UBu0fl

where F^ is the Frechet derivative. We call problem (1) well-posed, if sup ||[-P'(w)]_1|| < m(R).

Otherwise we call (1) ill-posed. The dynamical systems method (DSM) for solving (1) is the method consisting of the following steps:

a) finding a map u) such that the problem dxL

has the following properties:

3!u(i) Vt > 0; 3u(oo) := lim u{t); F(u(oo)) = 0, (5)

b) solving (4), then taking t —> oo, and finding the solution as the limit u(oo).

Note that u(oo) = u(oo,uo) if (1) has more than one solution. How does one find <£>? There are many ways [13] to find $ such that (5) holds.

We assume that $ is locally Lipschitz with resepct to u and continuous with respect to t. This implies local existence and uniqueness of the solution to (4).

Our aim is to review some of the results obtained recently [10]—[22]. These results demonstrate the power of the DSM, both as a theoretical tool for proving the existence of a solution to equation (1), proving that F is a global homeomorphism (under suitable additional assumptions), deriving convergent iterative schemes for solving (1), and developing numerical methods for solving a very wide class of linear and nonlinear operator equations, especially ill-posed. There is a large literature on linear ill-posed problems (e.g., see [6]), and a less extensive one on nonlinear ill-posed problems (e.g., [27], [10]).

Let us describe briefly the scope of the results obtained by the DSM in [10]—[22], assuming (2) unless otherwise stated:

(1) Every solvable well-posed equation (1) can be solved by the DSM method which converges exponentially fast.

(2) Every solvable ill-posed linear equation (1) with a bounded operator A can be stably solved by the DSM. Here F(u) := Au — f = 0, and stably means that if the noisy data f$ are given, ||/i — /|| <5, then there is a tg such that lim^o ||w(i<5) — = 0.

(3) Every solvable ill-posed equation (1) with monotone operator F, i.e., (F(u) — F(v),u — v)>0, can be stably solved by the DSM.

(4) Every solvable equation (1) can be stably solved by the DSM provided that the operator T := A*A, A := F'(y), maps the set

{u : |M| < r}, where r > 0 is sufficiently small, into the set 0:={u: 0 < |M| < 1}.

(5) If F is monotone, hemicontinuous, defined on all of H, but (2) is not assumed, then if equation (1) is solvable, possibly nonuniquely, then its minimal-norm solution y can be stably found by the DSM.

(6) If F = L + g, where L is a closed, linear, densely defined in H operator, which has a bounded inverse, g is a nonlinear operator satisfying (2), and equation (1) is solvable, then its solution can be stably found by the DSM, provided, for example, that supueB(U0,fl) HI7 + V(«)]-1II < m(R) and supR>0 ^ = oo.

(7) A sufficient condition for the surjectivity of a map F is

(8) A sufficient condition for the map F to be a global homeomorphism is IH-F^u)]-1!! < h(\\u\\), where ^fj = oo, h(s) > 0 is a continuous function on [0, oo).

(9) A method for constructing convergent iterative schemes for solving equation (1) is given.

(10) A method for constructing a Newton-type method without inverting the derivative operator is developed.

2. Well-posed problems

Assume (2) and (3). Take $ = -[F'(u)]_:1F in equation (4):

This is a continuous analog of the Newton's method. From (2) and (3) it follows that $ is Lipschitz, so (6) is locally solvable. It will be globally solvable, i.e., solvable Vi > 0, if supt>0 ||u(i)|| < oo. Let us prove this estimate. Let ||F(u(t))|| := g(t). Then gg = -g2 by (6), so g(t) = g(0)e~t. From (6) one gets ||u|| < g(0)e_tm(i?). Thus ||u(i) - u(0)|| < g(0)m(R) as long as g(0)m(R) < R, (7)

that is, as long as the trajectory u(t) stays in the ball B(uo, R) for all t > 0. Assume (7). Then 3u(oo), sup ||u{t) - u0|| < R, ||u(t) - u(oo)|| < Re~\ (8)

t> o and from (6), as t —> oo, one gets F(u(oo)) = 0. We have proved that equation (1) can be solved by the DSM (6).

Theorem 2.1. If (2), (3) and (7) hold, then equation (6) has a unique global solution, and (5) holds. Moreover, estimates (8) hold, i.e., u(t) converges to the solution zt(oo) at the exponential rate and the trajectory u(t) stays in the ball B(uq, R) for all t> 0.

Remark 2.1. Many other choices of $ are discussed in [13], [14]. These choices include continuous analogs of the modified Newton's method, Gauss-Newton method, gradient method, method of simple iterations, etc.

3. Surjectivity of F

Theorem 3.1. Assume (2), (3) and let the following condition hold:

Then F is swrjective.

Proof. The proof of Theorem 2.1 shows that if (9) holds, then, for any fixed u0, (7) holds for some R > 0. Thus (1) is solvable. The same argument holds for the equation F(u)-f = 0, for any / € H. Theorem 3.1 is provedO

4. When is F a global homeomorphism?

Theorem 4.1. Assume (2), (3) and let h : 1+ -» R+, R+ := (0,oo), be a continuous function such that

Then F : H H is a global homeomorphism.

Proof. From (3) it follows that F is a local homeomorphism. To prove that F is a global homeomorphism one has to prove that: a) F is surjective, and b) if F(u) = F(v) then u = v. Let us prove a). As in Section 2, one gets

< INI < MIMDSoe-4, go:=g(0), where the inequality ||u||" < ||w|| was used. Thus

This and (10) imply supt>0 ||it(í)|| < oo, so supt>0 /i(||u(i)||) < oo, and (11) yields ||ú|| < ce_t. Thus ||w(i) — uo|| < c, u{oo) exists, and, as in Section 2, F(u(oo)) = 0. Since this argument remains valid if F(u) is replaced by F(u) — f with an arbitrary / e H, the surjectivity of F follows. Thus, a) is proved.

Below we consider the equation F(u) — f = 0. Let us prove b).

If ||uo — fo|| is sufficiently small, and sup||t¿(í,it0)-it(i,i;o)|| <c||uo-uo||, (12)

where c > 0 stands for various constants, then b) follows.

Indeed, let u := u(oo,uo) = limt_>oo i¿(t,i¿o), F(u) = f = F(v), where v = limt-Kx, v(t, vq). Define w(s) := u0+s(vo — uo). If s is sufficiently small, then ||iu(s) — uo|| is as small as one wishes, and ||u(i, w(s)) — it(i,i¿o)|| is as small as one wishes uniformly for all t > 0. Thus ||u(oo, w(s)) — w(oo,uo)|| is small, F(u(oo,uo)) = f = F(u(oo,w(s))). Since F is a local homeomorphism, it follows that u(oo,w(s)) = m(oo, uq) = u. Finitely many small steps are needed to get to ui(l) = vq and conclude that v — u(OO, lo(l)) = lt(OO, lío) = u.

To complete the proof, let us check (12).

Let ú = — |F'(ii)]-1F(ii), u(0) = u0,v = -[F»]-1^), u(0) = v0, ip := u(t) - v(t). Then i = -([f(ti)]"1 - [f»]"1)^) - /) - [Ff(v)]~1(F(u) - F(v)). (13)

Let g(t) := ||^(i)||. Multiply (13) by ip, use the formula F(u) - F(v) = F'(v)(u -v) + K, ||K\\ < VII2, and get

where the estimate ||F(it(i)) — /|| < ce_t was used. Since g(t) > 0, one can consider instead of (14) the following one:

Let g = e-t/i, so that g0 := g{0) = /i(0). Then h, < c(h2 + /i)e_t. This implies In ^ < In + c, i.e., ^ < where g0 = ||u(0) - v(0)||.

Therefore, if go is sufficiently small, so that go < -¡¿j, then h(t) < ch(0) = cg(0). Therefore g{t) < ce_t||it(0) - u(0)||, so (12) follows. Theorem 4.1 is proved. □

5. Linear ill-posed equations

Assume

where A is a bounded linear operator, Ay = /, y is the minimal-norm solution, B := A*A, h ~ A*f, — /|| < 5, e(t) > 0 is a monotonically decreasing function, limt-»oo e(i) = 0, e(s)ds = oo. A solvable equation Au = f is equivalent to

Consider the equation u = -u+[B + e(t)]~1h, u(0) = u0, h = A*f. (18)

Theorem 5.1. Any solvable linear equation (16) with ||yl|| < oo can be stably solved by the DSM.

Proof. A solvable linear equation (16) is equivalent to (17): every solution to (16) solves (17) and vice versa. Consider a DSM (18). Since this is a linear equation, clearly u(t) exists for all t > 0 and is unique. The solution to (18) is: u(t) = uoe-t + /0f +e(s)]~1hds, and, since h = Bu, it is easy to check that limt_oo u(t) = y. Indeed, lime_o[Be]"1 Bu = y, where u = u(t) and e = e(t), and lim^oo J* e~<-t~s^g(s)ds = g(oo) provided there exists g(oo) := lim^^ g(t). Thus (5) holds and the DSM (18) is justified. If fs is given in place of /, then h& will replace h in (18). Without loss of generality one may assume that — /i|| < S. Let us(t) solve (18) with h replaced by kg. Let us prove:

Claim: There exists t$, lim^o t$ = oo, such that lim ||ua(i4) - y|| = 0.

To prove the claim, note that ||u,5(i,5)-2/|| < ||w«(ii)— u(i,5)||+||«(i<5) — 2/||-Since limi_o \\u(ts) — V|| = 0, it is sufficient to prove lima-.o ^¿(ia) — u(t,s)|| = 0. Let us(t) - u(t) := v(t), hs - h := p. Then v = —v + (B + e(t))~1p, v(0)=0.

Thus

Therefore any t$ such that lim^o tg = oo and lim^o = 0 is suitable for the claim to hold. Theorem 5.1 is proved. □

6. Nonlinear equations with monotone operators

Assume that F is monotone in the sense

where y is the minimal-norm solution to (19). Note that if F is monotone and (2) holds, then N(F) := {u : F(u) - / = 0} is a closed and convex set. Such sets in Hilbert spaces have a unique minimal-norm element, as is well known.

Consider the problem u = -A-^Fiu) +eu — f], ii(0) = u0, (20)

where Ae := F'(u) + el, I is the identity operator, e = £(t) > 0, e(t) < 0, lim £(t) = 0, — < (21)

Theorem 6.1. If (2) and (21) hold, and uo € H is arbitrary, then (5) holds (with F(u) — f in place of F(u)) and u(oo) = y. If f5 replaces f in (19) and (20), || fs — /|| < 5, then there exists tg, lim^-vo i<5 = 00, such that lims_o ¡|ii<s(i5) — y\\ = 0 where ug(t) solves (20) with fs in place of f.

Proof. Consider the equation

It is known (e.g., [13]) that if F is monotone and (2) and (21) hold, then there exists a unique solution V = V(t) to (22) and sup \\V\\ < ||y||, || < \\yfi, lim ||V(i) - y|| = 0. (23) t> 0 £ i-»00

Since ||u(i)-y|| < ||u(i)-V(i)|| + ||y(i)-y||, it follows that limt-oo y\\ = 0 provided that limt_oo ||w(i) - V(t)\\ = 0. Let g(t):=\\u(t)-V(t)\\, w:=u(t)-V(t).

We want to prove the relation: limt-,oo g(t) = 0. Prom (21) and (20) one gets w = -V-A^lF^-F^ + ew}. Moreover, F(u)-F(V) = Aw + K, where ||ii|| < by Taylor's formula. Thus w = -w - A~lK - V.

Multiply this by w and get g < -g + ^fcg2 + ||y||f. Let ^ := Co, llvll :=ci. Then

It follows from (24), (21), and Example 7.1 in the next section, that g(t) < ce(t) —> 0 as t —> oo. Theorem 6.1 is proved. □

We have used in (20) the operator A~l := (A + e/)-1 as an approximation to the inverse of the operator A := F'(u), where e > 0 tends to zero. This is natural because A > 0 is a bounded selfadjoint operator. For such an operator one can derive the relation lim£_»o A~lAu = u, using the spectral theorem for selfadjoint operators, and assuming that u G D(A), u _L N(A) := {u : Au = 0}, and otherwise u is arbitrary. For non-monotone F, and also for monotone F, one may use other approximations for the inverse of A when this inverse is unbounded or does not exist. If A is selfadjoint, but not necessarily non-negative, or if A is normal, or, more generally, when A is a spectral operator, then one can use spectral theory to approximate the inverse of A. For instance, if A is selfadjoint, then one may use a function <t>e(A) such that lim£^0 4)e{A)Au = u for all u G D(A), u _L N(A). When A is non-selfadjoint, one may use 4>t{A*A) to approximate the inverse of A, when this inverse is unbounded or does not exist. The regularized Gauss-Newton method is based on such an approximation. A motivation for this is the formula {A*A)~XA* = A"1 valid for a boundedly invertible linear operator A.

One can derive a stopping rule for problem (20) as in Section 5. If \\fs — f || < 8, and us(t) solves (20) with fs replacing /, then the error E can be estimated as follows:

E := |Mi) - y\\ < ||«4(i) - V5(t)\\ + ||V4(i) - V(t)|| + || V(t) - y||, where Vs(t) solves (22) with fs replacing /. One has || V(t) -y|| :— a(t) —» 0 as t —> oo. Using monotonicity of F, one derives the estimate

If ws(t) := us(t) — Vs(t) and gs(t) := ||w,s(f)||, then, as in the proof of Theorem 6.1, one derives the inequality similar to (24):

where the last new term comes from the estimates

\\Vs\\<^\m, \\V5\\<\\V\\ + 61, and \\V\\ < ||y||.

Applying Theorem 7.1, one gets the inequality gs(t) < ce(t), where c > 0 is a constant, provided that t < ts, where the stopping time t$ is found from the equation Sa = e(t), for a fixed 5 and 0 < a < 1. This equation has a unique solution ts because e decays to zero monotonically. Moreover, ts —> oo as 6 —> 0. The error estimate is:

E < gs(ts) + ^y + a(ts) 0 as <5 -» 0, because, due to the inequality 0 < a < 1, one has = ¿1-a —> 0 as J —> 0.

A novel discrepancy principle for nonlinear equations with monotone operators is formulated and justified in [15].

7. A differential inequality

9(t)<-l(t)9 + <*(t)92 + l3(t), 9(0) = go, t> 0. (25)

Assume that 7, a, and 0 are continuous nonnegative junctions, and there exists a fx(t) > 0 limt—00 M^) = 00, such that

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