fractionsof the phases in a two-phase composite but also determining the volume and shape of aninclusion in a body from exterior boundary measurements. This is shown in Sections 9 and 10.It is a classical and important inverse problem with a long history and many contributions:see [10, 26, 27, 47, 46, 22, 1, 19, 9, 11, 4, 3, 25, 44, 28] and references therein. The present section contains the main result which provides the theoretical foundation ofour explorations. As explained in the introduction, we try to balance the computationalaccessibility and simplicity with the loss of sharp bounds. A few comments about theversatility of the following theorem are elaborated after its proof.
Theorem 1
Letting d ( z k ) = min λ ∈ [ − , | λ − z k | (3.1) denote the distance from z k to the line segment [ − , , and assuming d min = min k d ( z k ) > / , (3.2) one can find complex constants α , α , . . . , α m each depending on m , such that (1.4) holdswith (cid:15) m → as m → ∞ . In particular, with α k = − T m ( z k )2 m − (cid:81) j (cid:54) = k ( z k − z j ) , (3.3)8 here T m ( z ) is the Chebyshev polynomial of the first kind, of degree m , (1.4) holds with (cid:15) m = 2 / (2 d min ) m which tends to zero as m → ∞ . Proof
We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k f ( z k ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) − dµ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.4)Since the right hand side of (3.4) is linear in dµ its maximum over all probability measuresis achieved when µ ( λ ) is an extremal measure, namely the point mass µ ( λ ) = δ ( λ − λ ) , (3.5)where λ is varied in [ − ,
1] so as to get the maximum value of the right hand side of (3.4).In fact, for this extreme measure one has equality in (3.4). Equivalently, we haveinf α sup µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k f ( z k ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = inf α sup µ (cid:90) − dµ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = inf α sup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.6)Thus, we seek a set of constants α , α , . . . , α m (each dependent on m ) and sequence (cid:15) m suchthat (cid:15) m → m → ∞ and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m for all λ ∈ [ − , . (3.7)More clearly, direct substitution of (3.7) into (3.4) shows that (1.4) holds.Now we may write m (cid:88) k =1 α k λ − z k = p ( λ ) q ( λ ) = R ( λ ) , (3.8)where q ( λ ) is the known monic polynomial q ( λ ) = m (cid:89) j =1 ( λ − z j ) (3.9)of degree m , and p ( λ ) is a polynomial of degree at most m − α k can then be identified with the residues at the poles λ = z k of R ( λ ): α k = p ( z k ) (cid:81) j (cid:54) = k ( z k − z j ) . (3.10)9hen, the problem becomes one of choosing p ( λ ) such thatsup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12) p ( λ ) q ( λ ) − (cid:12)(cid:12)(cid:12)(cid:12) = sup λ ∈ [ − , | p ( λ ) − q ( λ ) || q ( λ ) | (3.11)is close to zero. Clearly, the problem is now one of polynomial approximation of the monicpolynomial q ( λ ) of degree m by the polynomial p ( λ ). A natural choice is to take p ( λ ) = q ( λ ) − T m ( λ ) / m − , (3.12)where T m ( λ ) / m − is the Chebyshev polynomial T m ( λ ) of degree m , normalized to be monic.This choice minimizes the sup-norm of | p ( λ ) − q ( λ ) | over the interval λ ∈ [ − ,
1] and | p ( λ ) − q ( λ ) | = | T m ( λ ) / m − | ≤ / m − (3.13)provides a bound on the numerator in (3.11). To bound the denominator, we have | q ( λ ) | = m (cid:89) k =1 | λ − z k | ≥ m (cid:89) k =1 d ( z k ) , (3.14)where d ( z k ) is given by (3.1). Using (3.2) and the bounds (3.13) and (3.14) we see that (1.4)is satisfied with (cid:15) m = 2 / (2 d min ) m . Finally, with p ( λ ) given by (3.12) we see that the residues α k at the poles λ = z k of g ( λ ), given by (3.10) correspond to those given by (3.3). Remark 1
The use of Chebyshev polynomials is convenient as bounds on their sup-norm over theinterval [ − ,
1] are readily available. An alternative approach, also accessible from the nu-merical/computational point of view, is to work with the L norm and find the polynomial p ( λ ) of degree m − q ( λ ) of degree m in theprecise sense that (cid:90) − | ( p ( λ ) − q ( λ ) | dν ( λ ) , with dν ( λ ) = dλ/ | q ( λ ) | (3.15)is minimized. Subsequently, one has to invoke Bernstein-Markov’s inequality which boundsan L norm by uniform norm. This first step is a standard problem in the theory of or-thogonal polynomials: one chooses p ( λ ) − q ( λ ) to be the monic polynomial of degree m that is orthogonal to all polynomials of degree at most m − dν ( λ ). Separating the contribution of the denominator, by selecting ν to be the measure dλ/ √ − λ we recover the Chebyshev polynomials we have advocated in the proof of themain result. Remark 2 n of all datawe set q n ( z ) = ( z − z ( n ))( z − z ( n )) . . . ( z − z n ( n )) , n ≥ , and w n ( z ) = 1 | q n ( z ) | . For the proof of Theorem 1 above we only needlim sup n (cid:107) w n (cid:107) /n ∞ < . That is, there exists r <
2, so that for large n , the inequality w n ( λ ) ≤ r n , λ ∈ [ − , , holds true.By taking the natural logarithm, we are led to enforce the conditionlim sup n sup λ ∈ [ − , n n (cid:88) j =1 ln 1 | λ − z j ( n ) | < ln 2 . In other terms, an evenly distributed probability mass on the points z ( n ) , . . . , z n ( n ) shouldhave its logarithmic potential asymptotically bounded from above by a prescribed constant,on the interval [ − , q n to belong to some Jordan curve surrounding[ − , ±
1, see also [45, 6].
Remark 3
We can gain more flexibility in the choice of the input signal if we replace T m ( z k ) in theformula (3.3) for the residues α k with ( z k − z ) T m − ( z k ), where z is a prescribed (possiblycomplex) zero of p ( λ ) − q ( λ ) = ( λ − z ) T m − ( λ ). In particular, we may choose z to, say,minimize max t ≤ t | v ( t ) | / | v ( t ) | ≈ max t ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34) m (cid:88) k =1 α k f ( z k ) e − iω k ( t − t ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.16)to help ensure that the output signal is not too wild. If we are only interested in Re[ v ( t )]so that the z k come in complex conjugate pairs, then we may replace T m ( z k ) in (3.3) with( z k − z )( z k − z ) T m − ( z k ), and choose z to, say, minimizemax t ≤ t | Re[ v ( t )] | / | Re[ v ( t )] | ≈ max t ≤ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Re (cid:34) m (cid:88) k =1 α k f ( z k ) e − iω k ( t − t ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.17)In the first case, note that the signal u ( t ) (2.1) is linear in z while in the second case it islinear in the real coefficients of the quadratic ( λ − z )( λ − z ). So in either case we have alinear space of possible signals (though | z | should not be too large for the approximationto hold at time t ). Also α k → z → z k so in this limit the frequency ω k is absent fromthe input and output signals. More generally, to help minimize (3.16) or (3.17) one mightreplace T m ( z k ) with s M ( z k ) T m − M ( z k ) where s M ( λ ) is a polynomial of fixed degree M < m .11
Incorporating moments of the measure
Here we assume that the first n moments M , M , . . . , M n of the probability measure dµ ,given by (1.1), are known, in addition to M = 1 and that m (possibly complex) points z , z , . . . , z m not on the interval [ − ,
1] are given. We seek complex constants α , α , . . . , α m and γ , γ , . . . γ n , with say γ n = 1 such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k f ( z k ) − n (cid:88) (cid:96) =0 γ (cid:96) M (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.1)is small for all probability measures µ with the prescribed n moments. The analysis proceedsas before, only now we introduce the polynomial r ( λ ) = n (cid:88) (cid:96) =0 γ (cid:96) λ (cid:96) , (4.2)and set p ( λ ) and q ( λ ) to be the polynomials defined by (3.8) and (3.9). The goal is nowto choose polynomials p ( λ ) and r ( λ ) of degrees m − n , respectively, such that r ( λ ) ismonic and sup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12) p ( λ ) q ( λ ) − r ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) = sup λ ∈ [ − , | p ( λ ) − q ( λ ) r ( λ ) || q ( λ ) | (4.3)is close to zero. We choose p ( λ ) and r ( λ ) such that T m + n ( λ ) / m + n − = q ( λ ) r ( λ ) − p ( λ ) . (4.4)This is simply the Euclidean division of the normalized Chebyshev polynomial T m + n ( λ ) / m + n − by q ( λ ) with r ( λ ) being identified as the quotient polynomial and − p ( λ ) being identified asthe remainder polynomial. Then, assuming (3.2) and using (3.14), we havesup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12) p ( λ ) q ( λ ) − r ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) ( n ) m , with (cid:15) ( n ) m = 22 n (2 d min ) m (4.5)satisfying (cid:15) ( n ) m → m → ∞ , with n being fixed. With constants α k given by (3.10) andconstants γ (cid:96) being the coefficients of the polynomial r ( λ ), as in (4.2), it follows thatsup µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k f ( z k ) − n (cid:88) (cid:96) =0 γ (cid:96) M (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup µ (cid:90) − dµ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − n (cid:88) (cid:96) =0 γ (cid:96) λ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) ( n ) m . (4.6) Remark A = (cid:90) σ ( A ) λd P λ , (4.7)where σ ( A ) is the spectrum of A , assumed to be contained in the interval [ − ,
1] and d P λ is an orthogonal projection valued measure satisfying I = (cid:90) σ ( A ) d P λ . (4.8)Then, we have inf A (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m (cid:88) k =1 α k [ A − z k I ] − − n (cid:88) (cid:96) =0 γ (cid:96) A (cid:96) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup d P λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) σ ( A ) (cid:34) m (cid:88) k =1 α k λ − z k − n (cid:88) (cid:96) =0 γ (cid:96) λ (cid:96) (cid:35) d P λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ sup d P λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:90) σ ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − n (cid:88) (cid:96) =0 γ (cid:96) λ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d P λ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (4.9)Choosing constants α , α , . . . , α m and γ , γ , . . . γ n , with γ n = 1, as in the previous section,the bound (4.5) substituted in (4.9) implies that the desired bound (2.9) holds with (cid:15) ( n ) m =2 / [2 n (2 d min ) m ]. v ( t ) at any time t Supposing any constants α , α , . . . , α m are given, it is easy to get bounds on v ( t ) given by(2.2) at any time t that incorporate the n known moments M , M , . . . , M n . One introducesan angle θ and Lagrange multipliers γ , γ , . . . , γ n and takes the minimum value of (cid:90) − dµ ( λ ) Re (cid:34) e iθ m (cid:88) k =1 α k e − iω k ( t − t ) λ − z ( ω k ) + n (cid:88) (cid:96) =1 γ (cid:96) λ (cid:96) (cid:35) , (5.1)as µ varies over all probability measures supported on [ − ,
1] with unconstrained moments.The minimum will be achieved by the point masses µ = δ ( λ − λ ), where λ may take one ormore values. Typically we will need to choose the Lagrange multipliers γ , γ , . . . , γ n (thatdepend on θ ) so that the minimum is achieved at n values λ = λ ( (cid:96) )0 , (cid:96) = 1 , , . . . , n , andthen adjust the measure to be distributed at these points dµ ( λ ) = n (cid:88) (cid:96) =1 w (cid:96) δ ( λ − λ ( (cid:96) )0 ) , (5.2)with the non-negative weights w (cid:96) , that sum to 1, chosen so that the moments take theirdesired values. Then with this measure we obtain the boundRe[ e iθ v ( t )] ≥ a n (cid:88) (cid:96) =1 w (cid:96) Re (cid:34) e iθ m (cid:88) k =1 α k e − iω k ( t − t ) λ ( (cid:96) )0 − z ( ω k ) (cid:35) . (5.3)13y varying θ from 0 to 2 π we obtain bounds that confine v ( t ) to a convex region in thecomplex plane. Of course, if we are only interested in bounding Re[ v ( t )], then it suffices totake θ = 0 or π . (a) -15 -10 -5 0-12-10-8-6-4-2024 (b) Figure 4: (a) Bounds on the real part of the response of the system, Re[ v ( t )] (5.3), whenthe system is such that z ( ω ) = 2 + i/ω and the input signal Re[ u ( t )] is the one depicted in(b) (with c ( ω )=1). We choose the frequencies ω k to be [1 + 1 i ; 0 . . i ; 2 + 0 . i ], and weselect the coefficients α k according to (3.3) so that the bounds are extremely tight at t = 0,whereas the point masses λ ( (cid:96) )0 and the weights w (cid:96) are chosen for each moment of time t suchthat the minimum value of (5.1) is attained while the moments of the measure take theirdesired values. Specifically, the bounds on Re[ v ( t )] are plotted for three different scenarios,as shown by the legend.Figure 4 and Figure 5 depict the lower and upper bounds on Re[ v ( t )] for two systems( z ( ω ) = 2 + i/ω in Figure 4, thus mimicking the low frequency dielectric response of a lossydielectric material, and z ( ω ) = 2 − /ω in Figure 5, thus mimicking the dielectric responseof a plasma), when the coefficients α k in (5.3) are chosen such that the bounds are extremely14 a) -10 -8 -6 -4 -2 0-14-12-10-8-6-4-202 (b) Figure 5: (a) Bounds on the real part of the response of the system, Re[ v ( t )] (5.3), when thesystem is such that z ( ω ) = 2 − /ω and the input signal Re[ u ( t )] is the one depicted in (b)(with c ( ω )=1). We choose the frequencies ω k to be [1 + 1 i ; 0 . . i ; 2 + 0 . i ], like in thecase depicted in Figure 4.tight at t = 0, according to (3.3). For both systems, the bounds on Re[ v ( t )] are tighterthe higher the amount of pieces of information on the system is incorporated. Notice thatthe bounds colored in black (the largest ones) correspond to the case where only the zerothorder moment M of the measure is known but not the value of a : in such a case, as shownby the zoomed graph in the blue box, at t = 0, the upper bound takes value 1 and the lowerbound takes value 0, that are the smallest and the highest values a can take. On the otherhand, when a is assigned, the value that the corresponding bounds take at t = 0 is exactly a = 0 .
6, as shown by the zoomed graph in the blue box. The graphs show clearly that,in order to estimate the system parameter a , one has just to measure the response of thesystem at a specific moment of time t (if the applied field is carefully chosen).15hese are the type of bounds used in [33] to bound the temporal response of two-phasecomposites in antiplane elasticity. It is not yet clear whether those bounds can be derivedfrom variational principles. In general, in the theory of composites, variational methods haveproven to be more powerful than analytic approaches. Variational methods produce tighterbounds that often easily extend to multiphase composites: see the books [13, 49, 37, 2, 48]and references therein. For example, the variational approach gives tighter bounds on thecomplex permittivity at constant frequency of two-phase lossy composites [24], than thebounds obtained by the analytic approach [35, 8]. It also produces bounds on the complexeffective bulk and shear moduli of viscoelastic composites [14, 42]. An exception is boundsthat correlate the complex effective dielectric constant at more than two frequencies [36]that have yet to be obtained by a systematic variational approach. Variational bounds inthe time domain are available [12, 32], but these are nonlocal in time. Naturally, if one is interested in the response v ( t ) at a given (possibly complex) frequency ω , the easiest solution is to take an input signal u ( t ) at that frequency. However, it mightnot be easy to experimentally generate a signal at that frequency or it might not be easyto measure the response at that frequency. The problem becomes: find complex constants α , α , . . . , α m such that sup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =1 α k λ − z k − λ − z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m , (6.1)with z k = z ( ω k ), k = 0 , , . . . , m . Defining the polynomials p ( λ ) and q ( λ ) as in (3.8) and(3.9) one needs to find p ( λ ) of degree m − λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12) ( λ − z ) p ( λ ) − q ( λ )( λ − z ) q ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m . (6.2)Proceeding as before we choose( λ − z ) p ( λ ) = q ( λ ) − b m T m − ( λ ) with b m = q ( z ) /T m − ( z ) , (6.3)where b m has been chosen so that the polynomial q ( λ ) − b m T m − ( λ ) has a factor of ( λ − z ).Then the residues of R ( λ ) = p ( λ ) /q ( λ ) are given by α k = − b m T m − ( z k )( z k − z ) (cid:81) j (cid:54) = k ( z k − z j ) = − T m − ( z k ) (cid:81) j (cid:54) =0 ( z − z j ) T m − ( z )( z k − z ) (cid:81) j (cid:54) =0 ,k ( z k − z j ) , (6.4)and sup λ ∈ [ − , | ( λ − z ) p ( λ ) − q ( λ ) | = sup λ ∈ [ − , | b m T m − ( λ ) | = | b m | , (6.5)16o that (6.1) holds with (cid:15) m = | b m | d inf λ ∈ [ − , | q ( λ ) | , (6.6)where d denotes the distance from z to the interval [-1,1]. Joukowski’s map yields: z = 12 (cid:18) ζ + 1 ζ (cid:19) , with R = | ζ | > , (6.7)whence T m − ( z ) = 12 (cid:18) ζ m − + 1 ζ m − (cid:19) . (6.8)Moreover, since ζ runs over a circle of radius R , we have d = inf λ ∈ [ − , | ζ − ζ λ + 1 | ≥ ( R − R , (6.9)and | T m − ( z ) | ≥
12 ( R m − − R − m ) , m ≥ , (6.10)implying | b m | ≤ | q ( z ) | R m − − R − m . (6.11)All in all, the relevant bound (cid:15) m satisfies | (cid:15) m | ≤ R ( R − R m − − R − m sup λ ∈ [ − , | q ( z ) || q ( λ ) | . (6.12)We obtain an exponential decay (cid:15) m → m → ∞ provided the geometry of the loci z , z , . . . , z m is subject to the following condition: for a positive constant r < R , each z j ∈ H ( r ) = H ( r ) ∪ H ( r ) ∪ H ( r ) where H ( r ) = (cid:26) z : (cid:12)(cid:12)(cid:12)(cid:12) z − z z + 1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, Re z ≤ − (cid:27) ,H ( r ) = (cid:26) z : (cid:12)(cid:12)(cid:12)(cid:12) z − z Im z (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, Re z ∈ [ − , (cid:27) ,H ( r ) = (cid:26) z : (cid:12)(cid:12)(cid:12)(cid:12) z − z z − (cid:12)(cid:12)(cid:12)(cid:12) ≤ r, Re z ≥ (cid:27) . (6.13)In other words, all of the z j must be close to z in the precise sense that z j ∈ H ( r ). Note that,as shown in Figure 6a, in case r < H and H are sectors of disks, while H is a portion ofan ellipse. For r ∈ (1 , R ) these regions are complements of disks/ellipse, containing the point z , as shown in Figure 6c. Some of these regions can be empty, depending on the positionof z .A conservative choice would be r = 1 (see Figure 6b), in which situation H and H arebounded by straight lines, while H is a parabola. To fix ideas, let us assume z = x + iy - (a) - - (b) - - (c) Figure 6: Representation of the loci z k for a system for which z = 0 . − . i ,and R = 2 . x ≥ y ≥
0. All other cases being symmetrical. Then the euclidean region H (1)where z , z , . . . , z m are allowed consists of points z = x + iy subject to the constraints: x ≥ z, z ) ≤ dist( z, , (6.14)union with x ∈ [ − ,
1] and ( x − x ) + y ≤ y y. (6.15)If y = 0 then necessarily x > H is simply the right-half plane x > x , while inthe case y > H (1) is the interior of a parabola with vertex at ( x , y ), within the band | x | ≤
1, union with the polygonal region defined by the first distance inequality (in x ≥ β k = α k /c ( ω k ), generating the outputfunction v ( t ) given by (2.2), (6.1) implies the bound | v ( t ) − v ( t ) | ≤ a (cid:15) m , (6.16)where v ( t ) = a F µ ( z ( ω )) (6.17)is the response at time t to the single frequency input signal u ( t ) = e − iω ( t − t ) /c ( ω ) . (6.18)Of course, because this response v ( t ) is for a single frequency, v ( t ) determines v ( t ) for all t . In Figure 7 we depict the response v ( t ) of a given system subject to an input signalat the frequency ω and we compare the value it takes at t = 0 with the value taken bythe bounds on the response v ( t ) of a system having the same values of the moments of themeasure but subject to a multiple-frequency signal with amplitudes α k chosen such that thebounds are extremely tight at t = 0: v ( t ) lies, as expected, between the bounds on v ( t )at t = t , -2 -1.5 -1 -0.5 0 0.5 1-2-1.5-1-0.500.5 (a) z = 2 − iω -2 -1.5 -1 -0.5 0 0.5 1-0.6-0.5-0.4-0.3-0.2-0.10 (b) z = 2 − ω Figure 7: Comparison between the response v ( t ) of a given system with point masses at-0.5 and 0.5, due to an input at the frequency ω = 0 . i , and the upper and lower boundson the response v ( t ) of a system having the same value of the moments of the measure M i ( M = 1 and M = 0 .
4) and subject to an input signal of the type (2.1), with ω k given by[1 + 1 i ; 0 . . i ; 2 + 0 . i ] and coefficients β k chosen accordingly to (2.3) and (6.4). Noticethat in both cases the value of v ( t ) at t = 0 lies between the bounds on v ( t ) at t = 0. Remark v ( t ) is known for a given ω but one wants to predict the derivative v ( t ) dω = a dF µ ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) z = z ( ω ) dz ( ω ) dω . (6.19)As dF µ ( z ) dz = (cid:90) − dµ ( λ )( λ − z ) , (6.20)the problem becomes: find complex constants α , α , α , . . . , α n such thatsup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) j =1 α j λ − z j + α λ − z − λ − z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m . (6.21)Defining the polynomials p ( λ ) and q ( λ ) as in (3.8) and (3.9) one needs to find p ( λ ) of degree m − λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12) ( λ − z ) p ( λ ) − q ( λ )[1 − α ( λ − z )]( λ − z ) q ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m . (6.22)We now choose ( λ − z ) p ( λ ) = q ( λ )[1 − α ( λ − z )] − b m T m − ( λ ) , (6.23)with b m = q ( z ) /T m − ( z ) , α = q (cid:48) ( λ ) − b m T (cid:48) m − ( λ ) q ( z ) (6.24)selected so that the polynomial on the right hand side of (6.23) has a factor of ( λ − z ) , inwhich q (cid:48) ( λ ) = dq ( λ ) /dλ and T (cid:48) m − ( λ ) = dT m − ( λ ) /dλ . So the residues α k , for k (cid:54) = 0, are nowgiven by α k = − b m T m ( z k )( z k − z ) (cid:81) j (cid:54) = k ( z k − z j ) = − T m ( z k ) (cid:81) j (cid:54) =0 ( z − z j ) T m ( z )( z k − z ) (cid:81) j (cid:54) =0 ,k ( z k − z j ) , (6.25)where b m is still given by (6.3) andsup λ ∈ [ − , | ( λ − z ) p ( λ ) − q ( λ )[1 − α ( λ − z )] | = sup λ ∈ [ − , | b m T m − ( λ ) | = | b m | (6.26)so that (6.21) holds with (cid:15) m = | b m | d inf λ ∈ [ − , | q ( λ ) | . (6.27)Apart from an extra factor of d this is exactly the same as the formula (6.6), and so theconvergence (cid:15) m → m → ∞ is assured provided for a positive constant r < R , with each z j ∈ H ( r ) = H ( r ) ∪ H ( r ) ∪ H ( r ). 20 Framework for a wide variety of time-dependent prob-lems
Suppose that, in some Hilbert (or vector space) H , one is interested in solving for J theequations J = LE , QJ = J , QE = E , (7.1)for a prescribed field E , where L : H → H is an operator satisfying appropriate boundednessand coercivity conditions, and Q is a selfadjoint projection onto a subspace S of H , so thatboth E and J lie in S . Note that we can rewrite (7.1) as J = LE (cid:48) − s , Γ E (cid:48) = E (cid:48) , Γ J = 0 , (7.2)with E (cid:48) = E − E , s = − LE being the source term, and Γ = I − Q being the projectiononto the orthogonal complement of S in H . These equations arise in the extended abstracttheory of composites and apply to an enormous plethora of linear continuum equations inphysics: see, for example, the books [37, 43] and the articles [38, 39, 40, 41].The simplest example is for electrical conductivity (and equivalent equations), where onehas j (cid:48) ( x ) = σ ( x ) e ( x ) − s ( x ) , Γ e = e , Γ j (cid:48) = 0 , with Γ = ∇ ( ∇ ) − ∇· , (7.3)where σ ( x ) is the conductivity tensor, while ∇ · s , j = j (cid:48) + s , and e are the current source,current, and electric field, and ( ∇ ) − is the inverse Laplacian (there is obviously considerableflexibility in the choice of s ( x ), the only constraints being square integrability and that ∇ · s equals the current source). As current is conserved, ∇ · j = ∇ · s , implying ∇ · j (cid:48) = 0, which isclearly equivalent to Γ j (cid:48) = 0. In Fourier space Γ ( k ) = k ⊗ k /k , and Γ e = e implies theFourier components (cid:98) e ( k ) of e satisfy (cid:98) e = − i k ( i k · (cid:98) e ) /k . So e is the gradient of a potentialwith Fourier components − i k · (cid:98) e /k . In antiplane elasticity one takes a material with a cross-section in the ( x , x )-plane that is independent of x , applies shearing in the x directionand observes warping of the cross section. The displacement u ( x ) in the x -direction thatis associated with this warping satisfies a conductivity type equation ∇ · G ∇ u = ∇ s , where ∇ s is a shearing source term (dependent on ( x , x )), G ( x , x ) is the shear modulus, andcorrespondingly e = −∇ u and j = G ∇ u . The antiplane response also governs the warpingof rods under torsion for rods that have a non-circular cylindrical shape and are composedof long fibers aligned with the cylinder axis and embedded in a matrix such that the fiberseparation is much less than the cylinder circumference.One approach to solving (7.1) is to apply Q to both sides of the relation E = L − J toobtain E = QL − QJ , giving J = [ QL − Q ] − E , (7.4)where the inverse is on the subspace S . In general the operator L depends on the frequency ω and E could depend on ω too. Then the response at this frequency is (cid:98) J ( ω ) = [ Q ( L ( ω )) − Q ] − (cid:99) E ( ω ) . (7.5)21e are interested in the response in the time domain when (cid:99) E ( ω ) = β ( ω ) E for some complexamplitude β ( ω ) and E ∈ S does not depend on ω . In particular, for a sum of a finite numberof (possibly complex) frequencies in the time domain the input signal is E ( t ) = m (cid:88) k =1 e − iω k ( t − t ) (cid:99) E ( ω k ) = m (cid:88) k =1 β k e − iω k ( t − t ) E with β k = β ( ω k ) . (7.6)The resulting field J ( t ) is then J ( t ) = m (cid:88) k =1 β k e − iω k ( t − t ) [ Q ( L ( ω )) − Q ] − E , (7.7)and we want this to have a simple approximate formula at time t .To make progress we use another approach to solving (7.1). We introduce a “referencemedium” L = c I where the real constant c is chosen so that L − L is coercive andintroduce the so-called “polarization field” G = ( L − L ) E = ( L − c I ) E = J − c E . (7.8)Applying the projection I − Q to this equation gives( I − Q ) G = − c ( E − E ) = c E − c ( L − c I ) − G , (7.9)and solving this for G yields G = c [( I − Q ) + c ( L − c I ) − ] − E . (7.10)Finally, applying Q to both sides gives J = c (cid:8) Q + Q [( I − Q ) + c ( L − c I ) − ] − Q (cid:9) E . (7.11)By comparing (7.4) and (7.10) we have[ QL − Q ] − = c Q + c Q [( I − Q ) + c ( L − c I ) − ] − Q = c (cid:8) Q − Q [ Ψ − ( L + c I )( L − c I ) − ] − Q (cid:9) , (7.12)where Ψ = 2 Q − I has eigenvalues ±
1. It is not obvious at all that the right hand side of(7.12) is independent of c but the preceding derivation shows this. This type of solutionusing a reference medium L (that need not be proportional to I ) is well known in the theoryof composites: see, for example Chapter 14 of [37], [57], and references therein.Now assume L takes the form L = c P + c ( I − P ) , (7.13)where P is a projection operator onto a subspace P of H . In the theory of composites fortwo phase composites one frequently has L = c I χ ( x ) + c I (1 − χ ( x )) , (7.14)22here the characteristic function χ ( x ) is 1 in phase 1 and 0 in phase 2, and c and c couldbe the material moduli. For the antiplane elasticity problem one has c = G and c = G ,where G and G are the shear moduli of the phases. We take the limit c → c and then(7.12) becomes [ QL − Q ] − = c Q + 2 c QP [ PΨP − z P ] − PQ , (7.15)where the operator inverse is to be taken on the subspace P and z = c + c c − c . (7.16)Note that PΨP , like Ψ , has norm at most 1. In general, the two moduli c and c dependon the frequency ω and hence z defined by (7.16) will also, i.e. z = z ( ω ). Given an inputfield of the form (7.6) and letting J ( t ) = Q m (cid:88) k =1 β k c ( ω k ) e − iω k ( t − t ) E (7.17)denote the response when P = 0, i.e. when L ( ω ) = c ( ω ) I , the corresponding output fieldcan be taken to be v ( t ) = J ( t ) − J ( t ) = Q m (cid:88) k =1 α k e − iω k ( t − t ) P [ PΨP − z k P ] − PE , (7.18)with z k = z ( ω k ) = c ( ω k ) + c ( ω k ) c ( ω k ) − c ( ω k ) , α k = β k c ( ω k ) , (7.19)and we arrive back at the problem we have been studying. In particular, with constants α k given by (3.3) the inequality (2 .
9) with n = 0 implies | J ( t ) − J ( t ) − QPE | ≤ | PE | / (2 d min ) m . (7.20)Alternatively, we could have chosen c = c and let J ( t ) = Q m (cid:88) k =1 β k c ( ω k ) e − iω k ( t − t ) E (7.21)denote the response when P = I , i.e. when L ( ω ) = c ( ω ) I . Then, similarly to (7.18), wewould have J ( t ) − J ( t ) = Q m (cid:88) k =1 α k e − iω k ( t − t ) P ⊥ [( P ⊥ ΨP ⊥ + z k P ⊥ ] − P ⊥ E , (7.22)where z k is still given by (7.19), but now with α k = β k c ( ω k ), where P ⊥ = I − P is theprojection onto the subspace perpendicular to P . The problem, with n = 0 and with thesame choice of coefficients α k , requires a different input signal, i.e. a different choice of the β k given by β k = β k /c ( ω k ), to ensure that | J ( t ) − J ( t ) + 2 QP ⊥ E | ≤ | PE | / (2 d min ) m . (7.23)23 Framework in the context of the theory of compos-ites and its generalizations
In the theory of composites and its generalizations, one can identify a subspace of S that wecall U of “source free” fields, and we may wish to confine E to this subspace. Then (7.1)can be rewritten as J = LE , Γ E = 0 , Γ J = 0 , Γ E = E , (8.1)where Γ is the projection onto U , Γ is the projection onto E , defined as the orthogonalcomplement of S , and Γ is the projection onto J , defined as the orthogonal complement of U in the subspace S . Then Q = Γ + Γ and the Hilbert space H has the decomposition H = U ⊕ E ⊕ J , (8.2)and the projections onto these three subspaces are respectively Γ , Γ , and Γ .In particular, as observed independently in Sections 2.4 and 2.5 of [17], and in Chapter3 of [43], the Dirichlet-Neumann problem can be reformulated as a problem in the theoryof composites. In the simplest case of electrical conductivity, where one has an inclusion D (not necessarily simply connected) of (isotropic) conductivity c in a simply connected bodyΩ having smooth boundary, with c being the (isotropic) conductivity of Ω \ D , we may take H as the space of vector fields that are square integrable with the usual normalized L innerproduct, ( A , A ) = 1 | Ω | (cid:90) Ω A ( x ) · A ( x ) d x , (8.3)where | Ω | is the volume of Ω, and take • U to consist of gradients of harmonic fields u = −∇ V with ∇ V = 0 in Ω, • E to consist of gradients e = −∇ V with V = 0 on the boundary ∂ Ω of Ω, • J to consist of divergent free vector fields j with ∇ · j = 0 and j · n = 0 on ∂ Ω, where n is the outwards normal to ∂ Ω.The conductivity of the body may be identified with L given by (7.13) where P is theprojection onto those fields that are zero outside D . As we are considering time-dependentproblems in the quasistatic limit, where the body is small compared to the wavelength andattenuation lengths of electromagnetic waves at the frequencies ω k , the moduli c and c andthe fields are typically complex and frequency-dependent. The fields in U can be identifiedeither by the values that V takes on the boundary ∂ Ω or by the values that the flux n · ∇ V takes on the boundary ∂ Ω. Thus the equations (8.1) are nothing other than the Dirichletproblem in the body Ω, j = Le , e = −∇ V, ∇ · j = 0 , e = −∇ V , ∇ V = 0 , V = V , on ∂ Ω , (8.4)and the mapping from Γ e to Γ j is nothing other than the Dirichlet to Neumann map giving n · j in terms of V on ∂ Ω.For periodic two-phase conducting composites, with unit cell Ω, the framework is similar.We take H as the space of vector fields that are Ω-periodic with the usual normalized L inner product, given by (8.3), and take 24 U to consist of gradients of constant fields u (that do not depend on x ), • E to consist of gradients e = −∇ V with V being an Ω-periodic potential, • J to consist of Ω-periodic divergent free vector fields j with ∇ · j = 0, having zeroaverage over Ω.The conductivity of the body may be identified with L given by (7.13) where P is the projec-tion onto those fields in H that are zero outside phase 1, and c is the (isotropic) conductivityof phase 1 while c is the (isotropic) conductivity of phase 2. Remark
More generally, the conductivity in the periodic composite could be anisotropic, with theconductivity tensor having the special form L ( ω ) = c ( ω ) L P + c ( ω ) L ( I − P ) , (8.5)where L is a constant positive definite tensor. As L commutes with Γ and P , we candefine new orthogonal spaces E (cid:48) = L / E , J (cid:48) = L − / J , U (cid:48) = L / U = L − / U = U , (8.6)and rewrite (8.1) in the form J (cid:48) = L (cid:48) E (cid:48) , Γ (cid:48) E (cid:48) = 0 , Γ (cid:48) J (cid:48) = 0 , Γ (cid:48) E (cid:48) = E (cid:48) , (8.7)where J (cid:48) = L − / J , E (cid:48) = L / E , E (cid:48) = L / E , L (cid:48) = L − / LL − / = c ( ω ) P + c ( ω )( I − P ) , (8.8)and Γ (cid:48) = Γ , Γ (cid:48) = L − / Γ ( Γ L Γ ) − , Γ (cid:48) = I − Γ (cid:48) − Γ (cid:48) (8.9)are the projections onto U (cid:48) = U , E (cid:48) , and J (cid:48) , in which the inverse in the formula for Γ (cid:48) is tobe taken on the subspace E . As L (cid:48) now takes the same form as (7.13) we are back to thesame problem.Similarly, in a body where the conductivity tensor has the special form (8.5) we may take • U (cid:48) to consist of gradients of fields u = − L / ∇ V with ∇ · L ∇ V = 0 in Ω, • E (cid:48) to consist of fields e (cid:48) = − L / ∇ V with V = 0 on the boundary ∂ Ω of Ω, • J (cid:48) to consist of fields j (cid:48) with ∇ · L / j (cid:48) = 0 and ( L / j (cid:48) ) · n = 0 on ∂ Ω, where n is theoutwards normal to ∂ Ωas our three orthogonal subspaces. Letting Γ , Γ , and Γ denote the projections onto thesethree subspaces, respectively, and setting L (cid:48) = c ( ω ) P + c ( ω )( I − P ), the equations (8.6)hold and we may proceed as before. 25 Application to solving the Calderon problem withtime varying fields
Let us now use ideas from the Calderon problem to solve the inverse problem of finding theinclusion D from boundary measurements on ∂ Ω. With Ω being a three-dimensional body,we can take V = e i k · x with k , k real and k = i (cid:113) k + k , (9.1)where the last condition implies k · k = 0 which ensures that V is harmonic. Then (7.20)implies ( J ( t ) − J ( t ) + 2 QPE , ∇ e i k (cid:48) · x ) ≤ | k || k (cid:48) | / (2 d min ) m (9.2)for all real or complex k (cid:48) . We now choose k (cid:48) with k (cid:48) = − k , k (cid:48) , k (cid:48) real and with ( k (cid:48) ) + ( k (cid:48) ) = k + k (9.3)to ensure that e i k (cid:48) · x is harmonic and so that(2 QPE , ∇ e i k (cid:48) · x ) = 2( PE , Q ∇ e i k (cid:48) · x ) = 2( PE , ∇ e i k (cid:48) · x )= 2( k k (cid:48) + k k (cid:48) − k − k ) 1 | Ω | (cid:90) D e i ( k − k (cid:48) ) x + i ( k − k (cid:48) ) x d x (9.4)only depends on the Fourier coefficients of the characteristic function associated with D .Then, using integration by parts,( J ( t ) − J ( t ) , ∇ e i k (cid:48) · x ) = 1 | Ω | (cid:90) ∂ Ω [ J ( t ) − J ( t )] · n e i k (cid:48) · x dS, (9.5)where J ( t ) · n can be measured, while J ( t ) · n can be computed. As there is nothingspecial about the x axis we may rotate the cartesian coordinates to get estimates of otherFourier coefficients of the characteristic function associated with D . We may also take E as constant and replace ∇ e i k (cid:48) · x by E to get( J ( t ) − J ( t ) + 2 QPE , E ) = ( J ( t ) − J ( t ) , E ) + | E | | D | / | Ω | ≤ | E | / (2 d min ) m , (9.6)thus giving an estimate of the volume fraction | D | / | Ω | that D occupies in the body (i.e., theFourier coefficient at k = 0).With Ω being a two-dimensional body, the situation is similar. We take k real and k = ik , k (cid:48) = − k , k (cid:48) = − ik , (9.7)and (9.4) is replaced by (2 QPE , ∇ e i k (cid:48) · x ) = − k (cid:90) D e ik x d x , (9.8)while (9.5) and (9.6) still hold. Again we approximately recover the Fourier coefficients of thecharacteristic function associated with D from measurements of J ( t ) · n and computationsof J ( t ) · n . 26n the usual Calderon problem one solves the inverse problem by taking | k | to be verylarge, according to the so-called complex geometric optics approach [47]. Here we see thatthere is no need to take | k | to be very large if we allow time dependent applied fields. Forelectromagnetism in non-magnetic media the measurements are difficult as the time responseis typically extremely rapid (From Table 7.7.1 in [18] we see that electromagnetic relaxationtimes in seconds for copper, distilled water, corn-oil, and mica are 1 . × − , 3 . × − ,0 .
55, and 5 . × respectively, and measurements would need to be taken on these timescales). On the other hand, for the equivalent magnetic permeability, fluid permeability, orantiplane elasticity problems the relaxation times are much more reasonable [5, 20, 30] andmeasurements in the time domain become feasible. Even in electrical systems one can getlong relaxation times, such as the time to charge a capacitor. Remark
Instead of taking E ( t ) = Γ E ( t ) and J ( t ) as our input and output fields, one could take J ( t ) = Γ J ( t ) and E ( t ) as our input and output fields. Then one has E = L − J , Γ J = 0 , Γ E = 0 , Γ J = J , (9.9)which is exactly of the same form as (8.1), but with L replaced by L − and the roles of Γ , Γ , and E and J , and E and J interchanged. So all the preceding analysis immediatelyapplies to this dual problem too.
10 Generalizations
In many problems of interest the fields in H take values in a, say, s -dimensional tensor space T and the operator L : H → H in (7.1), appropriately defined, is frequency dependent withthe properties that • L ( ω ) is an analytic function of ω in the upper half plane Im( ω ) > • Im[ ω L ( ω )] ≥ ω ) > • L ( ω ) = L ( − ω ) when Im( ω ) > L ( ω ) (and accordingly J ) may need to be multiplied by a function of ω , for example i , ω , or iω , to achieve these properties. In the case of materials where L acts locally in real space,i.e. if Q = LP , then Q ( x ) = L ( x ) P ( x ) for some L ( x ), the first property is a consequence ofcausality, the second a consequence of passivity (that the material does not generate energy- see, for example, [53]), and the third a consequence of L ( ω ) being the Fourier transform ofa real kernel. It follows that L is an analytic function of − ω with spectrum on the negativereal − ω axis (corresponding to real values of ω ) having the implied properties that • Im( L ) ≥ − ω ) ≤ • L is real and L ≥ ω is real and − ω ≥ L ( ω ) is an operator-valued Stieltjes function of − ω . The operator B =[ QL − Q ] − entering (7.4) has the property that it is an analytic function of L withIm( B ) ≥ L ) ≥ , B is real and B ≥ L is real and L ≤ . (10.1)Hence, the Stieltjes properties of L as a function of − ω pass to those of B as a function of − ω : Im( B ) ≥ − ω ) ≤ , B is real and B ≥ ω is real and − ω ≥ . (10.2)Introducing z = ω − cω + c = 1 − cω + c , (10.3)for some real c >
0, ensures that the spectrum of B ( z ) is on the interval [ − ,
1] andIm( B ( z )) ≥ z ) ≥ , B is real and B ≥ z is real and z > z < − . (10.4)Note that this choice of z is quite different to that in (7.16), and not restricted to two-phasecomposites. Thus, B ( z ) has the integral representation B ( z ) = B + (cid:90) − d M ( λ ) λ − z , (10.5)where B is a positive definite operator and d M ( λ ) is a positive definite real operator-valuedmeasure, satisfying the constraint (cid:90) − d M ( λ )1 − λ ≤ B . (10.6)To begin, suppose we are only interested in the quadratic form ( BE , E ) associated with B . Then,( B ( z ) E , E ) = k (cid:20) (cid:90) − (1 − λ ) dη ( λ ) λ − z (cid:21) = k (cid:26) (cid:90) − (cid:20) − − zλ − z (cid:21) dη ( λ ) (cid:27) , (10.7)where k = ( B E , E ) is real and positive and dη ( λ ) = ( d M ( λ ) E , E ) / [ k (1 − λ )] (10.8)is a positive real valued measure, satisfying the constraint (cid:90) − dη ( λ ) ≤ . (10.9)Note that k can be identified with ( B ( z ) E , E ) in the limit z → ∞ , i.e. as ω → i √ c .28f we are interested in finding complex coefficients ξ k , k = 1 , . . . , m , such that( B ( z ) E , E ) − m (cid:88) k =1 ξ k ( B ( z k ) E , E )= k (cid:40) (1 − m (cid:88) k =1 ξ k ) (cid:20) − (cid:90) − dη ( λ ) (cid:21) + (cid:90) − (cid:34) − z λ − z − m (cid:88) k =1 ξ k (1 − z k ) λ − z k (cid:35) dη ( λ ) (cid:41) (10.10)is small, we require thatsup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − z − m (cid:88) k =1 ξ k (1 − z k ) / (1 − z ) λ − z k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m . (10.11)In particular, with λ = 1, this implies | − m (cid:88) k =1 ξ k | ≤ | − z | (cid:15) m , (10.12)and so we obtain ( B ( z ) E , E ) − m (cid:88) k =1 ξ k ( B ( z k ) E , E ) ≤ k | − z | (cid:15) m . (10.13)By setting α k = ξ k (1 − z k ) / (1 − z ) we see this is exactly the problem encountered in Section6, and we may take the coefficients α k to be given by (6.4). The motivation for studying thisproblem is that the response at special frequencies can sometimes directly reveal informationabout the geometry. This is the case for elastodynamics in the quasistatic limit when onlytwo materials are present. The material parameters are the bulk moduli κ ( ω ), κ ( ω ) andshear moduli µ ( ω ), µ ( ω ) of the two phases. It may happen that µ ( ω ) = µ ( ω ) for certaincomplex frequencies ω and if κ ( ω ) (cid:54) = κ ( ω ) the response at frequency ω can reveal thevolume fraction of phase 1 in a composite, or more generally in a two-phase body. Remark 1
It is not much more difficult to treat bilinear forms. Then we have4( B ( z ) E , E (cid:48) ) = ( B ( z )( E + E (cid:48) ) , E + E (cid:48) ) − ( B ( z )( E − E (cid:48) ) , E − E (cid:48) )= k (1)0 (cid:26) (cid:90) − (cid:20) − − zλ − z (cid:21) dη ( λ ) (cid:27) − k (2)0 (cid:26) (cid:90) − (cid:20) − − zλ − z (cid:21) dη ( λ ) (cid:27) , (10.14)where k (1)0 = ( B ( E + E (cid:48) ) , E + E (cid:48) ) , k (2)0 = ( B ( E − E (cid:48) ) , E − E (cid:48) ) (10.15)29re both real and positive, while dη ( λ ) = ( d M ( λ )( E + E (cid:48) ) , E + E (cid:48) ) / [ k (1)0 (1 − λ )] ,dη ( λ ) = ( d M ( λ )( E − E (cid:48) ) , E − E (cid:48) ) / [ k (2)0 (1 − λ )] (10.16)are positive real valued measures, satisfying the constraints that (cid:90) − dη ( λ ) ≤ , (cid:90) − dη ( λ ) ≤ . (10.17)We seek complex coefficients ξ k , k = 1 , . . . , m , such that( B ( z ) E , E (cid:48) ) − m (cid:88) k =1 ξ k ( B ( z k ) E , E (cid:48) )= k (1)0 (cid:40) (1 − m (cid:88) k =1 ξ k ) (cid:20) − (cid:90) − dη ( λ ) (cid:21) + (cid:90) − (cid:34) − z λ − z − m (cid:88) k =1 ξ k (1 − z k ) λ − z k (cid:35) dη ( λ ) (cid:41) / − k (2)0 (cid:40) (1 − m (cid:88) k =1 ξ k ) (cid:20) − (cid:90) − dη ( λ ) (cid:21) + (cid:90) − (cid:34) − z λ − z − m (cid:88) k =1 ξ k (1 − z k ) λ − z k (cid:35) dη ( λ ) (cid:41) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( B ( z ) E , E (cid:48) ) − m (cid:88) k =1 ξ k ( B ( z k ) E , E (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( k (1)0 + k (2)0 ) | − z | (cid:15) m / . (10.19) Remark 2
Noting that ddz ( B ( z ) E , E ) = k (cid:90) − (1 − λ ) dη ( λ )( λ − z ) = k (cid:90) − λ − z (cid:20) − − zλ − z (cid:21) dη ( λ ) , (10.20)we can easily obtain bounds that correlate this derivative at z with the values of ( B ( z k ) E , E ), k = 0 , , . . . , m . We seek complex constants γ k , k = 0 , , . . . , m , such that (cid:20) ddz ( B ( z ) E , E ) (cid:21) − m (cid:88) k =0 ξ k ( B ( z k ) E , E )= k (cid:40)(cid:32) m (cid:88) k =0 ξ k (cid:33) (cid:20) − (cid:90) − dη ( λ ) (cid:21) + (cid:90) − (cid:34) − λ − z + 1 − z ( λ − z ) − m (cid:88) k =0 ξ k (1 − z k ) λ − z k (cid:35) dη ( λ ) (cid:41) (10.21)30s small, and this is ensured ifsup λ ∈ [ − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =0 ξ k (1 − z k ) / (1 − z ) λ − z k + ξ + [1 / (1 − z )] λ − z − λ − z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) m , (10.22)and (cid:15) m → m → ∞ . Observe that (10.22) with λ = 1 implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:88) k =0 ξ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | − z | (cid:15) m . (10.23)Comparing (10.22) with (6.21) we see that we should choose ξ = α − [1 / (1 − z )] , ξ k = α k (1 − z ) / (1 − z k ) , (10.24)and then, with b m and coefficients α k given by (6.24) and (6.25), (10.22) holds with (cid:15) m givenby (6.27). Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) ddz ( B ( z ) E , E ) (cid:21) − m (cid:88) k =0 ξ k ( B ( z k ) E , E ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | k || − z | (cid:15) m , (10.25)holds, and similarly one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) ddz ( B ( z ) E , E (cid:48) ) (cid:21) − m (cid:88) k =0 ξ k ( B ( z k ) E , E (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( k (1)0 + k (2)0 ) | − z | (cid:15) m . (10.26)The convergence of (cid:15) m to zero as m → ∞ is again ensured provided for a positive constant r < R , where R is defined by (6.7), each z k ∈ H ( r ) = H ( r ) ∪ H ( r ) ∪ H ( r ), wherethe regions H i , i = 1 , ,
3, are given by (6.13). The motivation for studying this problemis that the response may be trivial at certain frequencies ω while the derivative of theresponse with respect to ω at ω = ω directly reveals some information about the body. Thisis the case for electromagnetism when only two non-magnetic materials are present (withmagnetic permeabilities µ = µ = µ where µ is the permeability of the vacuum). It mayhappen that the electric permittivities of the two phases satisfy ε ( ω ) = ε ( ω ) for certaincomplex frequencies ω . At this frequency ω the body is homogeneous and its response canbe easily calculated. Using perturbation theory and assuming dε ( ω ) /dω (cid:54) = dε ( ω ) /dω the derivative of the response with respect to ω at ω = ω reveals information about thedistribution of the two phases in the body. Acknowledgements
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